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Transcript of Morfismos, Vol 6, No 2, 2002
VOLUMEN 6NÚMERO 2
JULIO A DICIEMBRE DE 2002 ISSN: 18706525
MORFISMOSComunicaciones EstudiantilesDepartamento de Matematicas
Cinvestav
Editores Responsables
• Isidoro Gitler • Jesus Gonzalez
Consejo Editorial
• Felipe Gayosso • Samuel Gitler• Onesimo HernandezLerma • Raul Quiroga Barranco
• Enrique Ramırez de Arellano • Francisco Ramırez Reyes• Jose Rosales Ortega • Enrique Torres Giese
• Mario Villalobos Arias • Heraclio Villarreal Rodrıguez
Editores Asociados
• Ricardo Berlanga • Emilio Lluis Puebla• Isaıas Lopez • Guillermo Pastor
• Vıctor Perez Abreu • Carlos Prieto• Carlos Renterıa • Luis Verde
Secretarias Tecnicas
• Roxana Martınez • Laura Valencia
Morfismos puede ser consultada electronicamente en “Revista Morfismos”de la direccion http://www.math.cinvestav.mx. Para mayores informes dirigirseal telefono 50 61 38 71.Toda correspondencia debe ir dirigida a la Sra. Laura Valencia, Departamentode Matematicas del Cinvestav, Apartado Postal 14740, Mexico, D.F. 07000 opor correo electronico: [email protected].
VOLUMEN 6NÚMERO 2
JULIO A DICIEMBRE DE 2002ISSN: 18706525
Informacion para Autores
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Author Information
MORFISMOS, the student journal of the Mathematics Department of Cinvestav, invitesundergraduate and graduate students to submit manuscripts to be published under thefollowing guidelines
• All manuscripts will be refereed by specialists. However, accepted papers will beconsidered to be “preliminary versions” in that authors may republish their papers inother journals, in the same or similar form.
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Lineamientos Editoriales
“Morfismos” es la revista semestral de los estudiantes del Departamento deMatematicas del CINVESTAV, que tiene entre sus principales objetivos el que losestudiantes adquieran experiencia en la escritura de resultados matematicos.
La publicacion de trabajos no estara restringida a estudiantes del CINVESTAV;deseamos fomentar tambien la participacion de estudiantes en Mexico y en el extranjero, ası como la contribucion por invitacion de investigadores.
Los reportes de investigacion matematica o resumenes de tesis de licenciatura,maestrıa o doctorado pueden ser publicados en MORFISMOS. Los artıculos queapareceran seran originales, ya sea en los resultados o en los metodos. Para juzgaresto, el Consejo Editorial designara revisores de reconocido prestigio y con experienciaen la comunicacion clara de ideas y conceptos matematicos.
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Si tienes alguna sugerencia sobre la revista hazlo saber a los editores y con gustoestudiaremos la posibilidad de implementarla. Esperamos que esta publicacion propicie, como una primera experiencia, el desarrollo de un estilo correcto de escribirmatematicas.
Morfismos
Editorial Guidelines
“Morfismos” is the journal of the students of the Mathematics Department ofCINVESTAV. One of its main objectives is for students to acquire experience inwriting mathematics. MORFISMOS appears twice a year.
Publication of papers is not restricted to students of CINVESTAV; we want toencourage students in Mexico and abroad to submit papers. Mathematics researchreports or summaries of bachelor, master and Ph.D. theses will be considered forpublication, as well as invited contributed papers by researchers. Papers submittedshould be original, either in the results or in the methods. The Editors will assignas referees well–established mathematicians.
Even though MORFISMOS is a refereed journal, the papers will beconsidered as preliminary versions which could later appear in othermathematical journals.
If you have any suggestions about the journal, let the Editors know and we willgladly study the possibility of implementing them. We expect this journal to foster, asa preliminary experience, the development of a correct style of writing mathematics.
Morfismos
Contenido
Approximation on arcs and dendrites going to infinity in Cn (Extended version)
Paul M. Gauthier and E. S. Zeron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Bayesian procedures for pricing contingent claims: Prior information on volatility
Francisco VenegasMartınez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Existence of Nash equilibria in discounted nonzerosum stochastic games withadditive structure
Heriberto HernandezHernandez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Existence of Nash equilibria in some Markov games with discounted payoff
Carlos Gabriel Pacheco Gonzalez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Morfismos, Vol. 6, No. 2, 2002, pp. 123
Approximation on arcs and dendrites going toinfinity in Cn (Extended version) ∗
Paul M. Gauthier 1 E. S. Zeron 2
Abstract
The StoneWeierstrass approximation theorem is extended to certain unbounded sets in Cn. In particular, on arcs which are oflocally finite length and are going to infinity, each continuous function can be approximated by entire functions.
2000 Mathematics Subject Classification: 32E30, 32E25.Keywords and phrases: Uniform and tangential approximation.
1 Introduction
This work is the original version of the paper: Approximation on arcsand dendrites going to infinity in Cn [14]. This version could not bepublished in its extended form because of size limitations. However, wewish to publish it because it contains a sketch of the proof of AlexanderStolzenberg’s theorem, which we announced in [14], and several lemmason tangential approximation by polynomial and meromorphic functionswhich could not be included on [14]. For example, we include a notsowellknown result of Arakelian in Proposition 3.1.
A famous theorem of Torsten Carleman [7] asserts that for eachcontinuous function f on the real line R and for each positive continuousfunction ϵ on R, there exists an entire function g on C such that
f(x)− g(x) < ϵ(x), for all x ∈ R.∗Invited article. In memoriam: Herbert James Alexander 19401999. Research
supported by NSERC (Canada), FCAR (Quebec) and Cinvestav (Mexico).1Centre de recherches mathematiques, Universite de Montreal, Canada.2Departamento de Matematicas del CINVESTAVIPN, Mexico.
1
Morfismos, Vol. 6, No. 2, 2002, pp. 123
2 Paul M. Gauthier and E. S. Zeron
Carleman’s theorem was extended to Cn by Herbert Alexander [3] whoreplaced the line R by a piecewise smooth arc going to infinity in Cn
and by Stephen Scheinberg [20] who replaced the real line R by the realpart Rn of Cn = Rn + iRn. In the present work, we approximate onclosed subsets of area zero in Cn and extend Alexander’s theorem toclosed connected subsets Γ ⊂ Cn which are of locally finite length andcontain no closed curves.
It would be a quite difficult task to give a complete description ofall the results before the words of Carleman, Alexander and Scheinberg.Nevertheless, we have added a special section (§4) at the end of thispaper, trying to compile the historic results which have drove us to thetheorems we are proving in this paper.
Let X be a subset of Cn. X is a continuum if it is a compactconnected set. The length and area of X are the Hausdorff 1measureand 2measure of X respectively. The set X is said to be of finite lengthat a point x ∈ X if this point has a neighbourhood in X of finite length,and X is said to be of locally finite length if X is of finite length at eachof its points. Notice that if X is a set of locally finite length, then eachcompact subset of X has finite length (though X itself need not be offinite length). We denote the polynomial hull of a compact set X by !X.The algebra of continuous functions defined on X is denoted by C(X).Finally, the definition and some properties of the first Cech cohomologygroup with integer coefficients H1(X) are presented in [10] and [23].
2 The AlexanderStolzenberg theorem
John Wermer laid the foundations of approximation on curves in Cn
and prepared the way for a fundamental result of Gabriel Stolzenberg[21] concerning hulls and smooth curves (for history see [22]). In [2],Alexander comments that Stolzenberg’s theorem can be improved toconsider continua of finite length instead of smooth curves. We shallrefer to the following version as the AlexanderStolzenberg Theorem.
Theorem 2.1 (AlexanderStolzenberg) Let X and Y be two compact subsets of Cn, with X polynomially convex and Y \X of zero area.Then,
A Every continuous function on X∪Y which is uniformly approximableon X by polynomials is uniformly approximable on X ∪ Y by rational functions.
Approximation on arcs going to infinity 3
Suppose, moreover, there exists a continuum Υ ⊂ Cn such that Υ \ Xhas locally finite length and Y ⊂ (X ∪Υ). Then:
B !X ∪ Y \ (X ∪ Y ) is (if nonempty) a pure onedimensional analyticsubset of Cn \ (X ∪ Y ).
C If the map H1(X ∪Y ) → H1(X) induced by X ⊂ X ∪Y is injective,then X ∪ Y is polynomially convex.
Notice that in this AlexanderStolzenberg Theorem, locally finitelength is required only for parts B and C. Moreover, the set Υ \X maybe of infinity length; we only need it to have locally finite length. On theother hand, the main ideas in the proof of pointsA andC are essentiallycontained in [22, pp. 187188]). However, we need to introduce severalchanges due to the new hypotheses.
2.1 Proof of part A of Theorem 2.1
We shall prove part A by considering two cases, depending on whetherY is itself of zero area or not. We only need to cite Stolzenberg’s ideaswhen Y has area zero (we obviously replace K by Y in the originalpaper).
“By the theory of antisymmetric sets (see[15]) it suffices to provethat if p ∈ Y \X then for each q = p in Y ∪X there is a realvalued f ,with f(q) = f(p), which is uniformly approximable by rational functionson Y ∪X.”
“Since X is polynomially convex there is a polynomial g such thatg(p) = 1 and ℜ(g) ≤ 0 on X ∪ q. Let c be a realvalued continuousfunction on g(Y ∪ X) which is identically 0 for ℜ(ζ) ≤ 1
2 and withc(1) = 1. The following argument of Wermer shows that c is a uniformlimit of rational functions on g(Y ∪X).”
“Namely, it suffices to prove that any measure µ on g(Y ∪X) whichannihilates all uniform limits of rational functions also annihilates c.This will be done if we can show that any such µ is supported on ℜ(ζ) ≤12. But Y has area zero and g is a polynomial, so g(Y ) has areazero and, hence,
!(z − ζ)−1dµ(z) = 0 for almost all ζ with ℜ(ζ) > 1
2 .Therefore, by Fubini’s Theorem, for almost all open disks ∆ ⊂ ℜ(ζ) >12, if ∂=the boundary of ∆ then
0 =−1
2πi
"
∂dζ
"dµ(z)
z − ζ=
4 Paul M. Gauthier and E. S. Zeron
=
!dµ(z)
2πi
!
∂
dζ
ζ − z=
!χ∆(z)dµ(z),
where χ∆ is the characteristic function of ∆. It follows that µ = 0 onℜ(ζ) > 1
2.”“Hence c is a uniform limit of rational functions on g(Y ∪X) and,
hence, f = c g is a continuous realvalued function on Y ∪ X, withf(q) = f(p), which is uniform limit of rational functions.”
This settles part A when Y has area zero.Now suppose we merely know that Y \ X has zero area. Let f be
a continuous function on X ∪ Y which is uniformly approximable on Xby polynomials and let ϵ > 0. There exists a polynomial p such thatf−p < ϵ/2 onX. SinceX is polynomially convex, it has a fundamentalsystem of neighbourhoods which are polynomial polyhedra [29, Lemma7.4]. From the continuity of f − p, it follows that f − p < ϵ/2 on somepolynomial polyhedron "X containing X in its interior. Extend p !X to a
continuous function p on Y \ "X so that f− p < ϵ/2 on "X∪Y . It is easyto see that the closure K of Y \ "X has area zero because K ⊂ Y \X,and so "X ∪ Y can be written as the union of "X with a compact set ofarea zero K; it follows from the first part of this proof that there is arational function h such that p − h < ϵ/2 on "X ∪ Y . By the triangleinequality, f − h < ϵ on X ∪ Y which concludes the proof of A.
2.2 Deduction of part C from part B in Theorem 2.1
Here we also need to replace Lemma 1 of [22, p. 188]) by the followingproposition.
Proposition 2.2 Let X and Y be two compact subsets of Cn, with Xrationally convex and Y \ X of zero area. Then, X ∪ Y is rationallyconvex. If, moreover, X is polynomially convex, then given a point p inthe complement of X ∪ Y , there is a polynomial f such that f(p) = 0,0 ∈ f(X ∪ Y ) and ℜf(z) < −1 for z ∈ X.
Proof: The set X has a fundamental system of neighbourhoods whichare rational polyhedra [21, p. 283] or [23]. Given a point p in thecomplement of X ∪ Y , choose a compact rational polyhedron "X whichcontains X in its interior, but p ∈ "X. Along with "X, the closure Kof Y \ "X is also rationally convex because it has zero area (notice thatK ⊂ Y \ X and see [10, p. 71] recalling that projections preserve the
Approximation on arcs going to infinity 5
zero area condition), so there are two polynomials g and h such that0 ∈ g(K), 0 ∈ h( !X) and g(p) = h(p) = 0.
The rational function (h/g) is smooth on K, and so (h/g)(K) haszero area. Thus, we can find a complex number λ ∈ (h/g)(K) whoseabsolute value λ is so small that the polynomial f = h − λg has nozeros on !X ∪ K. Since X ∪ Y ⊂ !X ∪ K and f(p) = 0, it follows thatX ∪ Y is rationally convex.
If, in addition, X is polynomially convex, one has just to choose !Xto be a compact polynomial polyhedron (see [21] or [29, Lemma 7.4])and the polynomial h to satisfy ℜ(h) < −1 on !X; and so, for sufficientlysmall λ, ℜ(f) < −1 on X.
Now we can conclude the proof of point C by citing Stolzenberg’sideas.
“Consider any p ∈ Y ∪ X and choose an f as in Proposition 2.2.Then f is a continuous invertible function on Y ∪X with a continuouslogarithm on X. But, for any T , H1(T ) is isomorphic to the group ofall continuous invertible complexvalued functions of T modulo thosewith continuous logarithms. Therefore, since H1(Y ∪ X) → H1(X) isinjective, there is a continuous branch of log(f) on all of Y ∪X. However,
by part B, !X ∪ Y \(X∪Y ) is (if nonempty) a onedimensional analyticsubset of Cn \ (Y ∪X); so by the argument principle (see, for instance
[21, p. 271]) f has no zeroes on !X ∪ Y \ (X ∪ Y ). Hence, any such p is
not in !Y ∪X, so Y ∪X is polynomially convex.”
2.3 Proof of part B of Theorem 2.1
This proof is implicitly contained in Alexander’s paper [2], but we needto make several remarks.
Set Γ = X ∪ Y and suppose there is a point p ∈ "Γ \ Γ. FromProposition 2.2, there is a polynomial f such that f(p) = 0, 0 ∈ f(Γ)and ℜ(f) < −1 on X. Fix the compact set L = f(Γ), the closedhalfplane H = ℜ(z) ≥ −1/2 and the open set Ω to be the connectedcomponent of C\L which contains the origin. Alexander’s arguments [2]can be slightly modified to show that "Γ ∩ f−1(Ω) is a onedimensionalanalytic subset of f−1(Ω). Notice that p ∈ f−1(Ω). Alexander usesthe hypothesis that the set L has finite length in the whole plane C.However, his argument works even if we restrict the set L to have finitelength just in the halfplane H. Indeed, the intersection L ∩ H is thepolynomial image of the compact set Γ ∩ f−1(H) of finite length; recall
6 Paul M. Gauthier and E. S. Zeron
that Γ ∩ f−1(H) = (Y \ X) ∩ f−1(H) has finite length because it iscompact and contained in the set Υ \ X of locally finite length. Nowwe shall rewrite the preparatory lemmas of [2] with their respectivemodifications.
Lemma 2.3 (Lemma 1 of [2]) Let X be a secondcountable topological space, Y a set, f : X → Y a function, σ a nonzero positive measureon Y such that if V is open in X , then f(V ) is σmeasurable. Then forσalmost all y ∈ Y, the image under f of each neighbourhood in X ofeach point of f−1(y) has positive σmeasure.
Lemma 2.4 (Lemma 2 of [2]) Let D be a closed Jordan domain inC with boundary of finite length, K a compact subset of ∂D of positivelength, Q a polynomially convex set in Cn, f a polynomial in Cn, s apositive integer. Assume that Q = (f−1(∂D) ∩ Q)∧ and that f Q is atmost sto1 over points of K (i.e., if λ ∈ K then f−1(λ)∩Q has at mosts points). Then f−1(Do) ∩Q is a (possibly empty) pure 1dimensionalanalytic subset of f−1(Do). Here Do stands for the interior of D.
The hypotheses of the previous two lemmas need not be changed, sowe refer to their original proofs in Alexander’s paper [2, p. 66]. In thefollowing lemmas, the notation #(E) stands for the number (≤ ∞) ofelements of the set E.
Lemma 2.5 (Lemma 3 of [2]) Let Γ be a compact set in Cn and fa polynomial in Cn such that Γ ∩ f−1(H) has finite length. For x ∈ R,set N(x) = #p ∈ Γ;ℜf(p) = x. Then
!∞−1/2N(x)dx < ∞.
For the proof that N is a Lebesgue measurable function, see [18,p. 216].Proof: By replacing Γ by its homeomorphic image in Cn+1 under themapping z '→ (f(z), z), a Lipschitz mapping preserving the finiteness oflength, we may assume that f(z) = z1, the first coordinate projection.
Let ϵm ↓ 0. Then for each m there exists a finite collection Cmof closed balls in Cn each of diameter less than ϵm such that Cm coversΓ∩f−1(H) and if αm denotes the sum of the diameters of the members ofCm, then αm ↑ length(Γ∩f−1(H)). Let Nm(x) = #B;B ∈ Cm and x ∈ℜz1(B). Then clearly
!∞−∞Nm(x)dx = αm. Also limNm(x) ≥ N(x)
whenever x ≥ −1/2; in fact, if N(x) ≥ k, and p1, p2, . . . pk are distinct
Approximation on arcs going to infinity 7
points in Γ∩(ℜz1)−1(x), then Nm(x) ≥ k as soon as ϵm < min∥pi−pj∥;i = j. Thus, by Fatou’s lemma
! ∞
−1/2N(x)dx ≤ lim
! ∞
−1/2Nm(x)dx ≤
≤ limαm = length(Γ ∩ f−1(H)) < ∞.
Lemma 2.6 (Lemma 4 of [2]) Let I = [0, 1] be the closed unit interval of the real line and F ∈ C(I) be such that ℜF is of boundedvariation. Define for x ∈ R, N(x) = #t ∈ I;ℜF (t) = x. Then"∞−1/2N(x)dx < ∞.
Proof: Let Γ ⊂ C1 be the set (ℜF (t), t); t ∈ I and take f(z) = z inLemma 2.5.
Definition. Let L be a closed subset of C. Let Ω1 and Ω2 be components of C \ L. We shall say that the pair (Ω1,Ω2) is amply adjacentprovided the following holds: there exist real numbers b > a > −1/2and c2 > c1, and a compact subset K1 ⊂ [a, b] of positive length suchthat [a, b] × cj ⊂ Ωj for j = 1, 2 and K = (K1 × [c1, c2]) ∩ L is asubset of ∂Ω1 ∩ ∂Ω2 such that the projection π1 maps K homeomorphically (and so 1to1) onto K1 (we are identifying C and R × R, soπ1(x, y) = x).
Lemma 2.7 (Lemma 5 of [2]) Let L ⊂ C be compact and such that"∞−1/2N(x)dx < ∞ where N(x) = #q ∈ L;ℜ(q) = x. Then, for every
component Ω of C\L which meets the halfplane H, there exists a finitesequence Ω0, Ω1, . . .Ωm of components of C \ L with Ω0 equal to theunbounded component, Ωm = Ω and (Ωj−1,Ωj) amply adjacent throughrectangles Rj = [a, b]× [cj−1, cj ] contained in H for j = 1, 2, . . . ,m.
The proof of this lemma is exactly the same as the original onepresented by Alexander in his paper [2, p. 69]; he chooses a line segment[a, b]× c ⊂ Ω and uses the fact that
" ab N(x) < ∞. Thus, we shall have
exactly the same result by choosing a horizontal line segment [a, b]×c ⊂Ω ∩H and following the original proof word for word.
