Morfismos, Vol 5, No 2, 2001

VOLUMEN 5NÚMERO 2

JULIO A DICIEMBRE DE 2001 ISSN: 1870-6525

MORFISMOSComunicaciones EstudiantilesDepartamento de Matematicas

Cinvestav

Editores Responsables

• Isidoro Gitler • Jesus Gonzalez • Isaıas Lopez

Consejo Editorial

• J. Rigoberto Gabriel • Onesimo Hernandez-Lerma• Francisco Hernandez Zamora • Maribel Loaiza Leyva• Raquiel Lopez Martınez • Raul Quiroga Barranco

• Enrique Ramırez de Arellano

Editores Asociados

• Ricardo Berlanga • Samuel Gitler• Emilio Lluis Puebla • 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 57 47 38 71.Toda correspondencia debe ir dirigida a la Sra. Laura Valencia, Departamentode Matematicas del Cinvestav, Apartado Postal 14-740, Mexico, D.F. 07000 opor correo electronico: [email protected].

VOLUMEN 5NÚMERO 2

JULIO A DICIEMBRE DE 2001ISSN: 1870-6525

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Morfismos

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Morfismos

Contenido

Degree and fixed point index. An account

Carlos Prieto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Existence of Nash equilibria in nonzero-sum ergodic stochastic games in Borelspaces

Rafael Benıtez-Medina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Monte Carlo approach to insurance ruin problems using conjugate processes

Luis F. Hoyos-Reyes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Sobre la estrechez de un espacio topologico

Alejandro Ramırez Paramo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Medida de colisin de un (a, d, b)-superproceso con su medida inicial

Jose Villa Morales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Morfismos, Vol. 5, No. 2, 2001, pp. 1-17

Degree and fixed point index.An account ∗

Carlos Prieto

Abstract

In this account, a development of the concepts of Brouwer degreeand Lefschetz-Hopf fixed point index is discussed in the light ofwork done mainly by A. Dold, H. Ulrich and the author. General-izations to certain coincidence situations including the equivariantcases are presented, as well as how to deal with the infinite di-mensional cases. In two appendices a proof of the Lefschetz-Hopftheorem for these indices is referred to, as well as a generalizationof Dold’s fixed point transfer is sketched.

2000 Mathematics Subject Classification: 55M20, 54H25.Keywords and phrases: Generalized fixed point problems, Brouwer de-gree, Lefschetz-Hopf fixed point index.

1 Introduction

1.0 Consider a system of equations

g1(x1, . . . , xk) = a1

......

...(1.1)

gl(x1, . . . , xk) = al

where the unknown are restricted by some conditions. These restrictionscan be more precisely described by saying that the point (x1, . . . , xk)has to belong to a certain subset V of the euclidean space Rk. Thus,we may see the system as an equation of the form

(1.2) g(x) = a ,∗Invited article.

1

2 Carlos Prieto

where g : V −→ Rl, and V ⊂ Rk.In the case k = l, V open and bounded in Rk and g continuous, such

that it can be extended to the boundary of V and has no solution inthis boundary, Brouwer [2] defined in 1911 the concept of degree, whichto such g assigns an integer, deg(g), such that if it is nonzero, then theequation has a solution.

The problem can be modified as follows. We shall consider two cases.1. k ≤ l. In this case, the system (1.1) can be rewritten as

f1(x1, . . . , xk) = g1(x1, . . . , xk)− a1 + x1 = x1...

fk(x1, . . . , xk) = gk(x1, . . . , xk)− ak + xk = xk

fk+1(x1, . . . , xk) = gk+1(x1, . . . , xk)− ak+1 = 0...

fl(x1, . . . , xk) = gl(x1, . . . , xk)− al = 0

or, written in vector form, we have a map

f : V −→ Rl = R

k × Rl−k , V ⊂ R

k ,

such that g(x) = a if and only if f(x) = (x, 0); hence, we look forsolutions x ∈ V for the equation

f(x) = (x, 0) ∈ Rk × R

l−k .

This is a generalized fixed point problem. For the classical problem,k = l, Lefschetz [17] defined in 1926 an invariant, L(f), with integralvalues, called the Lefschetz number, defined for V a polyhedron and fsuch that f(V ) ⊂ V . This number, which is easy to compute, has theproperty that L(f) = 0 implies the existence of a fixed point of f , i.e.a solution for the equation f(x) = x.