8 Paul M. Gauthier and E. S. Zeron
Lemma 2.8 (Lemma 6 of [2]) Let Γ be a compact subset of Cn and fa polynomial in Cn. Set L = f(Γ) ⊂ C. Suppose that
!∞−1/2N(x)dx < ∞
for N(x) = #p ∈ Γ;ℜf(p) = x, and that L ∩ H is contained in acontinuum L1 whose intersection L1∩H is of finite length. Let (Ω1,Ω2)be a pair of components of C\(L∪L1) which are amply adjacent througha rectangle R = [a, b]× [c1, c2] with b > a > −1/2. Suppose "Γ∩ f−1(Ωi)is a (possibly empty) pure 1dimensional analytic subset of f−1(Ωi) fori = 1. Then, the same is true for i = 2.
Again, the proof of this lemma follows word for word the originalone presented by Alexander in [2, p. 70], we only need to add the newtrivial condition b > a > −1/2. Alexander proves that "Γ ∩ f−1(Do) isa pure 1dimensional analytic subset of f−1(Do), where Do is an openset contained in R ∩ Ω2. He deduces then that "Γ ∩ f−1(Ω2) is also apure 1dimensional analytic set in f−1(Ω2) by using Lemma 11 of [22].This lemma is quite amazing because the component Ω2 may not becompletely contained in H.
The following lemma needs no changes in its hypotheses, so we referits proof to the original paper [2, p. 71].
Lemma 2.9 (Lemma 7 of [2]) Let Γ ⊂ S be two compact sets in Cn
and suppose that "S \S is a pure 1dimensional analytic subset of Cn \S.Then so is "Γ \ S (if nonempty).
We conclude the proof of part B of Theorem 2.1 following Alexander’s arguments. If the equality X ∪ Υ = Γ = X ∪ Y holds, we letL = f(Γ) and Ω be the connected component of C \ L which containsthe origin 0 = f(p). Apply Lemmas 2.5 and 2.7 to get a sequence Ω0,Ω1, . . .Ωm = Ω. Notice that Υ∩f−1(H) = Y ∩f−1(H) has finite lengthbecause it is compact and contained in the set Υ \ X of locally finitelength; so we can take the continuum L1 = f(Υ) in Lemma 2.8 becauseL1∩H = L∩H has finite length. We conclude inductively that "Γ∩f−1(Ω)is either empty or a pure 1dimensional analytic subset of f−1(Ω), for
L = L1 ∪ L and "Γ ∩ f−1(Ω0) = ∅. Hence !X ∪ Y \ (X ∪ Y ) is analytic
(empty or pure 1dimensional) at an arbitrary point p ∈ !X ∪ Y \(X∪Y ).Now suppose that X∪Y is strictly contained in X∪Υ. Let p ∈ "Γ\Γ
as above. ModifyΥ to obtainΥ0 such that p ∈ Υ0 butΥ0 is a continuumwith Y ⊂ X ∪Υ0 and Υ0 \X of finite length (say by radial projection tothe boundary inside a ball containing p in its interior, centered off Υ, anddisjoint from Γ). By the previous paragraph, !X ∪Υ0\(X∪Υ0) is a pure
Approximation on arcs going to infinity 9
1dimensional analytic subset of Cn\(X∪Υ0), and so is !X ∪ Y \(X∪Υ0)
because of Lemma 2.9. Therefore, the set !X ∪ Y \ (X ∪ Y ) is analytic(pure 1dimensional) at p.
An arc Υ is the homeomorphic image of a closed interval of the realline. A direct consequence of the AlexanderStolzenberg theorem is thatevery compact arc Υ which is of locally finite length everywhere exceptperhaps at finitely many of its points is polynomially convex and theapproximation condition C(Υ) = P (Υ) holds; notice that Υ may be ofinfinity length.
It is natural to ask whether the connectivity can be dropped in theseconsiderations. In fact, Alexander [4] gave an example of a compactdisconnected set Y of finite length in C2 for which !Y \Y is not a pure onedimensional analytic subset of C2 \Y . Thus, the connectivity cannot bedropped in the AlexanderStolzenberg theorem. Moreover, the followingexample shows that we cannot finesse Theorem 2.1 by enclosing Y ina continuum of finite length, although it is known that one can alwaysconstruct a compact arc Υ which meets every component of Y (so Y ∪Υis connected) and Υ \ Y is of locally finite length.
Example 2.10 There exists a discrete bounded set in C\0 such thatno continuum containing this sequence has finite length.
Consider the set E consisting of the complex numbers wj,k = k/j2+√−1/j, for j = 1, 2, . . . and k = 0, 1, . . . , j. It is easy to see that E is
contained in the disjoint union of the closed balls Bj,k with respectivecenters wj,k and radii 1
2(j+1)2 . Hence, each continuum which contains E
has to meet the center and the boundary of each ball Bj,k, so its lengthhas to be greater than
"j>1
j+12(j+1)2 = ∞.
3 Approximation on unbounded sets
Now we shall analyse approximation on closed subsets of Cn ratherthan on compact sets. Let Γ be a closed subset of Cn and F be asubclass of C(Γ). We say that a function f : Γ → C can be uniformly(resp. tangentially) approximated by functions in F if for each positiveconstant ϵ > 0 (resp. positive continuous function ϵ : Γ → R) thereis g ∈ F such that f − g < ϵ on Γ. We are mainly interested in twosubclasses F , that which is the restriction to Γ of the class O(Cn) of
10 Paul M. Gauthier and E. S. Zeron
entire functions, and that which is the restriction to Γ of the class ofmeromorphic functions on Cn whose singularities do not meet Γ.
Recall that any meromorphic function on Cn whose singularities donot meet Γ can be expressed as a quotient p/q of two entire functions pand q with q(z) = 0 for all z ∈ Γ, for the second Cousin problem can besolved in Cn. If Γ is compact, then of course, uniform and tangentialapproximation are equivalent; and we may even replace the classes ofentire and meromorphic functions on Cn by the classes of polynomialsand rational functions respectively.
We say that Γ is a set of uniform (resp. tangential) approximationby functions in the class F if each f ∈ C(Γ) can be uniformly (resp.tangentially) approximated by functions in F . Of course, as we havedefined them, such sets Γ cannot have any interior. In the literature,one also finds a more generous notion of sets of uniform or tangentialapproximation, which allows some sets having interior.
Before going any further, we should point out that, sets of uniformapproximation and sets of tangential approximation by holomorphicfunctions are in fact the same. This was proved by Norair Arakelianin his doctoral dissertation [5] in C. His proof works verbatim in Cn.Since this fact is not well known and the proof is short we include it.
Proposition 3.1 (Arakelian) Let Γ be a closed subset of Cn and letF be either the class of functions holomorphic on Γ or the class of entirefunctions. Then, Γ is a set of uniform approximation by functions in theclass F if and only if it is a set of tangential approximation by functionsin the same class.
Proof: Suppose Γ is a set of uniform approximation, f ∈ C(Γ) andϵ : Γ → R is a positive continuous function. Set ψ = ln ϵ. There existsa function g1 ∈ F such that ψ− g1 < 1 on Γ. Setting h = exp(g1 − 1),consider the functions f/h ∈ C(Γ). There exists a function g2 ∈ F suchthat f/h − g2 < 1 on Γ. Then, f − hg2 < h = exp(ℜ(g1) − 1) <expψ = ϵ. This completes the proof.
The following is a noncompact version of the StoneWeierstrassTheorem.
Proposition 3.2 A closed set Γ ⊂ Cn is a set of tangential approximation by entire functions if and only if one can approximate (in thetangential sense) the real part projections ℜ(zm) for m = 1, . . . , n.
Approximation on arcs going to infinity 11
Proof: The necessity is trivial. Moreover, if one can approximate thereal part ℜ(zm), one can approximate the imaginary part ℑ(zm) as well,since ℑ(zm) = i(ℜ(zm) − zm). Let I be the natural diffeomorphism ofCn onto the real part R2n of C2n. That is: I1(z) = ℜ(z1), I2(z) = ℑ(z1),I3(z) = ℜ(z2), I4(z) = ℑ(z2), etc., for z ∈ Cn. Given two continuousfunction f, ϵ ∈ C(Γ) with ϵ real positive, we may extend both of themcontinuously to all of Cn while keeping ϵ positive. By the theorem ofScheinberg (see introduction), there is an entire function F ∈ O(C2n)such that f(z)− F I(z) < ϵ(z)/2 for z ∈ Cn.
Since F is uniformly continuous on compact subsets of C2n, and thediffeomorphism I is proper, there is a positive continuous function δ onCn such that F I(z) − F (w) < ϵ(z)/2, for each z ∈ Cn and eachw ∈ C2n for which I(z)− w < δ(z).
By hypotheses, we can approximate each component of I on Γ byentire functions and so there exists an entire mapping h : Cn → C2n
with I − h < δ on Γ. Thus, F I −F h < ϵ/2 on Γ. By the triangleinequality, f − F h < ϵ on Γ. The function F h is entire because hand F are holomorphic.
An interesting consequence of this result is that neither projectionℜ(z) nor ℑ(z), in the complex plane z ∈ C, can be tangentially approximated on the classical examples where the tangential approximationfails to hold, although uniform approximation may sometimes be possible.
Example 3.3 Let
Γ =∞!
j=0
Γj ,
where Γ0 = [0,+∞)× 0, and for j = 1, 2, · · ·,
Γj =
"[0, j]× 1
2j,
1
2j + 1#∪"j× [
1
2j,
1
2j + 1]
#.
Then, both functions ℜ(z) and ℑ(z) can be approximated uniformly, butnot tangentially, by entire functions on Γ.
Proof: In his doctoral thesis, Arakelian [5] gave a complete characterization for sets of uniform approximation, from which it follows that Γis not a set of uniform approximation and a fortiori not a set of tangential approximation. Thus, by Proposition 3.2, the functions ℜ(z) and
12 Paul M. Gauthier and E. S. Zeron
ℑ(z) cannot be approximated tangentially. We show that they can beapproximated uniformly.
Fix ϵ > 0 and set Zϵ = z ∈ C : ℑ(z) ≤ ϵ and Wϵ = Γ \ Zϵ. Wemay assume Zϵ and Wϵ disjoint (by choosing an appropriate smaller ϵif necessary). Now, define the function
f(x) =
!ϵ for x ∈ Zϵ,
ℑ(x) for x ∈ Wϵ.
Invoking again Arakelian’s work (see [5], [13, p.245] or [9]), we deduce the existence of an entire function g such that f − g < ϵ onZϵ ∪Wϵ. Hence, ℑ − g < 2ϵ on Γ. So ℑ(z) and ℜ(z) = z − iℑ(z) canboth be approximated uniformly on Γ by entire functions.
It is interesting to compare Propositions 3.1 and 3.2 in the light ofthe previous example.
We should also notice that Proposition 3.1 also holds if we considerapproximation by functions holomorphic in a neighbourhood of Γ instead of approximation by entire functions. That is, we have that eachcontinuous function f ∈ C(Γ) can be approximated (in the tangentialsense) by functions holomorphic in a neighbourhood of Γ if and only ifevery projection ℜ(zm) can. This result suggests the following:
Proposition 3.4 Every closed set Γ ⊂ Cn of area zero is a set of tangential approximation by meromorphic functions in Cn. That is, everycontinuous function F defined on Γ can be tangentially approximated bymeromorphic functions whose singularities do not meet Γ.
Proof: Let f, ϵ ∈ C(Γ) be two continuous functions with ϵ real and positive. We must construct a meromorphic function F such that F (z)−f(z) < ϵ(z) on Γ. Let B0 be the empty set and Bk closed balls of radiusk and center in the origin.
Lemma 3.5 Each continuous function h ∈ C(Bk ∪ Γ) which can beuniformly approximated by polynomials in Bk can be uniformly approximated on D = Bk ∪ (Γ∩Bk+1) by rational functions whose singularitiesdo not meet Γ.
Proof: From Theorem 2.1.A, and for each δ > 0, there exists a rationalfunction (a/b)(z) such that (a/b)(z)− h(z) < δ for z ∈ D and 0 ∈b(D). Notice that b(Γ) has zero area, so we may choose a complex
Approximation on arcs going to infinity 13
number λ ∈ b(Γ) with absolute value so small such that λ ∈ b(D) and!!! a(z)b(z)−λ − h(z)
!!! < δ for z ∈ D.
The proof of Proposition 3.4 now follows a classical inductive process. There exists a rational function F1 whose singularities do not meetΓ and such that F1(z) − f(z) < (23 − 2−1)ϵ(z) for z ∈ Γ ∩ B1 by theprevious Lemma 3.5. Proceeding by induction, we shall construct a sequence of rational functions Fk which converges uniformly on compactsets to a meromorphic function with the desired properties.
Given a rational function Fk whose singularities do not meet Γ andsuch that Fk(z)−f(z) < (23−2−k)ϵ(z) in Γ∩Bk, let hk be a continuousfunction identically equal to zero on Bk and such that hk(z) +Fk(z)−f(z) < (23 − 2−k)ϵ(z) for z ∈ Γ ∩ Bk+1 as well. Fix a real number0 < λk < 1 strictly less than ϵ(z) for every z ∈ Γ ∩Bk+1.
Applying Lemma 3.5, there exists a rational function Rk whose singularities do not meet Bk ∪ Γ and such that Rk(z)− hk(z) < 2−1−kλk
for z ∈ Bk ∪ (Γ ∩ Bk+1). Thus, the singularities of the rational function Fk+1(z) = Fk(z) + Rk(z) do not meet Γ and Fk+1(z) − f(z) <(23 − 2−1−k)ϵ(z) for z ∈ Γ ∩Bk+1 by the triangle inequality.
Notice that Fk+1(z)−Fk(z) is holomorphic and its absolute value isless than 2−1−k inside Bk, so the sequence Fk converges to a meromorphic function with the desired properties.
Similar inductive processes were originally employed to prove Carleman’s theorem, stated in the introduction, which asserts that the realline R in C is a set of tangential approximation by entire functions.Alexander [3] extended Carleman’s theorem to piecewise smooth arcs Γgoing to infinity in Cn. That is, Γ is the the image of the real axis undera proper continuous embedding (a curve without selfintersections, goingto infinity in both directions). We should mention that this problem hadbeen considered independently by Bernard Aupetit and Lee Stout (seeAupetit’s book [1]). As a consequence of the AlexanderStolzenbergTheorem, we also have the following further extension of Carleman’stheorem, which was conjectured by Aupetit in [1] and announced byAlexander in [3].
Proposition 3.6 Let Γ be an arc which is of finite length at each oneof its points, except perhaps in a discrete subset, and going to infinityin Cn. Besides, let ϵ be a strictly positive continuous function on Γ.Then, for each f ∈ C(Γ), there exists an entire function g on Cn such
14 Paul M. Gauthier and E. S. Zeron
that f(z)− g(z) < ϵ(z) for all z ∈ Γ. That is, Γ is a set of tangentialapproximation by entire functions.
Alexander’s proof (see also [1]), for the case that Γ is smooth, reliesingeniously on the topology of arcs and the original Stolzenberg Theorem for smooth curves. It also works when the arc Γ is of locally finitelength everywhere except perhaps in a finite subset. One only needs torewrite Lemma 1 of [3], using the following corollary of Theorem 2.1.
Corollary 3.7 Let X and Y be two compact subsets of Cn such that Xis polynomially convex, Y is connected and Y \X is of finite length ateach one of its points, except perhaps at finitely many of them. If themap H1(X∪Y ) → H1(X) induced by X ⊂ X∪Y is injective, then X∪Yis polynomially convex and every continuous function f ∈ C(X ∪ Y )which can be approximated by polynomials in X can be approximated bypolynomials on the union X ∪ Y .
Proof: Let Y0 be the points where Y \X is not of locally of finite length.Notice that the inclusion mapping X → X ∪ Y can be decomposed asthe composition of the two mappings X → X ∪Y0 and X ∪Y0 → X ∪Y .Hence, the induced injective function H1(X ∪ Y ) → H1(X) can alsobe decomposed as the composition of H1(X ∪ Y ) → H1(X ∪ Y0) andH1(X ∪ Y0) → H1(X). It is easy to see that the last two functions areinjective as well. Now, suppose f ∈ C(X∪Y ) and f can be approximatedby polynomials on X. We have that X ∪ Y0 is polynomially convexbecause of the OkaWeil theorem or Theorem 2.1. Moreover, we alsohave that X ∪ Y is polynomially convex and f can be approximated bypolynomials on X ∪ Y by Theorem 2.1 again.
We can also approximate by entire functions on unbounded setswhich are more general than arcs, but first, we need to introduce thepolynomially convex hull of noncompact sets:
Definition. Given an arbitrary subset Y of Cn, its polynomially convex
hull is defined by !Y ="#
!K : K ⊂ Y is compact$.
Proposition 3.8 Let Γ be a closed set in Cn of zero area such that!D ∪ Γ \ Γ is bounded for every compact set D ⊂ Cn. Let B1 be an openball with center in the origin which contains the closure of !Γ \ Γ. Thatis, the set B1 ∪ Γ contains the hull !K of every compact set K ⊂ Γ.
Approximation on arcs going to infinity 15
Then, given two continuous functions f, ϵ ∈ C(Γ) such that ϵ is realpositive and f can be uniformly approximated by polynomials on Γ∩B1,there exists an entire function F such that F (z) − f(z) < ϵ(z) forz ∈ Γ.
Proof: Let B0 be the empty set, B1 as in the hypotheses and Bk openballs with center in the origin such that each Bk contains the closure
of !Γ ∪Bk−1 \ Γ. That is, the set Bk ∪ Γ contains the hull !K of everycompact set K ⊂ (Γ∪Bk−1). Define Xk to be the polynomially convexhull of Bk+1 ∩ (Γ∪Bk−1), so Xk ⊂ (Bk ∪Γ). The compact sets Xk andXk ∩Bk are both polynomially convex.
The given hypotheses automatically imply that there exists a polynomial F1 such that F1(z) − f(z) < (23 − 2−1)ϵ(z) on Γ ∩ B1. Proceeding by induction, we shall construct a sequence of polynomials Fk
which converges uniformly on compact sets to an entire function withthe desired properties.
Given a polynomial Fk such that Fk(z)− f(z) < (23 − 2−k)ϵ(z) onΓ ∩ Bk, let hk be a continuous function equal to Fk on Bk and suchthat hk(z)− f(z) < (23 − 2−k)ϵ(z) for z ∈ Γ ∩Bk+1 as well. Fix a realnumber 0 < λk < 1 strictly less than ϵ(z) for every z ∈ Γ ∩Bk+1.
Notice that Xk = (Xk∩Bk)∪(Γ∩Bk+1). Hence, by Theorem 2.1.A,the function hk can be approximated by rational functions on Xk becauseXk∩Bk is polynomially convex and Γ has zero area. Moreover, thefunctions hk can be approximated by polynomials by the OkaWeil theorem. Thus, there exists a polynomial Fk+1 such that Fk+1(z)−hk(z) <2−1−kλk for z ∈ Xk, and so Fk+1(z) − f(z) < (23 − 2−1−k)ϵ(z) onΓ ∩Bk+1.
Finally, the inequality Fk+1(z)−Fk(z) < 2−1−k holds for z ∈ Bk−1,so the sequence Fk converges to an entire function with the desiredproperties.
On the other hand, if the equality !Γ = Γ holds as well in the lastproposition, we can choose the empty set instead of the open ball B1
(because the proof is an inductive process); and so Γ becomes a setof tangential approximation by entire functions. There are many closedsets Γ which satisfy the hypotheses of the last proposition. For example,we have the following.
Theorem 3.9 Let Γ be closed connected set of locally finite length inCn whose first cohomology group H1(Γ) vanishes (Γ contains no simple
16 Paul M. Gauthier and E. S. Zeron
closed curves). Then, Γ is a set of tangential approximation by entirefunctions.