On the other hand, Hopf [12] and [13], a couple of years later definedanother integral invariant for the case k = l, V open and bounded andsuch that f can be extended to the boundary of V without fixed points,called the fixed point index, I(f), which fulfills the same theorem asthe Lefschetz number, namely, I(f) = 0 implies that f has fixed points.This index deals with more general situations, but it is also more difficultto compute. Their relationship is given by the so-called Lefschetz-Hopftheorem which states that in the case that both L(f) and I(f) aredefined, then I(f) = L(f).

Degree and fixed point index 3

The other case of our more general set up is the following:2. k ≥ l. In this case, the system (1.1) can be written as

f1(x1, . . . , xk) = g1(x1, . . . , xk)− a1 + x1 = x1...

...fl(x1, . . . , xk) = gl(x1, . . . , xk)− al + xl = xl

or, put in vector form, we have a map

f : V −→ Rl , V ⊂ R

k = Rl × R

k−l ,

such that, if x = (x′, x′′) ∈ V , g(x) = a if and only if f(x) = x′; hence,we look for solutions x = (x′, x′′) ∈ V for the equation

f(x′, x′′) = x′ .

This is another generalized fixed point problem.Both cases 1. and 2. can be put together into the following problem.

Take

f : V −→ Rk × R

m , V ⊂ Rk × R

n(1.3)

and we ask for the existence of generalized fixed points, namely, points(x, x′) ∈ V such that f(x, x′) = (x, 0).

We shall describe in the next sections, for cases with increasing gen-erality, fixed point indices which decide the existence of solutions forthis problem.

The first case we shall consider is when f not only is a map as in(1.3), but a family fb parametrized by the points b in a metric space B,in whose case we substitute the space Rk also by a family of more generalspaces Eb, which include finite polyhedra and smooth manifolds, whichin their time were considered by Lefschetz and Hopf. The problem isnow the following. Let

(1.4) f : V −→ E ×M , V ⊂ E ×N open ,

where E is a euclidean neighborhood retract over B, an ENRB forshort, namely a continuous family given by p : E −→ B, of retractsEb = p−1(b) of open sets in Rk (see 2.0), M and N are euclidean spaces(M = Rm, N = Rn), f preserves parameters (i.e. f(v) ∈ Eb × Mif v ∈ Eb × N) and is properly fixed, namely the solutions Fix(f) =(e, y) ∈ V | f(e, y) = f(e, 0) lie properly over B; in particular, the

4 Carlos Prieto

fixed point set of the restriction fb of f to each fiber over b is compact(see 2.0). In this case there is an invariant I(f), which lives in the (gen-eralized) cohomology –or homology– of B in dimension m− n and has,among others, the property that I(f) = 0 implies Fix(f) = ø.

The case M = N = R0 = 0 = 0 was studied by Dold in [7], wherehe generalized the work of Lefschetz and Hopf as well as previous workof himself, [4] and [6]. In these last, he studied the case B = ∗ (seealso [5]). The general case was studied by the author in [20]. This casewill be discussed in section 2.

Very frequently the problem presents symmetries, that is, all thespaces E,B,M,N admit group actions for a group G, and p and f arecompatible with those actions, i.e. they are equivariant. The solutionof the problem in this case is sharper, and if M = N = 0 it has beengiven basically by Dold in [9] and by Ulrich in [30, 31], although tomDieck has said something about it too [3]. Its generalization for realG-modules of finite dimension M and N was given by Ulrich and theauthor in [26]. This case we shall discuss in section 3.

There are generalizations of the problem in another direction, namely,in the case that E has infinite dimension, of great importance in sev-eral questions in nonlinear analysis. The development of this problemis as follows. Leray and Schauder [18] 1934 defined an index for thecase B = ∗, M = N = 0 and E a separable Banach space, requiringf to be such that the closure of the image of V under f , f(V ) ⊂ Eis compact. Granas [11] generalized this to the case in which E is anabsolute neighborhood retract (an ANR) and Ulrich [29] did it in theparametrized case (B = ∗). The general case (ANRBs and M and Nfinite dimensional G-modules) will be shortly discussed below in 3.4.

2 Fixed point index

2.0 Let B be a metric space. We shall be concerned with the followingcommutative diagrams, called fixed point situations over B

(2.1) E ×N ⊃ Vf

!!

pproj1 ""

E ×M

pproj1##

B ,


where p : E −→ B is an ENRB, i.e. a vertical retract (meaning thatthe retraction commutes with the projections p and projB) of an openset in B ×K, K a euclidean space (K = Rk), M and N are euclideanspaces (M = Rm, N = Rn) too and f is properly fixed over B, i.e. therestriction of the projection into B, p proj1, to the fixed point set,Fix(f) = (e, y) ∈ V | f(e, y) = (e, 0), is proper, in other words, foreach compact set C ⊂ B, the set (p−1(C)×N) ∩ Fix(f) is compact.