Proof: The proof strongly uses the topology of Γ. We show thateach point of Γ has finite order; that is, has a basis of neighbourhoodsin Γ having finite boundaries. Given a point z ∈ Γ, let Br be theopen ball in Cn of radius r and center z. Since Γ is locally of finitelength, the intersection of Γ with the closed ball Br has finite length, sothe intersection of Γ with the boundary of Bs must be a finite set foralmost all radii 0 < s < r. Whence, each subcontinuum of Γ is locallyconnected [17, p. 283]. On the other hand, there are no simple closedcurves contained in Γ because H1(Γ) = 0; so each subcontinuum of Γ isa dendrite, that is, a locally connected continuum containing no simpleclosed curves. In particular, if Γ is compact, then it is a dendrite.
Notice the following lemma.
Lemma 3.10 Each compact subset K ⊂ Γ is contained in a subcontinuum (dendrite) of Γ.
Proof: Since Γ is locally connected, the set K is contained in a finiteunion of subcontinua of Γ. The lemma now follows since Γ is arcwiseconnected (see Theorem 3.17 of [16]).
Let D be a compact set in Cn. Notice that D∪Γ may contain simpleclosed curves Υ with D∩Υ = ∅ but Υ ⊂ D. We shall call such a simpleclosed curve Υ ⊂ (D ∪ Γ) a loop. We show there exists a ball whichcontains all of these loops. Henceforth, let Br be open balls of radii rand center in the origin, and choose a radius s > 0 such that D ⊂ Bs.Recall that Γ ∩ Bs+1 has finite length, so there exists a ball Bt withs < t < s + 1 such that Γ meets the boundary of Bt only in a finitenumber of points Q = q1, . . . , qm. Let Υj be the possible loopswhich meet the complement of Bt. The set
!Υj \Bt is contained in
Γ and can be expressed as the union of compact arcs (not necessarilydisjoint) which lie outside of Bt except for their two end points whichlie in Q. Since Γ cannot contain simple closed curves, two different arcscannot share the same end points, and there can only be finitely manysuch arcs. Hence, there exists a ball Bδ which contains all the loops Υ,and D ⊂ Bδ.
We shall show that !D ∪ Γ\Γ is bounded. Without loss of generality,we may suppose that D is a closed ball. Since Γ is connected, the hull!D ∪ Γ is equal to
!r≥δ
"Kr, where Kr is the connected component of
Approximation on arcs going to infinity 17
Br ∩ (D ∪ Γ) which contains D. We can prove that !Kr = !Kδ ∪Kr, forevery r ≥ δ, using Alexander’s original argument. The following lemmais a literal translation of Lemma 1.(a) of [3], to our context.
Lemma 3.11 For every r ≥ δ, !Kr = !Kδ ∪ τr where τr = Kr \Kδ.
Since the notation is quite complicated and different from Alexander’s, and we need to invoke Theorem 2.1.B, we shall include the proofof Lemma 3.11, but first we conclude the proof of the theorem.
By Lemma 3.11, the set !D ∪ Γ\Γ is bounded because !Kr = !Kδ∪τr =!Kδ ∪ Kr and !D ∪ Γ = !Kδ ∪ Γ. Moreover, the equality !Γ = Γ holds aswell because each compact subset of Γ is contained in a dendrite of finitelength and is polynomially convex (see Lemma 3.10 and Alexander’swork [2]), so we can deduce from Proposition 3.8 that Γ is a set oftangential approximation.
Proof: [Proof of Lemma 3.11] Let Tr = !Kδ ∪ τr be the set on the righthand side of the asserted equality. Clearly, we have Tr ⊂ !Kr ⊂ !Tr (thesecond inclusion is in fact equality). Thus it suffices to show that Tr ispolynomially convex. Arguing by contradiction, we suppose otherwise.By Theorem 2.1.B, !Tr \ Tr is a 1dimensional analytic subvariety ofCn \ Tr.
Let V be a nonempty irreducible analytic component of !Tr \Tr. Weclaim that V \Kr is an analytic subvariety of Cn\Kr. Since Tr = !Kδ∪τr,it suffices to verify this locally at a point x ∈ V ∩Q where
Q = !Kδ \Kδ.
By Theorem 2.1.B, both !Kr and Q are analytic near x, where nearx refers to the intersection of sets with small enough neighbourhoodsof x, here and below. Furthermore, near x, V ⊂ !Kr, V ⊂ !Kr \ Q andQ ⊂ !Kr. Thus, near x, Q is a union of some analytic components of!Kr. It follows that near x, V is just a union of some of the other localanalytic components of !Kr at x; in fact, near x, V = V ∪ x. Put
W = V \Kr.
Then W is an irreducible analytic subset of Cn \Kr and moreover,
W \W ⊂ Kδ ∪ τr = Kr.
Thus W ⊂ !Kr by the maximum principle.
18 Paul M. Gauthier and E. S. Zeron
Fix a point p ∈ V ⊂ W . Since p ∈ Tr, we have p ∈ !Kδ and thereforethere exists a polynomial h such that h(p) = 0 and ℜh < 0 on !Kδ. Bythe open mapping theorem, either h(W ) is an open neighbourhood of0 or h ≡ 0 on W . In the latter case, h ≡ 0 on W and so W \ W isdisjoint from Kδ. This implies that W \W ⊂ τr so W ⊂ τr. We havea contradiction because τr is contained in a dendrite of finite lengthand is polynomially convex (see Lemma 3.10 and Alexander’s work [2]),and moreover, a dendrite cannot contain a 1dimensional analytic set.Hence, the former case holds. Since h(τr) is nowhere dense in the plane(recall that it is of finite length), there is a small complex number α ∈h(W ) such that α ∈ h(τr). Now put g = h−α. If α is sufficiently small,we conclude that (i) ℜg < 0 on !Kδ, (ii) g(q) = 0 for some q ∈ W and(iii) 0 ∈ g(τr).
Now (i) implies that the polynomial g has a continuous logarithmon !Kδ and so, by restriction, on Kδ. We can extend this logarithm of gon Kδ to a continuous logarithm of g on Kr because of (iii), since theball Bδ was chosen such that every simple closed curve (loop) Υ ⊂ Kr
is contained in Bδ and hence in Kδ. But Kr contains W \W . Applyingthe argument principle [21, p. 271] to g on the analytic set W gives acontradiction to (ii).
We remark that the condition of having zero area is essential inPropositions 3.4 and 3.8, as the following example (inspired by [8])shows.
Example 3.12 Let I be the closed unit interval [0, 1] of the real lineand K ⊂ I the compact set K =
"0, 1, 12 ,
13 ,
14 , . . .
#. It is easy to see
that the (2+ ϵ)dimensional Hausdorff measure of the closed connectedset Y = (I × 0) ∪ (K × C) in C2 is equal to zero for every ϵ > 0;moreover, the equality !Y = Y holds. However, the following continuousfunction f ∈ C(Y ) cannot be uniformly approximated by holomorphicfunctions in O(Y ):
f(w, z) =
$z if w = 10 otherwise
Suppose there exists a real number ϵ > 0 and a holomorphic functiong ∈ O(Y ) such that f−g < ϵ on Y . We automatically have that g(w, z)is bounded, holomorphic and constant on each complex line 1
j × C,j = 2, 3, . . .. Hence, the holomorphic function ∂g
∂z vanishes on each
complex line 1j ×C, j = 2, 3, . . . as well. Since the zero set of ∂g
∂z is an
Approximation on arcs going to infinity 19
analytic set, this derivative must be zero in a neighbourhood of 0×Cand hence on the connected set Y . The last statement is a contradictionto the fact that g(1, z)− z < ϵ for every z ∈ C.
On the other hand, to see that !Y = Y , notice that Y ="
r>0 Yr,where Yr = (I × 0)∪ (K ×∆r) and ∆r ⊂ C are closed disks of radiusr. The set K × ∆r is polynomially convex because it is the Cartesianproduct of two polynomially convex sets in C; and so Yr is polynomiallyconvex because of Theorem 2.1.
Although connectivity, as we have emphasized, plays a crucial rolein this paper, similar results can be obtained for sets whose connectedcomponents form a locally finite family. Finally, we remark that, ona Stein manifold, analogous results also hold by simply embedding theStein manifold into some Cn. A possible exception is Proposition 3.2,since ℜ(z) is not welldefined on a manifold.
4 Historical notes
In this section we recapitulate and supplement some of the historicalremarks which are dispersed throughout this paper.
Of course, the foundation of approximation theory is the Weierstrasstheorem (1885), which affirms that each closed interval is a set of uniform approximation by polynomials. This is essentially a real result.In the complex setting, the most beautiful approximation theorem is adeep theorem of Walsh [24, in 1926] which lifts the Weierstrass theoremto the complex domain by asserting that each Jordan arc (homeomorphic image of a closed interval) in the complex plane is a set of uniformapproximation by polynomials. For a survey on this result of Walsh andits impact, see [12].
Just as Walsh’s theorem is the most beautiful result of uniformapproximation in the complex plane C, the outstanding open problem incomplex approximation is to extend Walsh’s theorem to higher dimensions (Cn). Any compact set firstly needs to be polynomially convex(see [21]) in order to be a set of uniform approximation by polynomials. In high dimensions, Wermer [25, in 1955] and Rudin [19, in 1956]gave examples of Jordan arcs which are not polynomially convex, andhence they are not sets of uniform polynomial approximation. The mainproblem can be then formulated more precisely: Is it true that eachpolynomially convex Jordan arc is a set of polynomial approximation?This problem has remained open for over half a century.
20 Paul M. Gauthier and E. S. Zeron
The following Jordan arcs are known to be sets of uniform approximation: analytic arcs (Wermer [26], [27] and [28] in 1958), C1smootharcs (Stolzenberg [22] in 1966), rectifiable arcs (Alexander [2] in 1971);and in the present paper we allow arcs which are of finite length at eachpoint, except perhaps at a finite set of points.
One can also consider rational approximation and; here again, anycompact set firstly needs to be rationally convex in order to be a setof uniform rational approximation. It is known that any compact setof area zero is a set of rational approximation. Bagby and one of theauthors [6] have given an example of an arc of finite area which is notrationally convex and, a fortiori, it is not a set of rational approximation.
We have seen, on one side, that we have polynomial approximationon Jordan arcs whose length is locally finite except perhaps at a finitesubset of points. On the other hand, we do not have rational approximation on a certain Jordan arc of finite area. It is quite natural to askabout the intermediate cases, namely, Jordan arcs whose dimension liesbetween 1 and 2. This question was in fact posed by Gamelin [11].
As mentioned earlier in this paper, the Weierstrass theorem was alsoextended in a different way by Carleman [7, in 1927], who showed thatthe realline in C is a set of tangential approximation by entire functionsin C. This result was also generalized to several complex variables intwo ways. First of all, Scheinberg [20, in 1976] showed that the realpart of Cn is a set of Carleman approximation by entire functions in Cn.Secondly, and this is the generalization which concerns us in the presentpaper, Carleman himself had conjectured and Keldysh proved that eachunbounded Jordan arc in C is a set of tangential approximation by entirefunctions as well. This result was extended by Alexander [2, in 1979]to unbounded Jordan arcs in Cn which are piecewise C1smooth. Wehave shown that Alexander’s result also holds for unbounded Jordanarcs which are of locally finite length. This had been conjectured byAupetit [1, in 1978] and announced by Alexander [2]. But we showed astronger result, by allowing a discrete subset of exceptional points, andalso by allowing more general sets than only unbounded arcs.
Paul M. GauthierDepartement de mathematiqueset de statistique et Centre derecherches mathematiques,Universite de Montreal,CP 6128 Centre Ville,Montreal, H3C 3J7, [email protected]
Eduardo Santillan ZeronDepartamento de Matematicas,CINVESTAV  I.P.N.,A. Postal 14740,Mexico D.F. 07000, [email protected]
Approximation on arcs going to infinity 21
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[9] Gaier, D., Lectures on complex approximation, Translated fromthe German by R. McLaughlin. Birkhauser, Boston Mass. 1987.
[10] Gamelin, T. W., Uniform algebras. PrenticeHall, EnglewoodCliffs N.J. 1969.
[11] Gamelin, T. W., Polynomial approximation on thin sets, Symposium on several complex variables (Park City, Utah 1970), 50–78.Lecture Notes in Math. 184, SpringerVerlag, Berlin 1971.
[12] Gauthier, P. M., Commentary, Walsh, Joseph L., Selected papers, 247–253. Edited by Theodore J. Rivlin and Edward B. Saff,SpringerVerlag, New York 2000.
22 Paul M. Gauthier and E. S. Zeron
[13] Gauthier, P. M.; Sabidussi, G., Complex potential theory. Proceedings of the NATO Advanced Study Institute and the Seminairede Mathematiques Superieures held in Montreal, Quebec. NATOAdvanced Science Institutes Series C: Mathematical and PhysicalSciences, 439. Kluwer, Dordrecht, 1994.
[14] Gauthier, P. M.; Zeron, E. S., Approximation on arcs and dendrites going to infinity in Cn, Can. Math. Bull., 45 (2002), No. 1,80–85.
[15] Glicksberg, I., Measures orthogonal to algebras and sets of antisymmetry, Trans. Amer. Math. Soc., 105 (1962), 415–453.
[16] Hocking, J. G.; Young, S. G., Topology. Dover Publications, NewYork 1988.
[17] Kuratowski, K., Topology Vol. II. Academic Press, New York andLondon 1968.
[18] Rado, T.; Reichelderfer, P., Continuous transformations in analysis. SpringerVerlag 1955.
[19] Rudin, W., Subalgebras of spaces of continuous functions, Proc.Amer. Math. Soc., 7 (1956), 825–830.
[20] Scheinberg, S., Uniform approximation by entire functions, J.Analyse Math., 29 (1976), 16–18.
[21] Stolzenberg, G., Polynomially and rationally convex sets, ActaMath., 109 (1963), 259–289.
[22] Stolzenberg, G., Uniform approximation on smooth curves, ActaMath., 115 (1966), 185–198.
[23] Stout, E. L., The theory of uniform algebras. Bogden & Quigley,TarrytownonHudson NY. 1971.
[24] Walsh, J. L., Uber die Entwicklung einer Funktion einer komplexen Veranderlichen nach Polynomien, Math. Ann., 96 (1926).
[25] Wermer, J., Polynomial approximation on an arc in C3, Ann ofMath., 62 (1955), 269–270.
[26] Wermer, J., Function rings and Riemann surfaces, Ann. Math.,67 (1958), 45–71.
Approximation on arcs going to infinity 23
[27] Wermer, J., Rings of analytic functions, Ann. Math., 67 (1958),497–516.
[28] Wermer, J., The hull of a curve in Cn, Ann. Math., 68 (1958),550–561.
[29] Wermer, J., Banach algebras and several complex variables. Graduate Texts in Mathematics No. 35, SpringerVerlag, New YorkHeidelberg 1976.
Morfismos, Vol. 6, No. 2, 2002, pp. 25–41
Bayesian procedures for pricing contingentclaims: Prior information on volatility ∗
Francisco VenegasMartınez 1
Abstract
This paper develops a Bayesian model for pricing derivative securities with prior information on volatility. Prior information isgiven in terms of expected values of levels and rates of precision:the inverse of volatility. We provide several approximate formulas,for valuing European call options, on the basis of asymptotic andpolynomial approximations of Bessel functions.
2000 Mathematics Subject Classification: 62F15, 60G15.Keywords and phrases: Bayesian inference, options pricing, stochasticvolatility, numerical methods.
1 Introduction
When studying market volatility, the standard procedure is to analyzedata. For example, we may explore plots and frequency histograms,or even examine how observations were collected. However, there isanother approach to study market volatility before data is observed,which is based on previous practical experience and understanding, theBayesian approach. In such a case, the parameters of a sampling modelare regarded as random variables, and all judgements are made in termsof the degree of belief on potential values of the parameters. In thisframework, a prior distribution is used to describe initial knowledge ofthe possible values of the parameters of a sampling model. For example, we may feel, based on earlier experience, that our degree of belief
∗Invited article.1 .ocixeM,yerretnoMedocigolonceT,saznaniFnenoicagitsevnIedortneC
25
26 Francisco VenegasMartınez
about the values of a given parameter may be expressed by a specificprobability distribution, which describes initial knowledge.
In pricing contingent claims it is of particular interest to draw inferences about unknown volatility or uncertain volatility parameters of theunderlying asset on the basis of prior information. Considering initialinformation before data is observed is not just a sophisticated extensionbut an essential issue to be taken into account for the theory and practice of derivatives. In this paper, we present a new Bayesian methodto price derivative securities when there is prior information on uncertain and changing volatility. In our proposal, investors are rational inthe sense that they use efficiently prior information by choosing a priordistribution that maximizes logarithmic utility among all admissible distributions describing available information. After all, the core of financetheory (mathematical or empirical) is the study of the rational behaviorof investors in an uncertain environment. A study for the behavior ofrational agents in the Mexican case can be seen in VenegasMartınez[20].
This paper is organized as follows. In the next section, we mentionsome of the limitations of the stochastic volatility approach, and discussthe need of considering prior information in pricing derivatives. In section 3, we review the Bayesian inference framework and its relationshipwith information theory. In section 4, we develop a Bayesian model toprice derivative securities and exploit its relationship with Bessel functions. In sections 5 and 6, we examine some asymptotic and polynomialapproximations of the basic Bayesian valuation problem. Through section 7, we carry out a comparison of our approach with other modelsavailable in the literature. Finally, In section 8, we draw conclusions,acknowledge limitations, and make suggestions for further research.
2 Limitations of the stochastic volatilityapproach
The most common setup of the stochastic volatility model consists in ageometric Brownian motion correlated with a meanreverting OrsteinUhlenbeck process. This approach for pricing derivatives has beenwidely studied with a remarkable theoretical progress; see, for instance,Ball and Roma [3], Heston [10], Renault and Touzi [18], Stein and Stein
Bayesian Procedures 27
[19], Wiggins [22], and Avellaneda et al. [2]. In particular, the stochastic volatility models allow us to reproduce in a more realistic way assetreturns, specially in the presence of fat tails (Wilmott [23]), asymmetry in the distribution (Fouque, Papanicolaou, and Sircar [7]), and thesmile effect (Hull and White [11]). However, there is a set of empiricalregularities (or stylized facts) that still need to be explained. In particular, the existing models do not explain how investors, ranging fromnon corporate individual to large trading institution, choose the bestpatterns of investment (rational behavior) if there is prior informationon volatility, and its implications when valuing derivatives.
3 The Bayesian approach to price derivativesecurities
In the real world, volatility is neither constant nor directly observed.Hence, it is natural to think of volatility as a nonnegative random variable with some initial knowledge coming from practical experience andunderstanding before data is observed. This is just the Bayesian wayof thinking about prior information. Under this approach, prior information is described in terms of a probability distribution (subjectivebeliefs) of the potential values of volatility. It is common in Bayesian inference, instead of studying volatility, σ > 0, to study precision, whichis defined as the inverse of the variance, h = σ−2; see, for instance,Leonard and Hsu [12], and Berger [4]. Thus, the lower the variance,the higher the precision. More precisely, from the Bayesian point ofview, we have a distribution, Ph, h > 0, describing prior information.We shall assume that Ph is absolutely continuous with respect to theLebesgue measure ν, so that the RadonNykodim derivative provides aprior density, π(h), i.e., dPh/ ν(h) = π(h) for all h > 0. Then, we maywrite
Phh ∈ A =!
Aπ(h)dν(h)
for all Borel sets A.
3.1 Maximum entropy priors
There are several wellknown methods reported in the Bayesian literature to construct densities that incorporate prior information by max
28 Francisco VenegasMartınez
imizing a criterion functional subject to a set of constraints in termsof expected values. Some of such methods are: noninformative priors(Jeffreys [14]); maximal data information priors (Zellner [24]); maximum entropy priors (Jaynes [13]); minimum crossentropy priors, alsoknown as relative entropy priors (Kullback [16]); reference priors (Good[8] and Bernardo [5])); and controlled priors (VenegasMartnez et al.[21]). We shall specialize in this paper in Jaynes’ maximum entropy forpragmatic and theoretical reasons that will appear later.
Let us suppose that there is initial information on volatility in termsof expected values, say
!ak(h)π(h)Ih>0dν(h) = ak, k = 0, 1, 2, ..., N,
where the functions ak(h) are Lebesguemesurable known functions andall the constants ak are known, as well. The maximum entropy principle states that from all densities satisfying the given information (constraints) we should choose the one that maximizes
H[π(θ)] = −"
h>0ln[π(h)]π(h)dν(h).
We define a0(h) ≡ 1 and a0 = 1 to ensure that the solution is indeed aproper density. Hence, we are interested in finding π(h) that solves thefollowing variational problem:
maxπ
H[π(θ)] = −"
h>0ln[π(h)]π(h)dν(h),
subject to C :"
ak(h)π(h)Ih>0dν(h) = ak, k = 0, 1, 2, ..., N.
In the sequel, we shall assume that the set of the constraints, C, form aconvex and compact set on π. Since H[π(h)] is strictly concave in π(h),the solution exists and is unique. In such a case, the necessary conditionfor π(h) to be a maximum, is also sufficient. By using standard necessaryconditions derived from calculus of variations (see, for instance, Chiang[6]), we found that if π(h) is optimal, then
π(h) = e1+λ0 exp
#N$
k=1
λkak(h)
%
,(1)
where λk, k = 0, 1, 2, ..., N , are the Lagrange multipliers associated withthe constraints C.
Bayesian Procedures 29
3.2 Relative entropy
Another useful inference method to estimate an unknown probabilitydensity, π(h), when there is an initial estimate p(h) of π(h), and information about precision h in terms of expectations, is based on determining π(h) that solves the following variational problem:
minπ
!
h>0π(h) ln
π(h)
p(h)dν(h),
subject to:!π(θ)Ih>0dν(h) = 1,
!ak(h)π(h)Ih>0dν(h) = ak, k = 1, 2, ..., N.
The quantity"h>0 π(h) ln(π(h)/p(h))dν(h) is called the relative en
tropy between π(h) and p(h), and satisfies a set of axioms of consistency:uniqueness of the final estimate; invariance under onetoone coordinatetransformations; system independence; and subset independence. Inthis case, if π(h) is optimal, we have that
π(h) = p(h)e1+λ0 exp
#N$
k=1
λkak(h)
%
,
where λk, k = 0, 1, 2, ..., N , are the Lagrange multipliers associated withthe constraints. Observe that when the initial estimate is a uniformdensity, then relative entropy becomes entropy, as defined in section3.1. Finally, it is important to mention the work of Avellaneda, Levy,and Paras [1] on derivative securities when modeling potential volatilityvalues occurring within an open interval using relative entropy.
3.3 Examples of priors on precision
Suppose that prior information on precision is given in terms of expectedvalues of levels and rates. That is, prior knowledge is expressed as:
!
h>0hπ(h)dν(h) =
β
α,(2)
and !
h>0ln(h)π(h)dν(h) = ψ(α)− ln(β),(3)
30 Francisco VenegasMartınez
where α > 0, β > 0, ψ(α) = dΓ(α)/dα, and Γ(·) is the Gamma function.Notice that for given expected values on levels and rates, equations (2)and (3) become a nonlinear system in the variables α and β. Sinceentropy is strictly concave and the Gamma distribution is the uniquedistribution that satisfies (2) and (3), we find that
π(hα,β) = hα−1βαe−βh
Γ(α), h > 0, α > 0, and β > 0,(4)
solves the maximum entropy problem. Another priors of interest, aftersome changes of variable, could be:
π
!1
h
"""""α,β
#
=hα+1βαe−βh
Γ(α), h > 0, α > 0, and β > 0,(5)
π
!1√h
"""""α,β
#
=2hα+
12βαe−βh
Γ(α), h > 0, α > 0, and β > 0,(6)
and
π
!
ln$1
h
% """""α,β
#
=βαe−βe− ln(1/h)−ln(1/h)
Γ(α),
h > 0, α > 0, and, β > 0, which stand, respectively, for prior distributions of σ2, σ, and ln(σ2). In any case, the best choice should reflectwhat has been learned from previous practical experience.
4 Statement of the basic Bayesian valuationproblem
Let us consider a Wiener process (Wt)t≥0 defined on some fixed filteredprobability space (Ω,F , (Ft)t≥0, IP), and a European call option on anunderlying asset whose price at time t, St, is driven by a geometricBrownian motion accordingly to
dSt = rSt dt+ h−1/2StdWt,
that is, (Wt)t≥0 is defined on a risk neutral probability measure IP.Notice that the stochastic differential equation driving the price of the
Bayesian Procedures 31
underlyng asset depends only on the riskfree rate of interest. The driftis independent of risk preferences about the expected return on the asset.In this case, investors do not require a premium as long as volatilityremains constant. Girsanov’s theorem can be used to remove a driftwith risk preferences by providing an equivalent risk neutral probabilitymeasure (see, for instance, Fouque, Papanicolaou, and Sircar [7].). The
option is issued at t0 = 0 and matures at T > 0 with strike price X.Under the Bayesian framework, we have that the price, at time t0 = 0,of the contingent claim when there is prior information on volatility, asexpressed in (4), is given by:
c(S0, T,X, rα,β) = e−rTE(π) E [max(ST −X, 0)S0]
= e−rT!
h>0
"!
s>X(s−X)f
ST S0(s)ds
#π(h)dν(h),(7)
where the conditional density of ST given S0 satisfies
fST S0
(s) =h1/2
s√2πT
exp
$
− h
2T
%G(s) +
T
2h
&2'
,
and
G(s) = ln%
s
S0e−rT
&.
If we assume that the required conditions to apply Fubinis’ theorem aresatisfied, so we can guarantee that integrals can be interchanged, then(7) becomes
c =e−rTβα
√2πTΓ(α)
!
s>X
%1− X
s
&I(sα,β)ds,(8)
where
I(sα,β) =!
h>0exp
$
− h
2T
%G(s) +
T
2h
&2'
hα−12 e−βhdν(h).(9)
Notice now that (9) can be, in turn, rewritten as
I(sα,β) = exp"−G(s)
2
#!
h>0exp
"−A(s)h− B
h
#hδ−1dν(h),(10)
32 Francisco VenegasMartınez
where
A(s) =
!G(s)2
2T+ β
"
> 0, B =T
8> 0, and δ = α+
1
2> 0.
The integral in (10) satisfies (see, for instance, Gradshteyn and Ryzhik[9])
#
h>0exp
$−A(s)h− B
h
%hδ−1dν(h) = 2
&B
A(s)
' δ2
Kδ
&2(BA(s)
',
(11)where Kδ(x), x = 2
)BA(s), is the modified Bessel function of order δ,
which is solution of the secondorder ordinary differential equation (see,for instance, Redheffer [17])
y′′ +1
xy′ −
!
1 +δ2
x2
"
y = 0, x > 0.(12)
We also have that Kδ(x) is always positive, and Kδ(x) → 0 as x →∞. Equation (11) is of noticeable importance since it says that all theadditional information on volatility provided by the prior distributionand the relevant information on the process driving the dynamics of theunderlying asset are now contained in Kδ.
4.1 Constant elasticity of return variance
In this section, we deal with the constant elasticity instantaneous variance case. Let us assume the underlying asset, St, evolves accordingto
dSt = rSt dt+ h−1/2Sb/2t dWt,
where the elasticity of return variance with respect to the price isdefined as b− 2. If b = 2, then the elasticity is zero and asset prices arelognormally distributed. In this section, we are concerned with the caseb < 2. After computing the Jacobian for transforming W. t ∼ N (0, t.)into ST , we find that the conditional density of ST given St satisfies
fST S0
(s) =h
δD[UV (s)1−2b]1/(4−2b)e−h[U+V (s)]Iδ
&2h(UV (s)
',
where
Bayesian Procedures 33
δ = 1/(2− b),
D =!
2r
(2− b)[er(2−b)T − 1]
"1/(2−b)
,
U =#DS0e
rT$2−b
,
V (s) = (Ds)2−b,
and Iδ(x), x = 2h%UV (s), is the modified Bessel function of the first
kind of order δ. If we assume that prior distribution is described by aGamma density, then
c =De−rTβα
δ√2πTΓ(α)
&
s>X(s−X)
'UV (s)1−2b
(1/(4−2b)J (sα,β)ds,
where
J (sα,β) =&
h>0hαe−h[β+U+V (s)]Iδ
)2h*UV (s)
+dν(h)
which is related with the noncentral chisquare density function. Moreover,
Iδ
)2h*UV (s)
+=
∞,
k=0
hδ+2k [UV (s)]k+(δ/2)
Γ(k + 1)Γ(δ + k + 1).
Hence,
J (sα,β) =∞,
k=0
[UV (s)]k+(δ/2)
Γ(k + 1)Γ(δ + k + 1)
&
h>0hα+δ+2ke−h[β+U+V (s)] dν(h)
=[UV (s)]δ/2
(β + U + V (s))α,
k=0
∞ [UV (s)]k Γ [α+ δ + 2k + 1]
Γ(k + 1)Γ(δ + k + 1).
In the particular case that there is not prior information, the solutionof maximizing H[π(θ)], subject only to the normalizing constraint, willlead to an improper uniform prior distribution, say π(h) ≡ 1 almosteverywhere with respect to ν, then if z = [V (s)/D]1/(2−b), equivalentlyV (s) = Dz2−b, we have
c = S0
& ∞
hDX2−beh(U+Dz2−b)
Dz2−b
U
.1/(4−2b)
Iδ#2h
√UDz2−b
$dz
+ Xe−rT& ∞
hDX2−beh(U+Dz2−b)
)U
Dz2−b
+1/(4−2b)
· Iδ#2h
√UDz2−b
$dz,
34 Francisco VenegasMartınez
where the following identity holds
! ∞
hDX2−beh(U+Dz2−b)
"#Dz2−b
U
$1/(4−2b)
+%
U
Dz2−b
&1/(4−2b) 'Iδ(2h
√UDz2−b
)dz = 1.
5 Asymptotic approximations for the basicBayesian valuation problem
In this section, we find an asymptotic approximate formula for pricingvanilla contingent claims according to equation (8)(11). In order to useasymptotic approximations for equation (11), we have to make someassumption on the strike price, X. Note first, that if the strike priceX is large, then x is large. In such a case, we may use the followingapproximation (see, for instance, Gradshteyn and Ryzhik [9]):
Kδ(x) ∼ *Kδ(x) =
+π
2xe−x
#
1 +4δ2 − 1
8x
$
,
which, in practice, performs well. In this case, we have the estimateprice
,c = S0*M1(S0, T,X, rα,β)− e−rTX *M2(S0, T,X, rα,β),
where
*M1 =
√2βα
S0
√πTΓ(α)
! ∞
Xe−(
12G(s)+r)
%T
8A(s)
& δ2 *Kδ(2
BA(s))ds,
and
*M2 =
√2βα
√πTΓ(α)
! ∞
X
1
se−
12G(s)
%T
8A(s)
& δ2 *Kδ(2
BA(s))ds.
The integrals *M1 and *M2 can be approximated with simple proceduresin MATLAB by using a large enough upper limit in the integral. Theupper limits of the integrals *M1 and *M2 are taken large enough so thatthe values of *M1 and *M2 have no sustantial change when larger uperlimits are used (within an error of 0.0001). Figure 1 shows the valuesof ,c as a function of α (δ = α+ 1
2) and β, with S0 = 42.00, X = 41.00,r = 0.11, and T = 0.25.
Bayesian Procedures 35
Figure 1. Values of !c as a function of α and β.
6 Polynomial approximations for the basicBayesian valuation problem
Polynomial approximations, for the basic Bayesian valuation problemstated in (8)(11), can be done only for some numerical values of theparameters. In this case, we apply the Frobenius’ method to obtain anapproximate polynomial of finite order. Let us consider the particularcase α = 0.5, i.e., δ = 1, in equation (9). The following polynomialapproximation is based on Frobenius’ method of convergent powerseriesexpansion:
K1(x) =1
x
"
x ln#x
2
$I1(x) +
6%
k=0
ak
#x
2
$2k
+ ϵ
&
, 0 < x ≤ 2,(13)
where a0 = 1, a1 = 0.15443144, a2 = −0.67278579, a3 = −0.18156897,a4 = −0.01919402, a5 = −0.0110404, a6 = −0.00004686, and
36 Francisco VenegasMartınez
I1(x) = x
[6∑
k=0
bk
(4x
5
)k
+ ϵ
]
, 0 < x ≤ 15
4,
where b0 = 1/2, b1 = 0.878900594, b2 = 0.51498869, b3 = 0.15084934,b4 = 0.02658733, b5 = 0.00301532, b6 = 0.00032411, and ϵ < 8 × 10−9.The complementary polynomial are given by
K1(x) =1√xex
ln(x
2
)I1(x) +
6∑
k=0
ak
(x
2
)−2k
+ ϵ, x > 2,(14)
where a0 = 1.25331414, a1 = 0.23498619, a2 = −0.03655620, a3 =0.01504268, a4 = −0.00780353, a5 = 0.00325614, a6 = −0.00068245,and
I1(x) = x
[8∑
k=0
bk
(4x
5
)−k
+ ϵ
]
, x >15
4,
where b0 = 39894228, b1 = −0.03988024, b2 = −0.00362018, b3 =0.00163801, b4 = −0.01031555, b5 = 0.02282967, b6 = −0.02895312,b7 = 0.01787654, b8 = −0.00420059, and ϵ < 2.2×10−7. It is importantto point out that K1(x) and I1(x) are linearly independent modifiedBessel functions, thus they determine a unique solution of Bessel differ
ential equation. If we denote by K(ϵ)1 (x) the polynomial approximation
in (13) and (14), we get from (8)(11) the following call option price:
c(ϵ) = S0M1(S0, T,X, rα = 0.5,β)− e−rTXM2(S0, T,X, rα = 0.5,β),
where
M(ϵ)1 =
β12
2S0π
∫ ∞
Xe−(
12G(s)+r) [A(s)]−
12 K(ϵ)
1
⎛
⎝
√TA(s)
2
⎞
⎠ ds,
and
M(ϵ)2 =
β12
2π
∫ ∞
X
1
se−
12G(s) [A(s)]−
12 K(ϵ)
1
⎛
⎝
√TA(s)
2
⎞
⎠ ds.
As before, integralsM(ϵ)1 andM(ϵ)
2 can be approximated by using simpleprocedures in MATLAB. Figure 2 shows the values of c(ϵ) as a functionof β with α = 0.5, S0 = 42.00, X = 41.00, r = 0.11, and T = 0.25.
Bayesian Procedures 37
Figure 2. Values of !c(ϵ) as a function of β.
7 Comparison with other models available inthe literature
In the Mexican case, there is not an exchange for trading options, andthe overthecounter market on options is an incipient market, so datais poor in both quantity and quality. Hence, it is impossible to carryout a reliable empirical analysis to compare market option prices withour theoretical prices. However, we work out an interesting numericalexperiment. In this experiment, we compare our prices with two otherprices from models available in the literature. In figure 3, the case ofthe classical Black and Scholes’ price, as a function of the strike price,is considered as a benchmark with parameter values S0 = 100, T = 0.5,r = 0.05, and σ = 0.2, and is represented by the solid line. The Kornand Wilmott’s [15] price with subjective beliefs on future behavior ofstock prices is represented by the dashed line. The parameter valuesin the Korn and Wilmott’s [15] model are µ = 0.1, α = 0.33, β =3.33, and γ = 0.1. Finally, the doted line shows our price !c(ϵ) withprior information on levels and rates. We examined, in this experiment,
38 Francisco VenegasMartınez
about 800 different combinations of the parameter values α and β, withparameter values β = 17 and α = 0.5. Notice that option prices withprior information on levels are higher than option prices with only priorinformation on future prices. As expected, Black and Scholes prices aresmaller than option prices with any prior information.
Figure 3. Option values as a function of the strike price.
8 Summary and conclusions
Prior information is a subjective issue, that is, different individuals havedifferent initial beliefs. It is difficult to accept that all individuals participating in a specific market can describe their initial knowledge withthe same functional form for the prior distribution, and it is still moredifficult to recognize as being true that all of such distributions havethe same parameters. The existence of a prior distribution is useful todescribe initial beliefs in much more complex markets than those in anaive BlackScholes. In a richer stochastic environment, we have developed a Bayesian procedure to value a European call option when thereis prior information on uncertain or changing volatility. In conclusion,the existence of a prior distribution is useful to describe initial beliefs
Bayesian Procedures 39
in much more complex markets than those in a naive BlackScholes.Needless to say, Monte Carlo methods should be developed and appliedin our proposed framework, and that will be our next goal.
Francisco VenegasMartınezC. de Investigacion en Finanzas,Tecnologico de Monterrey,14380 Mexico D.F., MEXICO,[email protected]
References
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[2] Avellaneda, M.; Friedman, C.; Holmes, R.; Samperi, D., Calibrating Volatility Surfaces RelativeEntropy Minimization, Appl.Math. Finance, 4 (1996), 3764.
[3] Ball, C.; Roma, A., Stochastic Volatility Option Prices, J. Financial and Quantitative Analysis, 24 (1994), 589607.
[4] Berger, J. O., Statistical Decision Theory and Bayesian Analysis,Second edition, SpringerVerlag, New York, 1985.
[5] Bernardo, J. M., Reference Posterior Distributions for BayesianInference, J. Roy. Statist. Soc., B41 (1979), 113147.
[6] Chiang, A. C., Elements of Dynamic Optimization, McGrawHillInc, 1992.
[7] Fouque, J.; Papanicolaou, G.; Sircar, K. R., Derivatives in Financial Markets with Stochastic Volatility, Cambridge UniversityPress, 2000.
[8] Good, I. J., What is the Use of a Distribution? In MultivariateAnalysis, (Krishnaia, ed.), Vol. II, 183203, Academic Press, NewYork, 1969.
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40 Francisco VenegasMartınez
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Bayesian Procedures 41
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Morfismos, Vol. 6, No. 2, 2002, pp. 43–65
Existence of Nash equilibria in discountednonzerosum stochastic games with additive
structure ∗
zednanreH–zednanreHotrebireH
Abstract
This work considers twoperson nonzerosum dynamic stochasticgames when the state and action sets are Borel spaces, with possibly unbounded (immediate) cost functions, and discounted costcriteria. The aim is to prove, under suitable assumptions, theexistence of a Nash equilibrium in stationary strategies. One ofthose assumptions is that the transition law and the cost functionshave an additive (or separable) structure.
2000 Mathematics Subject Classification: 91A15, 91A10.Keywords and phrases: Nonzerosum stochastic games, additive structure, discounted cost, Nash equilibria.
1 Introduction
In this work we study two–person nonzero–sum stochastic games whenthe state and action sets are uncountable Borel spaces. For such aclass of games, with either a discounted or an average cost criterion, theexistence of stationary Nash equilibria is still an open problem. Here wegive a solution to this problem for games with discounted cost criteriain the particular case when the cost functions and the transition lawhave an additive structure.
∗Research partially supported by a CONACyT scholarship. This paper is partof the author’s M. Sc. Thesis presented at the Department of Mathematics ofCINVESTAVIPN.
43
44 Heriberto Hernandez–Hernandez
The existence of Nash equilibria in stationary strategies for discounted stochastic games with additive (also known as “separable”)structure was first studied by Himmelberg et al. [7], Theorem 1, under the assumption that the action spaces are finite sets and the statespace is Borel. They showed that, given a probability distribution pof the initial state, there is a pair of stationary strategies which forma Nash equilibrium p–a.e. (a socalled p–equilibrium). This result wasstrengthened by Parthasarathy [15], Theorem 4, who showed that suchgames have a Nash equilibrium in stationary strategies. An extensionof Parthasarathy’s result in [15] was given in Nowak [12], Theorem 1.1.In the latter work the state set is a measurable space with a countably generated σ–algebra. Nowak also considered compact metric actionspaces and bounded reward functions. Our main result, Theorem 4.4,is also an improvement of Parthasarathy’s Theorem 4 in [15], becausethe state space and the action spaces that we consider are Borel spacesand the cost functions are possibly unbounded. Moreover, our approachto prove Theorem 4.4 is different from Nowak’s in [12]. Here we followthe approach developed in several works, via a fixedpoint theorem formultifunctions, as in Ghosh and Bagchi [5] and Parthasarathy [15], forinstance.