We first study the case E = B×K. The properness of Fix(f) −→ B,that is, the continuous compactness of F = Fix(f) implies the validityof a parametrized Heine-Borel theorem; namely, there exists a functionρ : B −→ R+ = (0,+∞), such that F ⊂ Bρ = (b, z, y) ∈ B ×K ×N |∥(z, y)∥ ≤ ρ(b) (the set Bρ can be described as a continuous family ofballs in K ×N = Rk+n of radius varying according to ρ).

Consider the following sequence of maps of pairs

(V, V − F )i−f

!!!"

(1) ""

B × (K ×M,K ×M − 0)

(E ×N,E ×N − Bρ)#!!!

!"

(2) ""

(E ×N,E ×N − F )

B × (K ×N,K ×N − 0)

B × (Rk+n,Rk+n− 0) !!❴❴❴❴❴❴❴❴❴ B × (Rk+m,Rk+m

− 0) ,

(2.2)

where (i − f)(b, z, y) = (b, (z, 0) − f2(b, z, y)), if f(b, z, y) = (b, f2(b, z,y)). The vertical inclusions are, respectively, (1) an excision and (2) ahomotopy equivalence (of the second spaces of the pairs), and all mapsare over B (i.e., they preserve the fibers). Thus they induce a stablemap

(2.3) If : B −→ B

of degree (k+m)−(k+n) = m−n (see [10], [24] or Appendix A (A.3)).Equivalently, (2.2) induces a homomorphism

(2.4) If : h∗(B) −→ h∗+m−n(B) .

for any cohomology theory h∗. More precisely, applying h∗ to (2.2) weget a homomorphism

hi+k+m(B × (Rk+m,Rk+m− 0)) −→ hi+k+m(B × (Rk+n,Rk+n

− 0)), i ∈ Z ,

which, after desuspending k +m times on the left side and k + n timeson the right side, gives

hi(B) −→ hi+m−n(B), i ∈ Z ,

6 Carlos Prieto

and thus (2.4). This homomorphism is called the index homomorphismof f . Important examples of h∗ are ordinary cohomology, K-theory,or stable cohomotopy. All these examples are multiplicative theorieshaving an element 1 ∈ h0(B); hence, for these theories, we may definethe fixed point index of f as

(2.5) I(f) = If (1) ∈ hm−n(B) .

Since the index map factors through the pair (E × N,E × N − F ), itvanishes when F = ø, therefore, it has the fundamental property

(2.6) I(f) = 0 =⇒ Fix(f) = ø .

Before passing to other important properties of the index, let us seesome special cases.

Let B = ∗ and m = n(= 0); the three cohomology theories men-tioned above are such that h0(∗) = Z. In this case, the index I(f) isan integer, which is the same in all cases; this is the classical fixed pointindex, or Lefschetz-Hopf index, [4].

If B = ∗ and n > m = 0, then, taking h∗ as stable cohomotopy,the index I(f) becomes an element of the n-stem, i.e. of the group Πst

n

of stable homotopy classes of maps Sk+n −→ Sk of spheres. In fact, in[6] and [20] it is proved that every element in Πst

n is the index of somef .

The fixed point index has, among others, the following properties.

Homotopy 2.7. Let f : V −→ E × M , V ⊂ E × N be properly fixedover B × I (I = [0, 1]). Then its restrictions f0 : V0 −→ E0 × M andf1 : V1 −→ E1 ×M to bottom B × 0 ≈ B and top B × 1 ≈ B of thecylinder B × I are properly fixed and I(f0) = I(f1) ∈ hm−n(B).

Additivity 2.8. Let f : V −→ E × M , V ⊂ E × N be properly fixedover B. Let V = V1 ∪ V2 with V1 and V2 open. If f1 = f |V1, f2 = f |V2

and f12 = f |V1 ∩ V2 are such that two of them are properly fixed, thenso is also the third and I(f) = I(f1) + I(f2)− I(f12) ∈ hm−n(B).

Excision 2.9. Let f : V −→ E ×M , V ⊂ E ×N be properly fixed overB. If V ′ ⊂ V is open and such that Fix(f) ⊂ V ′, then f ′ = f |V ′ isproperly fixed and I(f ′) = I(f) ∈ hm−n(B).

The next property allows us to define the index for general ENRBs.It is this property which constitutes the main difference between index


and degree and shows the convenience to work with the index ratherthan with the degree, which, in general, can not be defined for arbitraryeuclidean neighborhood retracts.