Other works dealing with an additive structure include [3], [8], [10],[16]. Examples may be found in [3] and [5]. A good survey of theexisting literature for both discounted and average criteria is given in[13].
The work is organized as follows. Section 2 presents standard material on stochastic games. The discounted optimality criteria we areinterested in is introduced in section 3. Our assumptions and main result, Theorem 4.4, are stated in section 4. Sections 5 and 6 are devotedto prove our main result. Section 7 presents some concluding remarks.
2 The stochastic game model
In this section we introduce the game model we are interested in. Westart with the following remark on terminology and notation —for further details see Bertsekas and Shreve [1], chapter 7, for instance.
Remark 2.1 (a) A Borel subset X of a complete and separable metrizable space is called a Borel space, and its Borel σ–algebra is denotedby B(X). We only deal with Borel spaces, and so “measurable” (for either sets or functions) always means “Borel measurable”. Given a Borel
Existence of Nash equilibria 45
space X, we denote by P(X) the family of all probability measures onX, endowed with the weak topology. In this case, P(X) is a Borel space.Moreover, if X is compact, then so is P(X).
(b) Let X and Y be Borel spaces. A measurable function φ : Y →P(X) is called a stochastic kernel on X given Y , and we denote byP(XY ) the family of all those stochastic kernels. If φ is in P(XY ),then we write its values either as φ(y)(C) or as φ(Cy), for all y ∈ Yand C ∈ B(X).
The game model. We shall consider the twoperson nonzerosumstochastic game model
(X,A,B,KA,KB, Q, c1, c2)
whereX is the state space, and A and B are the action spaces for players1 and 2, respectively. These spaces are all assumed to be Borel spaces.The sets KA ∈ B(X × A) and KB ∈ B(X × B) are the constraint sets.That is, for each state x ∈ X the x–section in KA, namely
A(x) := a ∈ A(x, a) ∈ KA,
represents the set of admissible actions for player 1 in the state x. Similarly, the x–section in KB,
B(x) := b ∈ B(x, b) ∈ KB,
stands for the set of admissible actions for player 2 in the state x. Let
K := (x, a, b)x ∈ X, a ∈ A(x), b ∈ B(x),
which is a Borel subset of X × A × B (see Lemma 1.1 in Nowak [11],for instance). Then Q ∈ P(XK) is the game’s transition law, and,finally, ci : K → R, i = 1, 2, is a measurable function that representsthe onestage cost function for player i.
The game is played as follows. At each stage (or time) t = 0, 1, 2, . . . ,the players 1 and 2 observe the current state x ∈ X of the system, andthen independently choose actions a ∈ A(x) and b ∈ B(x), respectively.As a result of this two things happen: (1) player i (i = 1, 2) incurs acost ci(x, a, b); and (2) the system moves to a new state according to theprobability distribution Q(·x, a, b). Cost accumulates throughout thecourse of the game, and the goal of each player is to minimize his/hercost.
46 Heriberto Hernandez–Hernandez
Strategies. Let H0 := X, and Ht := Kt ×X for t = 1, 2, . . . . For eacht, an element ht = (x0, a0, b0, . . . , xt−1, at−1, bt−1, xt) of Ht represents a“history” of the game up to time t. A strategy for player 1 is then definedas a sequence π1 = π1
t , t = 0, 1, . . . of stochastic kernels π1t ∈ P(AHt)
such thatπ1t (A(xt)ht) = 1 ∀ht ∈ Ht, t = 0, 1, . . . .
The set of all strategies for player 1 is denoted by Π1. Let S1 be the setof stochastic kernels φ ∈ P(AX) with the property
φ(A(x)x) = 1 ∀x ∈ X.
A strategy π1 = (π1t ) is called stationary if there exists φ1 ∈ S1 such
thatπ1t (·ht) = φ1(·xt) ∀ht ∈ Ht, t = 0, 1, . . . .
We shall identify S1 with the family of all stationary strategies forplayer 1. Let F1 be the set of measurable functions f : X → A suchthat f(x) is in A(x) for all x. Identifying f(x) with the Dirac measureδf(x)(·) concentrated at f(x) we see that
F1 ⊂ S1.
A stationary strategy φ ∈ S1 for player 1 is said to be deterministic ifthere exists f ∈ F1 such that
φ(·x) = δf(x)(·) ∀x ∈ X.
The sets of strategies Π2, S2 and F2 for player 2 are defined similarly.Let (Ω,F) be the canonical measurable space that consists of the
sample space Ω = (X×A×B)∞ and its product σ–algebra F . Then foreach pair of strategies (π1,π2) ∈ Π1×Π2 and each “initial state” x ∈ X,there exists, by a theorem of C. IonescuTulcea (see, for example, Bert
sekas and Shreve [1], pp. 140141), a unique probability measure P π1,π2
x
and a stochastic process (xt, at, bt), t = 0, 1, . . . defined on (Ω,F) in acanonical way, where xt, at, and bt represent the state and the actions ofplayers 1 and 2, respectively, at each stage t = 0, 1, . . . . The expectation
operator with respect to P π1,π2
x is denoted by Eπ1,π2
x .
3 The discounted cost
We now introduce the optimality criteria we are concerned with. Let αbe a number in (0, 1), a socalled “discount factor”, to be fixed throughout this work.
Existence of Nash equilibria 47
Definition 3.1 Let (π1,π2) ∈ Π1 ×Π2 and x ∈ X. The total expectedα–discounted cost for player i, i = 1, 2, when the players use the strategies π1,π2, given the initial state x0 = x is defined as
V i(π1,π2, x) := Eπ1,π2
x
! ∞"
t=0
αtci(xt, at, bt)
#.
Definition 3.2 A pair of strategies (π∗1,π∗2) ∈ Π1 × Π2 is called aNash equilibrium for the α–discounted cost criterion if, for each x ∈ X,we have
V 1(π∗1,π∗2, x) ≤ V 1(π1,π∗2, x) ∀π1 ∈ Π1,
V 2(π∗1,π∗2, x) ≤ V 2(π∗1,π2, x) ∀π2 ∈ Π2.
We shall establish, under certain assumptions, the existence of aNash equilibrium in S1 × S2.
Before proceeding we give some notation. For a given measurablefunctionf : K → R and probability measures φ ∈ P(A(x)) and ψ ∈ P(B(x)), let
f(x,φ,ψ) :=
$
A(x)
$
B(x)f(x, a, b)ψ(db)φ(da),
whenever the integrals are well defined. In particular,
ci(x,φ,ψ) :=
$
A(x)
$
B(x)ci(x, a, b)ψ(db)φ(da),
Q(·x,φ,ψ) :=$
A(x)
$
B(x)Q(·x, a, b)ψ(db)φ(da).
4 Existence of Nash equilibria
In this section we state our main result, Theorem 4.4, which requiresthe following assumptions.
Assumption 4.1 (a) For each x ∈ X, the sets A(x) and B(x) arecompact.
(b) There exists a measurable function w : X → R, with w(·) ≥ 1,such that
$
Xw(y)Q(dyx, ·, b) and
$
Xw(y)Q(dyx, a, ·)
48 Heriberto Hernandez–Hernandez
are continuous on the sets A(x) and B(x), respectively.(c) There exists a constant 1 ≤ β < 1/α such that
!
Xw(y)Q(dyx, a, b) ≤ βw(x) ∀(x, a, b) ∈ K.
The following assumption concerns the game’s additive (or “separable”) structure.
Assumption 4.2 (a) For i = 1, 2, there exist measurable functionsci1 : KA → R and ci2 : KB → R such that
ci(x, a, b) = ci1(x, a) + ci2(x, b) ∀(x, a, b) ∈ K.
Moreover, the functions ci1(x, ·) and ci2(x, ·) are continuous on A(x)and B(x), respectively, and
maxa∈A(x)
ci1(x, a) ≤ w(x), maxb∈B(x)
ci2(x, b) ≤ w(x) ∀x ∈ X,
where w(·) is the function in Assumption 4.1.(b) There exist substochastic kernels Q1 ∈ P(XKA), and Q2 ∈
P(XKB), such that
Q(·x, a, b) = Q1(·x, a) +Q2(·x, b) ∀(x, a, b) ∈ K.
Further, Q1(Cx, ·) and Q2(Cx, ·) are continuous on A(x) and B(x),respectively, for each C ∈ B(X).
Now we give our last assumption, in which we use again the functionw(·) in Assumption 4.1.
Assumption 4.3 There exists a probability measure µ ∈ P(X) and adensity function z on K×X such that for each (x, a, b) ∈ K,
Q(Cx, a, b) =!
Cz(x, a, b, y)µ(dy) ∀C ∈ B(X).
Moreover, for each x ∈ X,
limn→∞
!
Xz(x, an, bn, y)− z(x, a0, b0, y)w(y)µ(dy) = 0
whenever an → a0 in A(x) and bn → b0 in B(x). It is also assumed thatw ∈ L1(µ).
Existence of Nash equilibria 49
We now state our main result.
Theorem 4.4 Under the Assumptions 4.1, 4.2 and 4.3, there exists aNash equilibrium in S1 × S2.
Theorem 4.4 is proved in Section 6, after some technical preliminaries in Section 5.
5 Technical preliminaries
The w–norm. We denote by M(X) the family of real–valued measurable functions on X, and by B(X) the subfamily of bounded functionsin M(X).
If u ∈ M(X) we define its w–norm as
∥u∥w := supx∈X
u(x)w(x)
,
and u is w–bounded if ∥u∥w < ∞. Let Bw(X) be the Banach space ofall w–bounded functions in M(X). Thus Bw(X) contains B(X).
Lemma 5.1 Let x ∈ X, π1 ∈ Π1, and π2 ∈ Π2 be arbitrary. Then
V i(π1,π2, x) ≤ w(x)2
1− αβi = 1, 2.
Proof: Assumption 4.1(c) implies
Eπ1,π2
x [w(xt)] ≤ βtw(x) ∀t = 0, 1, . . . .
Also, by Assumption 4.2(a) we have, for i = 1, 2,
ci(xt, at, bt) ≤ 2w(xt) ∀t = 0, 1, 2, . . . .
HenceEπ1,π2
x ci(xt, at, bt) ≤ 2Eπ1,π2
x [w(xt)] ≤ 2βtw(x),
and it follows that
V i(π1,π2, x) ≤∞!
t=0
αtEπ1,π2
x ci(xt, at, bt)
≤∞!
t=0
αt2βtw(x)
= w(x)2
1− αβ. !
50 Heriberto Hernandez–Hernandez
Dynamic programming. We next develop the dynamic programmingresults needed to prove Theorem 4.4.
If one of the players, say player 2, fixes a strategy φ2 ∈ S2, then wehave a Markov control process for player 1, with control model
(1) (X,A,KA, Qφ2 , c1φ2),
whereQφ2(·x, a) = Q(·x, a,φ2(x)),
andc1φ2
(x, a) = c1(x, a,φ2(x)).
Let V 1φ2(·) be the α–discounted value function associated to (1). Note
that every (deterministic) stationary strategy for player 1 is a (deterministic) stationary policy for (1) and vice versa. Also, the correspondingα–discounted costs coincide. This fact, together with Lemma 5.1 andTheorem 8.3.6(b) in HernandezLerma and Lasserre [6], p. 47, implies
(2) V 1φ2w ≤ 2/(1− αβ) ∀φ2 ∈ S2.
We say that π∗1 ∈ Π1 is an optimal response to φ2 if
V 1(π∗1,φ2, x) = infπ1∈Π1
V 1(π1,φ2, x) ∀x ∈ X.
Similar considerations apply for each fixed φ1 ∈ S1.
Lemma 5.2 Let v : KA → R be a measurable function such that v(x, ·)is continuous on A(x) for each x ∈ X. Then there exists f ∈ F1 suchthat, for each x ∈ X,
(3) v∗(x) := mina∈A(x)
v(x, a) = v(x, f(x)),
and v∗ is a measurable function. Moreover, for each x ∈ X,
(4) v∗(x) = minλ∈P(A(x))
v(x,λ).
Proof: The existence of f ∈ F1 that satisfies (3) follows, for instance,from Lemma 8.3.8(a) in Hernandez–Lerma and Lasserre [6], p. 50. Onthe other hand, identifying each a ∈ A(x) with the Dirac measure δa(·) ∈P(A(x)) we get
(5) v∗(x) = mina∈A(x)
v(x, a) ≥ minλ∈P(A(x))
v(x,λ).
Existence of Nash equilibria 51
On the other hand, for each λ ∈ P(A(x)) we have
v(x,λ) =
!
A(x)v(x, a)λ(da) ≥ v∗(x) ∀x ∈ X.
Hence
(6) minλ∈P(A(x))
v(x,λ) ≥ v∗(x).
From (6) and (5) we obtain (4). !
Proposition 5.3 Let φ2 ∈ S2 be fixed. Then:(a) There exists f1 ∈ F1 such that f1 is an optimal response of player
1 to φ2.(b) The function V 1
φ2is the unique solution in Bw(X) to the dynamic
programming equation
(7) V 1φ2(x) = min
λ∈P(A(x))
"c1(x,λ,φ2(x))+α
!
XV 1φ2(y)Q(dyx,λ,φ2(x))
#.
(c) If φ∗1 ∈ S1 is an optimal response of player 1 to φ2 then, for each
x ∈ X,
(8) V 1φ2(x) = c1(x,φ
∗1(x),φ2(x)) + α
!
XV 1φ2(y)Q(dyx,φ∗
1(x),φ2(x)).
Similar results hold for each fixed φ1 ∈ S1.
Proof: (a),(b). If f1 ∈ F1 is an optimal policy for (1) then f1 is anoptimal response to φ2. This follows as in Theorem 3.1 in Maitra andParthasarathy [9], p. 295. Hence, to prove (a) it suffices to show theexistence of f1 ∈ F1 optimal for (1). To this end, let
v(x, a) := c1(x, a,φ2(x)) + α
!
XV 1φ2(y)Q(dyx, a,φ2(x)).
The function c1(x, ·,φ2(x)) is continuous on A(x) by Assumption 4.2(a).Also the function
$X V 1
φ2(y)Q(dyx, ·,φ2(x)) is continuous on A(x) by (2)
and Lemma 8.3.7 in Hernandez–Lerma and Lasserre [6], p. 48. Hence,by Lemma 5.2 there exists f1 ∈ F1 such that for each x ∈ X,
v∗(x) = mina∈A(x)
v(x, a) = v(x, f(x)).
52 Heriberto Hernandez–Hernandez
Therefore, by Theorem 8.3.6 in Hernandez–Lerma and Lasserre [6], p.47, the function f1 is an optimal policy for (1). This proves (a). Thesame theorem gives that, for each x ∈ X,
(9) V 1φ2(x) = v(x, f(x)) = v∗(x).
From (9) and (4) we obtain (7). This proves (b).(c) As in Theorem 3.1 in Maitra and Parthasarathy [9], p. 295, it
follows that
(10) V 1(φ∗1,φ2, x) = V 1
φ2(x) ∀x ∈ X.
On the other hand, as φ∗1 is a stationary strategy, it follows as in Remark
8.3.10 in HernandezLerma and Lasserre [6], p. 54, that V 1(φ∗1,φ2, ·) is
the unique solution in Bw(X) of the equation(11)
u(x) = c1(x,φ∗1(x),φ2(x)) + α
!
Xu(y)Q(dyx,φ∗
1(x),φ2(x)) ∀x ∈ X.
This fact and (10) give (8). !The proof of Proposition 5.3 also shows that, for each φ2 ∈ S2,
V 1φ2(x) = inf
φ1∈S1
V 1(φ1,φ2, x)
= infπ1∈Π1
V 1(π1,φ2, x).(12)
6 Proof of Theorem 4.4
The proof of Theorem 4.4 is based on a standard procedure; see Ghoshand Bagchi [5] or Parthasarathy [15], for example. This procedure consists of two steps:
(i) Topologize S1 and S2 so that they become metrizable and compactspaces.(ii) Show that the multifunction τ : S1 × S2 → 2S1×S2 defined by
τ(φ1,φ2) := (φ∗1,φ
∗2)φ∗
1 is an optimal response to φ2,
and φ∗2 is an optimal response to φ1
has a fixed point. That is, there exists (φ∗1,φ
∗2) ∈ S1 × S2 such that
(φ∗1,φ
∗2) ∈ τ(φ∗
1,φ∗2). In the latter case, we clearly have that (φ∗
1,φ∗2) is a
Nash equilibrium.
Existence of Nash equilibria 53
We next consider step (i). We follow Parthasarathy [15] to topologize S1 and S2 with the topology of relaxed controls introduced byWarga [17], chapter 4. Let B1 be the set of all measurable functionsh : KA → R such that h(x, ·) is continuous on A(x) for each x ∈ X,and maxa∈A(x) h(x, a) is a µ–integrable function on X. Here µ is theprobability measure in Assumption 4.3. Then B1 becomes a Banachspace if we define the norm of h ∈ B1 as
∥h∥ :=
!
Xmaxa∈A(x)
h(x, a)µ(dx).
We identify two stationary strategies φ,ψ ∈ S1 if φ(x) = ψ(x) µ–a.e.Further, we will identify each φ ∈ S1 with the bounded linear functionalΛφ : B1 → R given by
Λφ(h) :=
!
X
!
Ah(x, a)φ(dax)µ(dx)
=
!
Xh(x,φ(x))µ(dx) ∀h ∈ B1.
In this way we can view S1 as a subset of B∗1. Equip S1 with the weak∗
topology of B∗1. So a sequence (φn) in S1 converges to φ ∈ S1 if and
only if
Λφn(h) → Λφ(h) ∀h ∈ B1,
or, more explicitly, for each h ∈ B1
!
X
!
Ah(x, a)φn(dax)µ(dx) →
!
X
!
Ah(x, a)φ(dax)µ(dx)
which is the same as
(1)
!
Xh(x,φn(x))µ(dx) →
!
Xh(x,φ(x))µ(dx) ∀h ∈ B1.
Thus, it follows from Theorem IV.3.11 in Warga [17], p. 287, that S1
is metrizable and compact. Similarly, we define B2 to obtain that S2 ismetrizable and compact. In the remainder of this work, convergence inS1 or in S2 is understood with respect to the topology just described.
Before proceeding with step (ii), we establish a useful lemma.
54 Heriberto Hernandez–Hernandez
Lemma 6.1 Suppose that (φ1n,φ2n) → (φ1,φ2) in S1 × S2. Let h :K → R be such that
h(x, a, b) = h1(x, a) + h2(x, b) ∀(x, a, b) ∈ K,
for functions h1 ∈ B1, h2 ∈ B2 such that
maxa∈A(x)
h1(x, a), maxb∈B(x)
h2(x, b) ∈ Bw(X).
Then, as n → ∞,
h(x,φ1n(x),φ2n(x)) → h(x,φ1(x),φ2(x)) µ–a.e.
Proof: Choose an arbitrary f ∈ L1(µ). It is clear that
f(·)h1(·, ·)w(·) ∈ B1 and f(·)h2(·, ·)
w(·) ∈ B2.
Therefore, by (1),
!
Xf(x)
h1(x,φ1n(x))
w(x)µ(dx) →
!
Xf(x)
h1(x,φ1(x))
w(x)µ(dx)
and, similarly,!
Xf(x)
h2(x,φ2n(x))
w(x)µ(dx) →
!
Xf(x)
h2(x,φ2(x))
w(x)µ(dx).
If we add these two expressions we obtain!
Xf(x)
h(x,φ1n(x),φ2n(x))
w(x)µ(dx) →
!
Xf(x)
h(x,φ1(x),φ2(x))
w(x)µ(dx).
Therefore, by a Riesz’s representation theorem (see, for example, Folland [4], p. 182), the desired conclusion follows. !
To carry out step (ii) we next show that the multifunction τ isuppersemicontinuous and then we will apply Fan’s fixedpoint theorem(see Theorem 1 in Fan [2]). To prove that τ is uppersemicontinuous,suppose that
(2) (φ1n,φ2n) → (φ1,φ2) in S1 × S2,
and that
Existence of Nash equilibria 55
(3) (φ∗1n,φ
∗2n) → (φ∗
1,φ∗2) in S1 × S2
where, for each n ∈ N,
(4) (φ∗1n,φ
∗2n) ∈ τ(φ1n,φ2n);
then we have to show that
(5) (φ∗1,φ
∗2) ∈ τ(φ1,φ2).
To this end, we only prove that φ∗1 is an optimal response to φ2. The
proof that φ∗2 is an optimal response to φ1 is similar. As the proof is a
little bit long, we will split it into several results, the most importantbeing Proposition 6.7 and Proposition 6.10.
Lemma 6.2 The following holds µ–a.e. as n → ∞:
c1(x,φ∗1n(x),φ2n(x)) → c1(x,φ
∗1(x),φ2(x)).
Proof: This result is a consequence of Assumption 4.2 and Lemma 6.1.!
For notational ease, let
un(·) := V 1φ2n
(·) and un(·) :=un(·)w(·) ,
and define m := 2/(1− αβ). By (2) we have
un(·) ≤ m ∀n ∈ N.
Lemma 6.3 There exists a subsequence (unk) of (un) and a functionu0 ∈ Bw(X) such that (unk) converges µ–a.e. to u0. Therefore, withoutloss of generality we may assume that (un) converges µ–a.e. to u0.
Proof: Identify two functions in M(X) if they are equal µ–a.e., anddefine
U := u ∈ M(X) : u(·) ≤ m µ–a.e. ⊂ L∞(µ).
By a Riesz’s representation theorem (see, for example, Folland [4], p.182), we can view U as a subset of [L1(µ)]∗. More precisely, we mayidentify U with the set
U = u ∈ [L1(µ)]∗ : u ≤ m,
56 Heriberto Hernandez–Hernandez
where u is the norm of u in [L1(µ)]∗. Hence, by Alaoglu’s Theorem(see, for example, Folland [4], p. 162), U is a metrizable and compactsubset of [L1(µ)]∗ in the corresponding weak* topology. Then, as (un) isa sequence in U, there exists a subsequence (unk) of (un) and a functionu0 ∈ U such that (unk) converges to u0 in the weak* sense of [L1(µ)]∗.This implies that (unk) converges to u0 in L∞(µ). Then, letting u0(·) :=u0(·)w(·) we have u0 ∈ Bw(X) and (unk) converges µ–a.e. to u0. !
Lemma 6.4 For each x ∈ X it holds that, as n → ∞,
(6) maxλ∈P(A(x)),γ∈P(B(x))
!!!!!
"
X[un(y)− u0(y)]Q(dyx,λ, γ)
!!!!! → 0.
Proof: [Nowak [14], pp. 413414]. Clearly, (6) will follow if we provethat for each x ∈ X
(7) fn(x) := maxa∈A(x),b∈B(x)
!!!!!
"
X[un(y)− u0(y)]Q(dyx, a, b)
!!!!! → 0.
Before doing this, note that, for n ∈ N, we can write “max” in (6)because!!#
X [un(y)− u0(y)]Q(dyx, ·, ·)!! is continuous on the compact set
P(A(x))× P(B(x)) (see Hernandez–Lerma and Lasserre [6], p. 48, forthe continuity, and Remark 2.1(a) for the compactness of P(A(x)) ×P(B(x))). Now pick an arbitrary x ∈ X, and let fn(x) be as in (7).For each n ∈ N, let (an, bn) be a point in A(x) × B(x) at which themaximum in (7) is attained; such a point exists by Assumption 4.1(a).By the latter assumption we may assume without loss of generality thatan → a0 ∈ A(x) and bn → b0 ∈ B(x).
We obviously have
fn(x) =
!!!!!
"
X[un(y)− u0(y)]Q(dyx, an, bn)
!!!!!
=
!!!!!
"
X[un(y)− u0(y)]Q(dyx, a0, b0) +
"
Xun(y)Q(dyx, an, bn)
−"
Xun(y)Q(dyx, a0, b0)−
"
Xu0(y)Q(dyx, an, bn)
+
"
Xu0(y)Q(dyx, a0, b0)
!!!!!
≤ gn(x) + hn(x) + kn(x),
Existence of Nash equilibria 57
where
gn(x) :=
!!!!!
"
X[un(y)− u0(y)]Q(dyx, a0, b0)
!!!!!,
hn(x) :=
!!!!!
"
Xun(y)Q(dyx, an, bn)−
"
Xun(y)Q(dyx, a0, b0)
!!!!!,
kn(x) :=
!!!!!
"
Xu0(y)Q(dyx, an, bn)−
"
Xu0(y)Q(dyx, a0, b0)
!!!!!.
Assumptions 4.3 and 4.1(c) imply that"
Xw(y)z(x, a0, b0, y)µ(dy) =
"
Xw(y)Q(dyx, a0, b0) ≤ βw(x).
Hence, w(·)z(x, a0, b0, ·) is in L1(µ). Therefore, by Assumption 4.3 andbecause we may assume that (un) converges to u0 in the weak* sense of[L1(µ)]∗ (see the proof of Lemma 6.3), we have
gn(x) =
!!!!!
"
X[un(y)− u0(y)]w(y)Q(dyx, a0, b0)
!!!!!
=
!!!!!
"
X[un(y)− u0(y)]w(y)z(x, a0, b0, y)µ(dy)
!!!!! → 0
as n → ∞. Next, we have
hn(x) =
!!!!!
"
Xun(y)w(y)z(x, an, bn, y)µ(dy)
−"
Xun(y)w(y)z(x, a0, b0, y)µ(dy)
!!!!!
≤ m
"
Xz(x, an, bn, y)− z(x, a0, b0, y)w(y)µ(dy),
and so hn(x) → 0 by Assumption 4.3. Since u0(·) ≤ m, we conclude inthe same manner that kn(x) → 0. Thus we have proved that fn(x) → 0as n → ∞, i.e. (7) holds. !
Lemma 6.5 For each fixed u ∈ Bw(X) the following holds µ–a.e.:"
Xu(y)Q(dyx,φ∗
1n(x),φ2n(x)) →"
Xu(y)Q(dyx,φ∗
1(x),φ2(x)).
58 Heriberto Hernandez–Hernandez
Proof: Let u ∈ Bw(X). We first show that!X u(y)Q1(dy·, ·) ∈ B1.
This function is measurable on KA (see, for example, Bertsekas andShreve [1], p. 144), and continuous on A(x) for each x (see Hernandez–Lerma and Lasserre [6], p. 48). Also, for x ∈ X and a ∈ A(x),""""#
Xu(y)Q1(dyx, a)
"""" ≤#
Xu(y)Q1(dyx, a)
≤ uw#
Xw(y)Q1(dyx, a)
≤ uw$ #
Xw(y)Q1(dyx, a)
+
#
Xw(y)Q2(dyx, b)
%
= uw#
Xw(y)Q(dyx, a, b) ∀b ∈ B(x)
≤ uwβw(x) by Assumption 4.1(c),
and so
maxa∈A(x)
"""""
#
Xu(y)Q1(dyx, a)
""""" ≤ βuww(x).
Therefore, as w(·) ∈ L1(µ), we get#
Xu(y)Q1(dy·, ·) ∈ B1 and max
a∈A(x)
&#
Xu(y)Q1(dyx, a)
'∈ Bw(X).
Similarly#
Xu(y)Q2(dy·, ·) ∈ B2 and max
b∈B(x)
&#
Xu(y)Q2(dyx, b)
'∈ Bw(X).
Thus, by Lemma 6.1, the proof is complete. !
Lemma 6.6 For µ–almost all x ∈ X we have(8)
limn→∞
#
Xun(y)Q(dyx,φ∗
1n(x),φ2n(x)) =
#
Xu0(y)Q(dyx,φ∗
1(x),φ2(x)).
Proof: For any x ∈ X we have"""""
#
Xun(y)Q(dyx,φ∗
1n(x),φ2n(x))−#
Xu0(y)Q(dyx,φ∗
1(x),φ2(x))
"""""
≤ fn(x) + gn(x) + hn(x),
Existence of Nash equilibria 59
where
fn(x) :=
!!!!!
"
X[un(y)− u0(y)]Q(dyx,φ∗
1n(x),φ2n(x))
!!!!!,
gn(x) :=
!!!!!
"
X[un(y)− u0(y)]Q(dyx,φ∗
1(x),φ2(x))
!!!!!,
hn(x) :=
!!!!!
"
Xu0(y)Q(dyx,φ∗
1n(x),φ2n(x))
−"
Xun(y)Q(dyx,φ∗
1(x),φ2(x))
!!!!!.
By Lemma 6.4, both fn(x) and gn(x) tend to zero as n → ∞. It remainsto show that
(9) hn(x) → 0 µ–a.e. as n → ∞.
Lemma 6.5 implies that the following holds µ–a.e.:
(10)"
Xu0(y)Q(dyx,φ∗
1n(x),φ2n(x)) →"
Xu0(y)Q(dyx,φ∗
1(x),φ2(x)).
Also, the fact that un(·) = un(·)w(·) ≤ mw(·), together with Assumption 4.1(c), Lemma 6.3 and Lebesgue’s Dominated ConvergenceTheorem imply that, for each x ∈ X,
(11)
"
Xun(y)Q(dyx,φ∗
1(x),φ2(x)) →"
Xu0(y)Q(dyx,φ∗
1(x),φ2(x)).
Thus, (10) and (11) imply (9). !
The preceding lemmas are summarized in the following proposition.
Proposition 6.7 The following equality holds µ–a.e.:
(12) u0(x) = c1(x,φ∗1(x),φ2(x)) + α
"
Xu0(y)Q(dyx,φ∗
1(x),φ2(x)).
Proof: By Proposition 5.3(c) and (4) we have, for each n ∈ N,(13)
un(x) = c1(x,φ∗1n(x),φ2n(x)) + α
"
Xun(y)Q(dyx,φ∗
1n(x),φ2n(x)).
60 Heriberto Hernandez–Hernandez
Then, letting n → ∞ in (13), by Lemmas 6.2, 6.3 and 6.6 we obtain(12) µ–a.e. !
Similar arguments yield the following result.
Lemma 6.8 There exists a set C ∈ B(X) such that µ(C) = 1, and foreach x ∈ C and λ ∈ P(A(x)) we have
c1(x,λ,φ2n(x)) → c1(x,λ,φ2(x)),!
Xun(y)Q(dyx,λ,φ2n(x)) →
!
Xu0(y)Q(dyx,λ,φ2(x)).
We also need the following simple fact.
Lemma 6.9 Let fn, f : X → R be given functions, with f continuous,where X is a compact metrizable space. If fn → f pointwise, then
lim supn→∞
[infxfn(x)] ≤ min
xf(x).
Proof: Let x∗ ∈ X be such that
f(x∗) = minx
f(x).
Thus fn(x∗) → f(x∗) implies
lim supn→∞
[infxfn(x)] ≤ lim
n→∞fn(x
∗) = minx
f(x). !
Proposition 6.10 The following equality is satisfied µ–a.e.:
(14) u0(x) = minλ∈P(A(x))
"c1(x,λ,φ2(x)) + α
!
Xu0(y)Q(dyx,λ,φ2(x))
#.
Proof: Choose an arbitrary x ∈ C, where C is the set in Lemma 6.8.Define for λ ∈ P(A(x)) and n ∈ N
fn(λ) := c1(x,λ,φ2n(x)) + α
!
Xun(y)Q(dyx,λ,φ2n(x)).
The function fn is continuous on P(A(x)) because so are the functionsc1(x, ·,φ2n(x)) and
$X un(y)Q(dyx, ·,φ2n(x)), by Assumption 4.2 and
Existence of Nash equilibria 61
Lemma 8.3.7 in HernandezLerma and Lasserre [6], p. 48, respectively.By the same argument we have that the function
f(λ) := c1(x,λ,φ2(x)) + α
!
Xu0(y)Q(dyx,λ,φ2(x)),
is continuous on P(A(x)). We also have, by Lemma 6.8, that fn → fpointwise. Proposition 5.3(b) imply that
un(x) = minλ∈P(A(x))
fn(λ).
Therefore, because P(A(x)) is compact (see Remark 2.1(a)), we deducefrom Lemmas 6.3 and 6.9 that
u0(x) = limn→∞
un(x)
= limn→∞
"min
λ∈P(A(x))fn(λ)
#
≤ minλ∈P(A(x))
f(λ)
= minλ∈P(A(x))
"c1(x,λ,φ2(x)) + α
!
Xu0(y)Q(dyx,λ,φ2(x))
#.
Thus, since x ∈ C was arbitrary, the following holds µ–a.e.:
u0(x) ≤ minλ∈P(A(x))
"c1(x,λ,φ2(x)) + α
!
Xu0(y)Q(dyx,λ,φ2(x))
#.
The reverse inequality follows from Proposition 6.7. !
To conclude the proof of Theorem 4.4 we need to eliminate thequalifier “µ–a.e.” in Proposition 6.7 and Proposition 6.10.
Let D ∈ B(X) be such that µ(D) = 1 and such that (12) and (14)are satisfied in D. Define v0 : X → R such that v0(x) := u0(x) forx ∈ D, whereas for x ∈ Dc (the complement of D)
v0(x) := minλ∈P(A(x))
$c1(x,λ,φ2(x)) + α
!
Xu0(y)Q(dyx,λ,φ2(x))
%.
Clearly v0 ∈ Bw(X). Since µ(Dc) = 0, by Assumption 4.3 we have
Q(Dcx, a, b) =!
Dcz(x, a, b, y)µ(dy) = 0 ∀(x, a, b) ∈ K.
62 Heriberto Hernandez–Hernandez
Therefore, for each (x, a, b) ∈ K,
(15)
!
Xv0(y)Q(dyx, a, b) =
!
Xu0(y)Q(dyx, a, b).
Thus, from (14) and (15) we have, for all x ∈ X,
(16) v0(x) = minλ∈P(A(x))
"c1(x,λ,φ2(x)) + α
!
Xv0(y)Q(dyx,λ,φ2(x))
#.
From Proposition 5.3(b) and (16) we obtain
(17) v0(x) = V 1φ2(x) ∀x ∈ X.
Then, it follows from Proposition 5.3(a),(c) that there exists f1 ∈ F1
such that(18)
v0(x) = c1(x, f1(x),φ2(x)) + α
!
Xv0(y)Q(dyx, f1(x),φ2(x)) ∀x ∈ X.
Define
ψ∗1(x) :=
$φ∗1(x) if x ∈ D,f1(x) if x ∈ Dc.
Clearly ψ∗1(x) = φ∗1(x) µ–a.e. and ψ∗
1 ∈ S1. By Proposition 6.7, (15)and (18) we have(19)
v0(x) = c1(x,ψ∗1(x),φ2(x))+α
!
Xv0(y)Q(dyx,ψ∗
1(x),φ2(x)) ∀x ∈ X.
As we are identifying two elements in S1 if they are equal µ–a.e. thenall of the results in this section that involve φ∗1 hold if we replace φ∗1 byψ∗1. Hence, by (19) we may assume without loss of generality that
(20)
v0(x) = c1(x,φ∗1(x),φ2(x)) + α
!
Xv0(y)Q(dyx,φ∗1(x),φ2(x)) ∀x ∈ X.
Therefore (as in (11)), from (20) we get
(21) v0(x) = V 1(φ∗1,φ2, x) ∀x ∈ X.
From (17) and (21) we obtain
(22) V 1φ2(x) = V 1(φ∗1,φ2, x) ∀x ∈ X.
Existence of Nash equilibria 63
Thus, by (12), φ∗1 is an optimal response to φ2. In a similar way it can
be seen that φ∗2 is an optimal response to φ1. This establishes (5), and
so the multifunction τ is uppersemicontinuous. Finally, by Fan’s fixedpoint theorem (see Theorem 1 in Fan [2]), we conclude that there existsa Nash equilibrium in stationary strategies. This completes the proofof Theorem 4.4. !
7 Concluding remarks
In this work we have imposed an additive structure on the cost functionsand the transition law to establish the existence of a Nash equilibrium instationary strategies. An interesting and challenging open problem is toestablish a similar existence result without such an additive structure.
We have dealt with a twoperson game for notational convenience.The result can easily be extended to an N–person game, for any finiteN > 2.
AcknowledgementThe author thanks to Dr. Onesimo HernandezLerma for his com
ments.
Heriberto HernandezHernandezDepartamento de Matematicas,CINVESTAVIPN,A. Postal 14–740,07000 Mexico D.F., [email protected]
References
[1] Bertsekas, D. P.; Shreve, S. E., Stochastic Optimal Control: TheDiscrete Time Case, Academic Press, New York, 1978.
[2] Fan, K., Fixed point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci., U.S.A., 38 (1952),121126.
[3] Filar, J.; Vrieze, K., Competitive Markov Decision Processes,Springer–Verlag, New York, 1997.
64 Heriberto Hernandez–Hernandez
[4] Folland, G. B., Real Analysis: Modern Techniques and Their Applications, Wiley, New York, 1984.
[5] Ghosh, M. K.; Bagchi, A., Stochastic games with average payoffcriterion, Appl. Math. Optim., 38 (1998), 283301.
[6] HernandezLerma, O.; Lasserre, J. B., Further Topics on Discrete–Time Markov Control Processes, Springer–Verlag, New York,1999.
[7] Himmelberg, C.; Parthasarathy, T.; Raghavan, T. E. S.; VanVleck, F., Existence of p–equilibrium and optimal stationarystrategies in stochastic games, Proc. Amer. Math. Soc., 60 (1976),245–251.
[8] Kuenle, H.–U., Equilibrium strategies in stochastic games with additive cost and transition structure and Borel state and actionspaces, Internat. Game Theory Review, 1 (1999), 131147.
[9] Maitra, A.; Parthasarathy, T., On stochastic games, J. Optim.Theory Appl. 5 (1970), 289300.
[10] Mohan, S. R.; Neogy, S. K.; Parthasarathy, T.; Sinha, S., Verticallinear complementarity and discounted zero–sum stochastic gameswith ARAT structure, Math. Program., Ser. A, 86 (1999), 637648.
[11] Nowak, A. S., Measurable selection theorems for minimax stochastic optimization problems, SIAM J. Control Optim., 23 (1985),466476.
[12] Nowak, A. S., Nonrandomized strategy equilibria in noncooperativestochastic games with additive transition and reward structure, J.Optim. Theory Appl., 52 (1987), 429441.
[13] Nowak A. S.; Szajowski, K., Nonzero–sum stochastic games, Ann.Internat. Soc. Dynamic Games, 4 (1999), 297342.
[14] Nowak, A. S., Optimal strategies in a class of zerosum ergodicstochastic games, Math. Meth. Oper. Res., 50 (1999), 399419.
[15] Parthasarathy, T., Existence of equilibrium stationary strategiesin discounted stochastic games, Shankhya Series A, 44 (1982),114127.
Existence of Nash equilibria 65
[16] Raghavan, T. E. S.; Tijs, S. H.; Vrieze, O. J., On stochastic gameswith additive reward and transition structure, J. Optim. TheoryAppl., 47 (1985), 451464.
[17] Warga, J., Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.