Commutativity 2.10. Let E = B × L −→ B and E′ = B × L′ −→ Bwith L and L′ euclidean spaces, and let U ⊂ E, U ′ ⊂ E′ ×N be open.If ϕ : U ′ −→ E × M and ψ : U −→ E′ are maps over B such that thecomposite

(ψ × 1M )ϕ : ϕ−1(U ×M) −→ E′ ×M , ϕ−1(U ×M) ⊂ U ′ ⊂ E′ ×N

is properly fixed, then also the composite

(ψ × 1N )−1(U ′)ϕ(ψ×1N )−−−−→ E ×M , (ψ × 1N )−1(U ′) ⊂ U ×N ⊂ E ×N

is properly fixed and I((ψ × 1M )ϕ) = I(ϕ(ψ × 1N )) ∈ hm−n(B).

We show now how the commutativity allows us to generalize theindex:

Proposition and Definition 2.11. If p : E −→ B is an ENRB, Mand N are euclidean spaces and V ⊂ E × N is open, then every mapover B, f : V −→ E ×M admits a decomposition

f : Vα×1N !! U

β!! E ×M ,

where U is open in B×K×N for some euclidean space K, and α : E −→B ×K. If f is properly fixed, then

g = (α× 1M )β : U −→ B ×K ×M

is also properly fixed. Hence, I(g) ∈ hm−n(B) is defined and dependsonly on f and not on the factorization f = β(α × 1N ). Thus we definethe fixed point index of f as I(f) = I(g).

Proof: (sketch) Let

E

i

""W ⊂ B ×K

r

##

be a representation of E as an ENRB and define U = (r × 1N )−1V ⊂W × N ⊂ B × K × N . So let α = i (then α × 1N : V −→ U , since

8 Carlos Prieto

(r×1N )(i×1N ) = id : V −→ V ) and β = f(r×1N ) : U −→ E×M . Thenβ(α × 1N ) = f . The rest of the proof is a straightforward applicationof the commutativity property 2.10.

Properties 2.7 to 2.10 remain true for the general index.

Comment 2.12. There is a Lefschetz-Hopf formula relating the indexwith a trace (Lefschetz number) in the case m = n (= 0); see [8] or [10].For the case m > n (= 0), the formula holds trivially; see [20]. The casem < n has also a formula which follows from a more general one; see[24] and appendix A.

Examples 2.13.

(a) [7, 5.3] Let B = S1 = z ∈ C | ∥z∥ = 1 and consider the mapf : B × S1 −→ B × S1, f(b, z) = (b, b · z). This is a properly fixedmap over B (for the projection B × S1 −→ B and M = N = 0).If one takes stable cohomotopy as the cohomology theory, thenI(f) is the nontrivial element of π0

st(B) = Πst1 = Z/2 which also is

the Lefschetz trace of f∗ : π∗

st(B × S1) −→ π∗

st(B × S1), seen as ahomomorphism of π∗

st(B)-modules.

(b) [20, 4.27] Let S2 = C ∪ ∞ be the Riemann sphere. If k ∈ Z,then the map

S2 ⊃ Cf

!! S2 × C , f(z) = (z, zk)

is properly fixed over B = S2 (E = B, p = id, M = C = R2), andI(f) = k ∈ π2

st(B) = Πst0 = Z.

3 Generalizations of the index

3.0. Very often the situations one studies present some kind of symme-tries; if these are given by the action of a compact Lie group G, thereare cohomology theories which are fine enough to detect the symmetries.More precisely we shall be concerned in first place with the equivariantindex, which will be defined for situations like (2.1), but now assum-ing that G acts on all spaces involved and that every map in questioncommutes with the group action. To be precise, p : E −→ B will be aG-ENRB, i.e., G acts on both E and B, p is G-equivariant and E isa vertical equivariant retract of an open (invariant) set B × K, wherenow K is a G-module. In fact, K, M and N are now all G-modules,


that is, euclidean spaces with a linear action of G. Observe that in thiscase the fixed point set F of f is G-invariant and hence it is possible tochoose ρ : B −→ R+ also G-invariant, i.e. ρ(γb) = ρ(b) for all γ ∈ G,b ∈ B. Therefore, Bρ becomes G-invariant too. One has thus that thesequence of maps (2.2) consists of equivariant maps (over B) and thusproduces an equivariant stable map

(3.1) If : B −→ B

of degree [K⊕M ]− [K ⊕N ] = [M ]− [N ] ∈ RO(G) (A.3) where RO(G)denotes the real representation ring (or ring of G-modules) of the groupG, (see e.g. Appendix A).