Morfismos, Vol. 6, No. 2, 2002, pp. 67–87
Existence of Nash equilibria in some Markovgames with discounted payoff ∗
Carlos Gabriel Pacheco Gonzalez
Abstract
This work considers N−person stochastic game models with adiscounted payoff criterion, under two different structures. First,we consider games with finite state and action spaces, and infinitehorizon. Second, we consider games with Borel state space, compact action sets, and finite horizon. For each of these games, wegive conditions that ensure the existence of a Nash equilibrium,which is a stationary strategy in the former case, and a Markovianstrategy in the latter.
2000 Mathematics Subject Classification: 91A15, 91A25, 91A50Keywords and phrases: Stochastic game, Nash equilibrium.
1 Introduction
In this paper we study the existence of Nash equilibria for stochasticgames with a discounted payoff criterion. First we consider a game withfinite state and action spaces, and infinite horizon. In a more generalframework, we study games with a Borel state space, compact actionsets, and finite horizon. The purpose of this work is to present in a clear,selfcontained manner the proofs of these results. Our main source wasthe paper by Dutta and Sundaram [7].
The first studies of games in the economics literature were the papersby Cournot [6], Bertrand [3], and Edgeworth [8] on oligopoly pricing and
∗Research partially supported by a CONACyT scholarship. This paper is partof the author’s M. Sc. Thesis presented at the Department of Mathematics ofCINVESTAVIPN.
67
68 Carlos Gabriel Pacheco Gonzalez
production. The idea of a general theory of games was introduced byJohn von Neumann and Oskar Morgenstern in their famous 1944 bookTheory of Games and Economic Behavior [25], which proposed thatmost economic questions should be analyzed as games. Nash [15] introduced what came to be known as ”Nash equilibrium” as a way of extending gametheoretic analyses to nonzerosum games. Stochastic gameswith discounted payoffs have been widely studied. This class includesthe twoperson zerosum stochastic games, for which Nash equilibria areknown to exist under a variety of assumptions; see, for instance, Filarand Vrieze [10], Nowak [16] or RamırezReyes [19].
In this work we study nonzerosum games under two different setsof hypotheses. The first result (for games with finite state spaces) wasproved by Rogers [21] and Sobel [24], with an extension to countablestate spaces by Parthasarathy [17]. The second result was proved byRieder [20] (as an approximation to what he calls an ε−equilibrium) under some special assumptions on the structure of the game. The generalresult (that is, games with Borel state and action spaces, and infinitehorizon) is an open problem, even with compact action sets. However, ithas been solved imposing an additive structure in the reward functionsand the transition law; see HernandezHernandez [12] or Parthasarathyand Sinha [17], for instance.
The remainder of this work is organized as follows. Section 2 presentsstandard material on stochastic games, including the discounted optimality criteria and the definition of a Nash equilibrium. Sections 3 and4 are devoted to proving the two main results, Theorems 3.1 and 4.1respectively, that is, the existence of Nash equilibria for the games mentioned in the first paragraph. An appendix is included with some usefulfacts needed in the proofs of Theorems 3.1 and 4.1.
2 The stochastic game model
In this section we introduce the N −person stochastic game model. Westart with the following remark on terminology and notation (for furtherdetails see Bertsekas and Shreve [4], chapter 7).
Remark 2.1 a) A Borel subset X of a complete and separable metricspace is called a Borel space, and its Borel σ−algebra is denoted byB(X). A Borel subset of a Borel space is itself a Borel space.
Nash Equilibria 69
b) Let X and Y be Borel spaces. A stochastic kernel on X given Y ,is a function P (·  ·) such that
b.1) P (·  y) is a probability measure on X for each fixed y ∈ Y , andb.2) P (D  ·) is a measurable function on Y for each fixed D ∈ B(X).c) The set of all stochastic kernels on X given Y is denoted by
P(X  Y ). Moreover, P(X) denotes the set of probability measures onX.
Definition 2.2 A stochastic game model is described by
(1) GM := N , S, (Ai,Φi, ri)i∈N , Q, T ,
where:(1) N = 1, ..., N is the finite set of players.(2) S is the state space, a Borel space.(3) Each player i ∈ N is characterized by three objects (Ai,Φi, ri),
where:(a) Ai, a Borel space, is the action space of player i. Let A =
A1 × ...×AN and denote by a a generic element of A.(b) Φi, a multifunction from S to Ai, defines for each s ∈ S
the set Φi(s) of feasible actions for player i at state s. Let Φ(s) =Φ1(s)× ...× ΦN (s) and K = (s,a) : s ∈ S, a ∈ Φ(s) .
(c) ri, a bounded measurable function from K to R, specifies (foreach state s and action a ∈ Φ(s) taken by the players at s) a rewardri(s,a) for player i.
(4) Q, a stochastic kernel in P(S  K), specifies the game transitionlaw.
(5) T ∈ 0, 1, 2, ...∪ ∞ is the horizon of the game.
If T = 1, the game is static, and it is denoted by
N , S, (Ai,Φi, ri)i∈N .
The game is played as follows. At each time t = 0, 1, ..., each playerobserves the current state s ∈ S of the system, and, independently ofthe other players, chooses an action ai ∈ Φi(s). Then each player i ∈ Nobtains a reward ri(s,a), and the system moves to a new state according
70 Carlos Gabriel Pacheco Gonzalez
to the probability distribution Q(·  s,a). The objective of each playeris to win as much as possible.
Histories. A thistory of the game is a complete description ofthe evolution of the game up to the beginning of period t. Thus, athistory specifies the state sr that occurred in each previous periodr ∈ 0, 1, ..., t− 1 , the actions ar = (a1,r, ..., aN,r) taken by the playersin those periods (ai,r denotes de action taken by player i at periodr), and the state st in the period t. Let Ht be the set of all possiblet−histories, with ht denoting a typical element of Ht, i.e.
(2) ht = (s0,a0, s1,a1, ..., st−1,at−1, st) with ar ∈ Φ(sr).
Note that H0 = S and Ht = K ×Ht−1 for t = 1, 2, ....
Strategies. A strategy πi for player i is a vector πitT−1t=0 (or
sequence if T = ∞) of stochastic kernels πit ∈ P(Ai  Ht), where for eacht and each t−history ht up to t, πit specifies the action πit(ht) ∈ P(Ai)such that
πit(Φi(st)  ht) = 1 ∀ht ∈ Ht, t = 0, 1, ....
A strategy is also called a mixed or randomized strategy, which meansthat the player chooses an action in a random manner. The set ofmixed strategies includes the pure strategies, when the player choosesthe actions in a deterministic way.
Let Πi denote the set of all strategies for player i, and let Π: =Π1 × ... × ΠN . A generic element of Π is denoted by π, and it is saidto be a multistrategy. A strategy πi = πitT−1
t=0 for player i is calledMarkov if πit ∈ P(Ai  S) for each t = 0, 1, ..., T − 1, meaning thateach πit depends only on the current state st of the system. The setof all Markov strategies of player i will be denoted ΠiM . A Markovstrategy πi = πitT−1
t=0 is said to be stationary if πit = πi0 for eacht = 0, 1, ..., T − 1, where πi0 ∈ P(Ai  S). We denote by ΠiS the set ofall stationary strategies of player i. We have
ΠiS ⊂ ΠiM ⊂ Πi.
In a similar manner
ΠS ⊂ ΠM ⊂ Π,
Nash Equilibria 71
where ΠS := Π1S × ... × ΠNS is the set of stationary multistrategies,and ΠM := Π1M × ...×ΠNM is the set of Markov multistrategies.
Optimality criteria. Let δ be a fixed number in (0, 1), and definethe δ−discounted expected payoff function for player i as
(3) Ji,δ(s,π) := Eπs
! ∞"
t=0
δtri(st,at)
#
for each multistrategy π and each initial state s. It represents the expected present value of the rewards of player i under the multistrategyπ. The number δ is called a ”discount factor”.
Definition 2.3 For n = 1, 2, ..., we define the T stage expected discounted payoff function for player i as
Ji,δ,T (s,π) := Eπs
!T−1"
t=0
δtri(st,at)
#,
where 0 < δ < 1 is a discount factor. If T = ∞, we write Ji,δ,T (s,π) asJi,δ(s,π); see (3).
Now we are in position to define a Nash equilibrium. As usual inthe literature, the vector (πi,π−i) will signify the multistrategy π withits strategy πi replaced by πi.
Definition 2.4 A multistrategy π is a Nash Equilibrium of the T stage game GM if
Ji,δ,T (s,π) ≥ Ji,δ,T (s, (πi,π−i)) for all s ∈ S,πi ∈ Πi, i ∈ N .
Before proceeding we give some notation. First note that$A means$
A1...
$AN
and that%
A means%
A1...
%AN
. Let ν : K → R be ameasurable function, π0 ∈ P(Φ1(s))× ...×P(ΦN (s)) and πi0 ∈ P(Φi(s))for some s ∈ S and some i ∈ N , then
ν(s,π0) :=
&
Aν(s,a)π0(da)
(4) =
&
Aν(s,a)π1,0(da1)...πi0(dai)...πN,0(daN )
72 Carlos Gabriel Pacheco Gonzalez
and
(5) ν(s, (πi0,π−i0)) :=
!
Aν(s,a)π1,0(da1)...πi0(dai)...πN,0(daN )
In particular
ri(s,π0) :=
!
Ari(s,a)π0(da)
=
!
Ari(s,a)π1,0(da1)...πi0(dai)...πN,0(daN ),
ri(s, (πi0,π−i0)) :=
!
Ari(s,a)π1,0(da1)...πi0(dai)...πN,0(daN ),
Q(·  s,π0) :=!
AQ(·  s,a)π0(da)
=
!
AQ(·  s,a)π1,0(da1)...πi0(dai)...πN,0(daN )
and
Q(·  s, (πi0,π−i0)) :=
!
AQ(·  s,a)π1,0(da1)...πi0(dai)...πN,0(daN ).
Remark 2.5 If A1, ..., AN (and hence A) are finite or countable sets,then the integrals are replaced with summations.
The next two sections are devoted to proving the existence of a Nashequilibrium in games with
1) finite state and action spaces, and infinite horizon (Theorem 3.1);and
2) games with Borel state space, compact action sets, and finitehorizon (Theorem 4.1).
The proofs are based on a standard procedure; see, for instance,Dutta and Sundaram [7]. The procedure is to introduce a multifunctionwhich is a K−mapping (see Definition 5.5) on a nonempty compact convex set. Then we use Kakutani´s or Glicksberg´s fixedpoint theoremto ensure the existence of a fixed point (Definition 5.6), which yields aNash equilibrium.
Nash Equilibria 73
3 The equilibrium existence in finite spaces
Theorem 3.1 Suppose S and Ai are finite spaces for each i, and T =∞. Then the stochastic game model GM has a Nash equilibrium instationary strategies.
Proof: As was already mentioned, the idea is to prove the existence ofa fixed point of a certain multifunction; this fixed point is an equilibrium.To prove the existence of a fixed point of the multifunction, we useKakutani´s theorem (Theorem 5.7).
The proof is organized as follows:In Step 0 we define the multifunction. In Step 1 we prove that such
a multifunction is defined on a nonempty compact convex subset of Rn.In Step 2 we prove that the multifunction maps points to a nonemptyconvex set, and, finally, in Step 3, we prove that the multifunction isu.s.c. (see Definition 5.5). The last two steps prove that the multifunction is a K−mapping (Definition 5.5). Since Steps 1, 2 and 3 verify thehypotheses of Kakutani´s theorem, a fixed point exists.
Step 0: Definition of the multifunction BR.Let π := (π1, ...,πn) ∈ ΠS , and letBRi(π) be the set of best responses
of player i to π, that is
BRi(π) :=
!πi ∈ ΠiS : Ji,δ(s, (πi,π−i)) = sup
α∈ΠiS
Ji,δ(s, (α,π−i)) ∀s ∈ S
".
Let BR = BR1 × ....×BRN . Note that BR : ΠS ! ΠS .Step1: The set ΠS is a nonempty compact convex subset of a
normed space.Because Ai is a finite set, the set P(Ai) is simply the positive unit
simplex of dimension Ai − 1 (· means the cardinality). Note thatP(Ai) ⊂ RAi−1. Since S is also a finite set, the S−fold P(Ai  S) =P(Ai)S, the Cartesian product of this simplex, is a compact convexsubset of a finitedimensional Euclidean space. Now every stationarystrategy for player i can be associated in the obvious way with a uniquesequence of elements of
I Φ(s1)−1 × ...× IΦ(sS)−1,
where I := [0, 1] . Hence ΠiS is a nonempty compact convex subset ofRn for some n, and so is ΠS = Π1S × ...×ΠNS .
Step 2: BR(π) is a nonempty convex subset of ΠS for each π ∈ ΠS .
74 Carlos Gabriel Pacheco Gonzalez
Given π ∈ ΠS , the bestresponse problem faced by player i (that is,finding BRi(·)) is a discounted Markov decision problem
MDP :=!S,Ai,Φi, r
′i, Q
′, T = ∞"
where S is the state space and Ai is the action space. Furthermore, foreach s ∈ S, Φi(s) is the set of feasible actions in state s. The rewardfunction is
(6) r′i(s, ai) := ri(s, (ai,π−i))
Similarly, the transition law is
(7) Q′(s′  s, ai) := Q(s′  s, (ai,π−i))
By Remark 2.5, (6) and (7) represent finite sums. It is well known thatthere exists a nonempty set BRi (π) of optimal stationary strategies inresponse to π; see Filar and Vrieze [10], Theorem 2.3.1, for example.
Denote the value function of the MDP as J∗i (s). To prove that
BRi(π) is convex, let α,β ∈ BRi(π) and 0 ≤ λ ≤ 1; then we want toprove that µ := λα+ (1− λ)β is in BRi(π).
Since α,β ∈ BRi(π), we have that the Bellman equations
(8) J∗i (s) = r′i(s,α) + δ
#
SJ∗i (s
′)Q(ds′  s,α)
and
(9) J∗i (s) = r′i(s,β) + δ
#
SJ∗i (s
′)Q(ds′  s,β)
hold. Therefore, by (8) and (9),
r′i(s, µ) + δ
#
SJ∗i (s
′)Q(ds′  s, µ),
= λ
$r′i(s,α) + δ
#
SJ∗i (s
′)Q(ds′  s,α)%
+(1− λ)
$r′i(s,β) + δ
#
SJ∗i (s
′)Q(ds′  s,β)%
= λJ∗i (s) + (1− λ)J∗
i (s) = J∗i (s).
That is
r′i(s, µ) + δ
#
SJ∗i (s
′)Q(ds′  s, µ) = J∗i (s),
Nash Equilibria 75
and so µ is in BRi(π). This shows that BR (π) is a nonempty convexset for each π ∈ ΠS .
Step 3: BR is an u.s.c. multifunction on ΠS , and BR (π) is compactfor each π ∈ ΠS .
This will be established if we show that it holds for each BRi. Sofix i. Suppose that
πn := (π1n, ...,πNn) → π := (π1, ...,πN ),
and αin ∈ BRi(πn) is such that αin → αi ∈ ΠiS . We are going to showthat αi ∈ BRi(π).
Let J∗i,n(s) denote the value function of the player i in a best response
to πn. Since δ < 1, the sequence J∗i,n(s) is uniformly bounded by
M = (1− δ)−1max ri(s,a) : (s,a) ∈ S ×A .
So, because J∗i,n(s) ∈ [−M,M ] for each s ∈ S and for each n, J∗
i,nconverges pointwise (perhaps through a subsequence) to a limit J∗
i .On the other hand, we know that J∗
i,n(s) satisfies the Bellman equation
(10) J∗i,n(s) = ri(s, (αin,π−i)) + δ
!
SJ∗i,n(s
′)Q (ds  s, (αin,π−i)) ,
for each n and s. Moreover, for any β ∈ P(Φi(s)), we have
(11) J∗i,n(s) ≥ ri(s, (β,π−i)) + δ
!
SJ∗i,n(s
′)Q (ds  s, (β,π−i)) .
Since the integrals are finite sums (recall our Remark 2.5), whenn → ∞ we have
ri(s, (αin,π−i)) → ri(s, (αi,π−i)),!
SJ∗i,n(s
′)Q (ds  s, (αin,π−i)) →!
SJ∗i (s
′)Q (ds  s, (αi,π−i)) ,
and, similarly,!
SJ∗i,n(s
′)Q (ds  s, (β,π−i)) →!
SJ∗i (s
′)Q (ds  s, (β,π−i)) .
Hence, letting n → ∞ in (10) and (11) we get
J∗i (s) = ri(s, (αi,π−i, )) + δ
!
SJ∗i (s
′)Q (ds  s, (αi,π−i))
76 Carlos Gabriel Pacheco Gonzalez
and
J∗i (s) ≥ ri(s, (β,π−i)) + δ
!
SJ∗i (s
′)Q (ds  s, (β,π−i)) ,
respectively. These expressions establish precisely that J∗i (s) is the value
function in a best response of i to π, and that αi is a stationary bestresponse, that is, αi ∈ BRi (π) . Thus, BRi is an u.s.c. multifunctionfor each i, and so is BR.
Also note that the u.s.c. shows that BRi (π) is a closed set in ΠiS .Thus, since ΠiS is compact, BRi (π) is also compact.
Summarizing, we have that for each π, BR (π) is a nonempty compact convex subset, and thatBR(·) is u.s.c.; henceBR(·) is aK−mapping.An appeal to Kakutani´s fixedpoint theorem (Theorem 5.7) yields theexistence of π∗ ∈ ΠS such that π∗ ∈ BR (π∗) , completing the proof ofTheorem 3.1. !
4 The equilibrium existence in Borel spaces
Consider the stochastic game model GM in (1) with the followingassumptions.
Assumption 0 S and Ai are Borel spaces, and Ai is compact foreach i.
Assumption 1 For all i, Φi : S " Ai is a compact–valued multifunction on S.
Assumption 2 For all i, ri is bounded and jointly measurable in(s,a), and it is continuous in a for each fixed s ∈ S.
Assumption 3 For each Borel subset B of S, Q(B  s,a) is jointlymeasurable in (s,a), and setwise continuous in a for each fixed s; thatis, if an → a then Q(B  s,an) converges to Q(B  s,a).
Theorem 4.1 Suppose the assumptions 0,1,2,3 hold and T is finite.Then the stochastic game model GM has a Nash equilibrium in Markovian strategies (possibly nonstationary).
The result is an easy consequence of the following lemmata. The firstlemma essentially shows that the theorem is true for T = 1; the combination of the lemmata, together with a selection theorem establishesthe result for general T < ∞ through an induction argument.
Nash Equilibria 77
Lemma 4.2 For some fixed s ∈ S, consider a N player stochasticgame model in which the action sets are the compact metric spacesΦ1(s) , ..., ΦN (s) and the reward functions are r1(s, ·), ..., rN (s, ·) defined on Φ(s) = Φ1(s) × ... × ΦN (s). If ri(s, ·) is continuous on Φ(s)for each i ∈ N , the stochastic game model N , s , (Φi(s), ri)i∈N admits a Nash equilibrium. That is, there exist π∗ := (π∗
1, ...,π∗N ) ∈
P(Φ1(s))× ...× P(ΦN (s)) such that, for each i,
ri(s,π∗) ≥ ri(s, (πi,π
∗−i)) ∀πi ∈ P(Φi(s)).
In the following proof we consider
Π(s) := P(Φ1(s))× ...× P(ΦN (s)).
Proof: The idea is to prove the existence of a fixed point of a certainmultifunction; this fixed point is an equilibrium. To prove the existenceof such a fixed point, we use Glicksberg´s theorem (Theorem 5.8).
The proof is organized as follows: In Step 0 we define a multifunctionBR. In Step 1 we prove that BR is defined on a nonempty compactconvex subset of a locally convex Hausdorff space. In Step 2 and Step3 we prove that BR is a K−mapping (Definition 5.5). By the Steps 1,2 and 3 and Glicksberg’s theorem, a fixed point exists.
Step 0: We define the multifunction BR as in the Step 0 of theproof of the Theorem 3.1.
For each π := (π1, ...,πN ) ∈ Π(s) , let BRi(π) be the set of bestresponses of player i to π, that is
BRi(π) :=
!πi ∈ P(Φi(s)) : ri(s, (πi,π−i)) = sup
µi∈P(Φi(s))ri(s, (µi,π−i))
".
Let BR := BR1 × ....×BRN . Note that BR : Π(s) ! Π(s).Step1: Π(s) is a nonempty compact convex space.Convexity is obvious. Morever, by Theorem 5.4, P(Φi(s)) equipped
with the topology of weak convergence is a compact metric space; henceso is Π(s).
Step 2: BR(π) is a nonempty convex set for each π ∈ Π(s).Given π := (π1, ...,πN ) ∈ Π(s), the bestresponse problem faced by
player i is a discounted Markov decision problem MDP := s ,Φi, r′iwhere s is the state space, Φi(s) is the action space, and
(12) r′i(s, ai) := ri(s, (ai,π−i)),
78 Carlos Gabriel Pacheco Gonzalez
which is continuous in ai. It is well known that the set BRi(π) isnonempty (see Puterman [18], Theorem 6.2.10).
Let r∗i (s) be the value function of theMDP and let µ1, µ2 ∈ BRi(π),0 ≤ λ ≤ 1 and µ = λµ1+(1−λ)µ2 (i.e. µ(B) = λµ1(B)+ (1−λ)µ2(B)for each B ∈ B(Φi(s))). Then
(13) r′i(s, µ) = λr
′i(s, µ1) + (1− λ)r
′i(s, µ2).
Since µ1, µ2 ∈ BRi(π), the expression (13) is the same as
λr∗i (s) + (1− λ)r∗i (s) = r∗i (s),
i.e.r′i(s, µ) = r∗i (s).
Therefore, µ ∈ BRi(π), and so BRi(π) is convex.Step 3: BR is u.s.c. and BR(π) is compact for each π ∈ Π(s).Let πn := (π1n, ...,πNn) ∈ BR(π) such that
(14) πn → π := (π1, ...,πN )
in the weak topology, and αin ∈ BRi(πn) with
(15) αin → αi ∈ P(Φi(s)).
We are going to show that αi ∈ BRi(π). With this in mind, note thatαin ∈ BRi(πn) gives
(16) ri(s, (αin,π−in)) ≥ ri(s, (β,π−in)) ∀β ∈ P(Φi(s)) and n.
Since ri(s,a) is continuous in a, by Corollary 5.3, as n → ∞ (14) and(15) give
ri(s, (αin,π−in)) → ri(s, (αi,π−i))
andri(s, (β,π−in)) → ri(s, (β,π−i)) ∀β ∈ P(Φi(s)).
Then, as n → ∞, the inequality (16) yields
ri(s, (αi,π−i)) ≥ ri(s, (β,π−i)) ∀β ∈ P(Φi(s));
hence αi ∈ BRi(π), which shows that BRi is u.s.c.The u.s.c. proves that BRi (π) is a closed set in P(Φi(s)). Thus,
since P(Φi(s)) is compact, BRi (π) is compact.
Nash Equilibria 79
We have that for each π, BR (π) is a compact convex nonempty setand BR(·) is u.s.c., so BR(·) is a K−mapping. Finally, Glicksberg´sfixedpoint theorem implies the existence of π∗ ∈ Π(s) such that π∗ ∈BR (π∗) , completing the proof of Lemma 4.2, because π∗ is a Nashequilibrium. !
Lemma 4.3 For i = 1, ..., N, let vi : S → RN be a bounded measurablefunction. For each s ∈ S, i ∈ N , a ∈ Φ1(s)× ...× ΦN (s), define
Hi(s,a) := ri(s,a) + δ
!
Svi(s
′)Q(ds′  s,a).
Then the stochastic game model N , S, (Ai,Φi(s), Hi(s, ·))i∈N admits aNash equilibrium. That is, there exists π∗ := (π∗
1, ...,π∗N ) ∈ P(Φ1(s))×
...× P(ΦN (s)) such that, for each i ∈ N ,
Hi(s,π∗) ≥ Hi(s, (πi,π
∗−i)) ∀πi ∈ P(Φi(s)), s ∈ S.
In the following proof we consider
Π := P(A1)× ...× P(AN )
andΠ(s) := P(Φ1(s))× ...× P(ΦN (s)).
Proof: The idea is, using Lemma 4.2, to prove the existence of Nashequilibria for each s ∈ S (Step 0), and then use a selection theorem toget a Nash equilibrium (Step 1).
Step 0: It is easy to show that thet continuity condition in Assumption 3 is equivalent to the following: If an → a, then
!
Sf(s′)Q(ds′  s,an) →
!
Sf(s′)Q(ds′  s,a)
for each bounded measurable function f . Then
Hi(s,an) → Hi(s,a)
if an → a. This gives that Hi(s,a) is continuous in a. By Lemma 4.2,for each s ∈ S the stochastic game model
N , s , (Φi(s), Hi(s, ·))i∈N
has a Nash equilibrium.
80 Carlos Gabriel Pacheco Gonzalez
Since it is possible to find equilibria for each s ∈ S , consider themultifunction Θ : S ! Π which assigns the set of equilibria points toeach s ∈ S, i.e.
(17) Θ(s) := π∗ ∈ Π(s) : Hi(s,π∗) ≥ Hi(s, (πi,π
∗−i))
∀ πi ∈ P(Φi(s)), i = 1, ..., N
or equivalently
(18) Θ(s) := π∗ ∈ Π(s) : Hi(s,π∗) = sup
πi∈P(Φi(s))Hi(s, (πi,π
∗−i)),
i = 1, ..., N
Before proceeding with the next step, we make the following remark:without loss of generality we may assume that
ri(s,a) = θ − 1 if a /∈ Φ(s) for each i ∈ N ,
where θ := infi,s,a ri(s,a). So, if π∗ := (π∗1, ...,π
∗N ) ∈ Θ(s) then
(19) ri(s,π∗) ≥ ri(s, (µi,π
∗−i)) ∀µi ∈ P(Ai),
and, moreover, if θ = min(infi,s,a ri(s,a), infi,s vi(s)), then
(20) Hi(s,π∗) ≥ Hi(s, (µi,π
∗−i)) ∀µi ∈ P(Ai),
even if support(µi)∩Φi(s)c = ∅. So Hi(s,π) is well defined for everyπ ∈ Π.
We use this remark in the next step.Step 1: There is a measurable selector for Θ, i.e. a measurable
function ξ : S → Π, such that ξ(s) ∈ Θ(s) for each s ∈ S.We want to use Theorem 5.11to show the existence of a measurable
selection. To use such theorem, we need to prove that(i) Π satisfies the property S (Remark 5.10),(ii) Θ(s) is compact for each s ∈ S, and(iii) Θ−1(F ) is a Borel set in S for every closed set F in Π.Since Π is separable metric space (by Theorem 5.4), it is easy to see
(i) holds; it suffices to take a countable dense subset of Π and the familyof closed balls with radius a rational number and center an element ofthe countable dense set.
Nash Equilibria 81
In order to prove (ii), let π∗n ⊂ Θ(s) (where π∗
n := (π∗1n, ...,π
∗Nn))
be such that π∗n → π∗. Since π∗
n ∈ Θ(s), for each n we have
(21) Hi(s,π∗n) ≥ Hi(s, (µi,π
∗−in)) ∀µi ∈ P(Ai),
and, therefore, by Corollary 5.3, as n → ∞ we obtain
(22) Hi(s,π∗) ≥ Hi(s, (µi,π
∗−i)) ∀µi ∈ P(Ai).
It follows that π∗ ∈ Θ(s), and so Θ(s) is a closed subset of the compactspace Π(s). Hence Θ is a compactvalued multifunction.
To prove (iii), we use (18).First note thatHi(·, ·) and supπi∈P(Φi(s))Hi(·, (πi, ·)) are jointly mea
surable, then the function
(23) F : S ×Π → RN ,
defined by
F (s,π) :=
!Hi(s,π)− sup
µi∈P(Φi(s))Hi(s, (µi,π−i))
"
i=1,...,N
is jointly measurable.Then, the set
F−1((0, ..., 0)), with (0, ..., 0) ∈ RN
is a Borel set. Finally, note that F−1((0, ..., 0)) = Gr(Θ); hence, byTheorem 5.9, (iii) holds.
Since the assumptions of Theorem 5.11 hold, a measurable selectorfor Θ exists. !
Proof: [Proof of Theorem 4.1] Consider T = 1. By assumption, ri(s, ·)is continuous on Φ(s) for each s, and ri is measurable on S ×A. Thus,by Lemma 4.3 , there exists a strategy π :=(π1, ...,πN ) ∈ Π such thatfor each s ∈ S,
ri(s,π) ≥ ri(s, (β,π−i)) ∀β ∈ P(Φi(s)) and i ∈ N .
Denote π by π1, and let νi(s) = ri(s,π1) (i = 1, ..., N). Then π1 isa Nash equilibrium of the one period game N , S, (Ai,Φi, ri), with νi
82 Carlos Gabriel Pacheco Gonzalez
(i = 1, ..., N) the corresponding equilibrium payoffs. Now, let Hi(s,a)as in Lemma 4.3, then the stochastic game
N , S, (Ai,Φi(s), Hi(s, ·))i∈N
has an equilibrium. Denote π2 the equilibrium of this game, then(π1,π2) is the equilibrium of
N , S, (Ai,Φi, ri)i∈N , Q, 2 .
We now proceed by induction. Suppose the existence of a Nashequilibrium strategy (πT−1, ...,π1) of the stochastic game
N , S, (Ai,Φi, ri)i∈N , Q, T − 1
and let νi(s) be the corresponding discounted equilibrium payoff ofplayer i (i = 1, ..., N) and initial state s. By Lemma 4.3, the game
N , S, (Ai,Φi(s), Hi(s, ·))i∈N
admits an equilibrium πT ∈ Π. It follows that (πT ,πT−1, ...,π1) specifiesan equilibrium of the stochastic game N , S, (Ai,Φi, ri)i∈N , Q, T, andthat the discounted equilibrium payoff is given by
νi(s,πT ) =
!
AHi(s,a)πT (da  s), i = 1, ..., N.
Note that if π = (πT ,πT−1, ...,π1), then
νi(s,πT ) = Ji,δ,T (s,π).
Also note that π ∈ ΠM . This completes the proof. !
5 Appendix
In this section we summarize some facts used in the proof of Theorems3.1 and 4.1.
The topology of weak convergence
Definition 5.1 Let P(X) be the set of all probability measures on(X,B(X)) where X is a general metric space. The topology of weak
Nash Equilibria 83
convergence is the topology in the space P(X) which has the followingbasic neighborhoods of any element µ ∈ P(X),
Uϵ(µ, f1, ..., fk) :=!λ ∈ P(X) :
""""#
Sfidλ−
#
Sfidµ
"""" < ϵ, i = 1, ..., k
$
where ε is positive and f1, ..., fk are elements of C(X) (the space ofcontinuous bounded functions on X).
The following are some useful results.
Lemma 5.2 Let (X, d) be a metric space. Let µn ⊂ P(X) and µ ∈P(X). Then µn → µ in the topology of weak convergence if and only if%fdµn →
%fdµ for all f ∈ C(X).
For a proof of Lemma 10 see, for instance, Proposition 7.21 in Bertsekas and Shreve [10].
Corollary 5.3 Let X1, ..., XN be metric spaces. Let µin be a sequencein P(Xi) and µi ∈ P(Xi) for i = 1, ..., N. Each P(Xi) has the topologyof weak convergence. If
µin → µi for i = 1, ..., N,
then#
fdµ1n...dµNn →#
fdµ1...dµN ∀f ∈ C(X1 × ...×XN ).
Next, we have a result used in sections 3 and 4; for a proof seeProposition 7.22 in Bertsekas and Shreve [4].
Theorem 5.4 If (X, d) is a compact separable metric space, then thetopology of weak convergence in P(X) is compact, separable and metrizable.
Kakutani´s Theorem
If each point x of a space X is mapped into a nonempty set T (x)of a space Y , we call T a setvalued mapping, also known as a multifunction or correspondence. We write T : X ! Y to specify that it is amultifunction.
84 Carlos Gabriel Pacheco Gonzalez
Definition 5.5 Let T be a multifunction from a topological space Xto a topological space Y. We say that T is a K−mapping of X into Y if
i) for each x in X, T (x) ⊂ Y is a compact convex set; andii) the graph of T, which defined as
Gr(T ) = (x, y) : y ∈ T (x) ,
is closed in X × Y.If i) holds, then the condition ii) is equivalent to the upper semicon
tinuity (u.s.c.) condition, i.e. if xn → x in X, yn ∈ T (xn) and yn → y,then y ∈ T (x).
Definition 5.6 A fixed point for a multifunction T : X ! X is a pointx such that x ∈ T (x).
Theorem 5.7 (Kakutani theorem) If T : X ! X is a K−mapping,where X is a nonempty compact convex subset of Rm, then T has a fixedpoint.
See Smart [23], Chap. 9 for further details. Kakutani´s theoremwas extended to Banach spaces by Bohnenblust and Karlin [5], and tolocally convex spaces by Ky Fan [9] and Glicksberg [11].
Theorem 5.8 (Glicksberg theorem) Let T : X ! X be a Kmappingwhere X is a nonempty compact convex subset of a locally convex Hausdorff space. Then T has a fixed point.
The proof is in Corollary 16.51 in Aliprantis and Border [1].Correspondence with measurable graphLet T : X ! Y be a multifunction and A ⊂ Y , then the lower
inverse T−1 is defined by
T−1(A) := x ∈ X : T (x) ∩A = ∅ .
Theorem 5.9 Let X and Y be nonempty Borel spaces. Let T : X ! Ybe a compactvalued multifunction, then the following statements areequivalent:
a) Gr(T ) is a Borel subset of X × Y ;b) T−1(F ) is a Borel subset of X for every closed set F ⊂ Y.
The proof is in Aliprantis and Border [1] Theorem 14.84 (see Proposition D.4 in HernandezLerma and Lasserre [13] for reference).
Nash Equilibria 85
A selection theorem
Remark 5.10 A topological space X is said to satisfy condition S ifthere is a countable family Fn of closed sets which separates the pointsof X, that is, if x and y are any distinct points of X, there is a set Fn
which contains one of them but not both.
Condition S is trivially satisfied when X is a Borel space.
Theorem 5.11 (Selection theorem) Let (S,A) be a measurable space,and X be a topological space which satisfies condition S. If Θ : S ! Xis a multifunction such that Θ(s) is a nonempty compact set of X andΘ−1(F ) ∈ A for every closed set F in X, then there is a measurable selector ξ for Θ (that is, a measurable function from S to X withξ(s) ∈ Θ(s) for each s ∈ S).
See Leese [14] for the proof. For further details see Wagner [26].
Carlos Gabriel Pacheco GonzalezDepartamento de Mathematicas,CINVESTAVIPN,Apartado Postal 14–740,07000 Mexico D. F., [email protected]
References
[1] Aliprantis, C. D.; Border, K. C., Infinite Dimensional Analysis,SpringerVerlag, Heidelberg, 1999.
[2] Amir, R., On stochastic games with uncountable state and actionspaces, in Stochastic Games and Related Topics (Raghavan, T. E.S.; Ferguson, T. S.; Parthasarathy, T.; Vrieze, O., eds.), Kluwer,Boston, 1991.
[3] Bertrand, J., Theorie mathematique de la richesse sociale, Journaldes Savants (1883), 499508.
[4] Bertsekas, D. P.; Shreve, S. E., Stochastic Optimal Control: TheDiscrete Time Case, Academic Press, New York, 1978.
[5] Bohnenblust, H. F.; Karlin, S., On a theorem of Ville, in Contributions to the Theory of Games (Kuhn H. W.; Tuker, A. W., eds.),Vol. I, 15560, Princeton University Press, 1950.
86 Carlos Gabriel Pacheco Gonzalez
[6] Cournot, A., Recherches sur les Principes Mathematiques de laTheorie des Richesses. English edition (Bacon, N., ed.): Researchesinto the Mathematical Principles of the Theory of Wealth, Macmillan, 1897.
[7] Dutta, P. K.; Sundaram, R. K., The equilibrium existence problem in general markovian games, in Organizations with IncompleteInformation: Essays in Economics Analysis (M. Majumdar, ed.),Cambridge University Press, 1998.
[8] Edgeworth, F., La teoria pura del monopolio, Giornale degliEconomisti (1897), 1331.
[9] Fan, K., Fixed point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 1216.
[10] Filar, J.; Vrieze, K., Competitive Markov Decision Processes,SpringerVerlag, New York, 1997.
[11] Glicksberg, I., A further generalization of the Kakutani fixed pointtheorem, with application to Nash equilibrium points, Proc. Amer.Math. Soc., 3 (1952), 170174.
[12] Hernandez Hernandez, H., Existence of Nash equilibria in discounted nonzerosum stochastic games with additive structure, Tesisde Maestria, CINVESTAVIPN, Mexico D.F, 2002.
[13] HernandezLerma, O.; Lasserre, J. B., DiscreteTime Markov Control Processes: Basic Optimality Criteria, SpringerVerlag, NewYork, 1996.
[14] Leese, S. J.,Measurable selections and the uniformization of Souslinsets, Amer. J. Math., 100 (1978), no. 1, 1941.
[15] Nash, J., Equilibrium points in Nperson games, Proc. Nat. Acad.Sci. U.S.A., 36 (1950), 4849.
[16] Nowak, A. S., Optimal strategies in a class of zerosum ergodicstochastic games, Math. Meth. Oper. Res., 50 (1999), 399419.
[17] Parthasarathy, T.; Sinha, S., Existence of equilibrium in discountedstochastic games, Shankhya Series A, 44 (1982), 114127.
[18] Puterman, M. L., Markov Decision Processes: Discrete StochasticDynamic Programming, John Wiley & Sons, New York, 1994.
Nash Equilibria 87
[19] Ramırez Reyes, F., Existence of optimal strategies for zerosumstochastic games with discounted payoff, Morfismos, Vol. 5, No. 1(2001), 6383.
[20] Rieder, U., Equilibrium plans for nonzero sum Markov games,in Seminar on Game Theory and Related Topics (Moeschlin, O.;Pallaschke, D., eds.), Springer, Berlin, 1979.
[21] Rogers, P. D., Nonzero sum stochastic games, ORC Report no. 698, Operations Research Center, Univesity of California, Berkeley,1969.
[22] Royden, H. L., Real Analysis, Macmillan, New York, 1988.
[23] Smart, D. R., Fixed Point Theorems, Cambridge University Press,1974.
[24] Sobel, M. J., Noncooperative stochastic games, Ann. Math. Stat.,42 (1971), 19301935.
[25] Von Neumann J.; Morgenstern, O., Theory of Games and EconomicBehavior, Princeton University Press, 1944.
[26] Wagner, D., Survey of measurable selection theorems, SIAM J. Control Optim., 15 (1977), 8591003.
MORFISMOS, Comunicaciones Estudiantiles del Departamento de Matematicas del CINVESTAV, se termino de imprimir en el mes de enero de 2003 enel taller de reproduccion del mismo departamento localizado en Av. IPN 2508,Col. San Pedro Zacatenco, Mexico, D.F. 07300. El tiraje en papel opalinaimportada de 36 kilogramos de 34 × 25.5 cm consta de 500 ejemplares en pastatintoreto color verde.
Apoyo tecnico: Omar Hernandez Orozco.
Contenido
Approximation on arcs and dendrites going to infinity in Cn (Extended version)
Paul M. Gauthier and E. S. Zeron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Bayesian procedures for pricing contingent claims: Prior information on volatility
Francisco VenegasMartınez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Existence of Nash equilibria in discounted nonzerosum stochastic games withadditive structure
zednanreHzednanreHotrebireH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Existence of Nash equilibria in some Markov games with discounted payoff
Carlos Gabriel Pacheco Gonzalez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67