As before, this sequence induces, equivalently to (2.1), an equivariantindex homomorphism of f as

(3.2) If : h∗

G(B) −→ h∗+[M ]−[N ]G (B)

for any RO(G)-graded G-equivariant cohomology theory h∗G (see [16] or[21]). More precisely, applying h∗G to the, now equivariant, sequence(2.2), we get, for any ρ ∈ RO(G), a homomorphism

hρ+[K]+[M](B × (K ⊕M,K ⊕M − 0)) −→ hρ+[K]+[M](B × (K ⊕N,K ⊕N − 0)) ,

which, after desuspending, by K ⊕M on the left and by K ⊕N on theright, yields

hρ(B) −→ hρ+[M ]−[N ](B), ρ ∈ RO(G) ,

and thus (3.1).In analogy to the nonequivariant case, important examples for h∗G are

the equivariant ordinary cohomology of Lewis, May and McClure [19],equivariant K-theory ([1] or [27]), equivariant stable cohomotopy ([28],[16], [10]) or its approach via fixed point theory, FIX ([30, 31], [21]). Allthese theories are multiplicative and have an element 1 ∈ h0G(B). Wedefine the equivariant fixed point index of f as the element

(3.3) IG(f) = If (1) ∈ h[M ]−[N ]G (B) .

Once again, this index has the properties 2.5. through 2.10. andcan thus be extended to general G-ENRBs exactly in the same way asbefore, (2.11).

The case B = ∗ and M = N(= 0) is interesting. Equivariantstable cohomotopy π∗

G is such that π0G(∗)

∼= A(G), the Burnside ring of

10 Carlos Prieto

G (of finite G-sets, if G is finite; see [3] for a thorough study of A(G)for G a compact Lie group). Thus the equivariant index in this case isan element of A(G). In [8] it is proved that every element of A(G) isthe equivariant index of some equivariant f . In fact, more generally, in

[30, 31] and [21] it is proved that every element in π[M ]−[N ]G (B) is the

index of some equivariant f (as described in 3.0).In [8] it is proved that for M = N = 0 and B = ∗, the equivariant

index is determined by the “classical” indices I(fH) of the restrictionsfh : V H −→ EH of f to the spaces whose points remain fixed underthe action of the elements of the closed subgroups H ⊂ G. In [30],relationships between IG(f) and

!

I(fH)"

are thoroughly studied.

3.4. The adequate set up to speak about the fixed point index in infinitedimensions is that of (separable) Banach spaces or, more generally, thatof the absolute neighborhood retracts over B, the ANRBs. We shallsketch here in a very short way a generalization of Ulrich’s work [29] inthis direction.

An absolute neighborhood retract over B, an ANRB, p : E −→ B isdefined as a vertical retract of an open set in B × K, where K nowdenotes a separable Banach space. We consider fixed point situations,that is, commutative diagrams

(3.5) E ×N ⊃ Vf

!!

""

E ×M

##

B

where E −→ B is an ANRB , M and N are euclidean spaces (possiblywith a group action, in whose case f has to be equivariant) and fis strongly fixed, which means that besides being properly fixed, theclosure, f(V ), of its image (or at least to the image, f(W ), of someopen neighborhood W of Fix(f)) lies properly over B. In the casethat E = B × K −→ B, it is possible to approximate f by mapsf ′ : V ′ −→ B × P × M , properly fixed over B, where V ′ is open inB × P × N and P is a finite polyhedron contained in K. Since thenP is an ENR, then B × P is an ENRB and the index I(f ′) is defined.If two approximations f ′ and f ′′ are close enough to f then they arehomotopic (in the sense of 2.7; thus I(f ′) = I(f ′′). Therefore, we maydefine the index of f , I(f), as I(f ′) for f ′ close enough to f .


Since, if Fix(f) = ø we may assume V = ø and so V ′ = ø, we have

(3.6) Fix(f) = ø =⇒ I(f) = 0 .

Properties 2.7. through 2.10. remain true and so the possibility ofdefining the index of the situation (3.4) for a general ANRB p : E −→ Bholds.

With due care all this can be carried out equivariantly too.

4 Equivariant degree

4.0. In this section we describe a special case of the index which refersto an important class of equations (cf. [14, 15]). We shall discuss thedegree, which we shall define via the index, and using the properties ofthis last, we shall show its fundamental properties.

Let G be a compact Lie group and M , N (finite dimensional) realG-modules, and let K be a euclidean space. If B is a metric G-spaceand

(4.1) B ×K ×N ⊃ Vg

!! K ×M

is an equivariant map, with V open and invariant in B ×K × N , andis such that the set of solutions g−1(0) of the equation g(b, z, y) = 0 liesproperly over B (e.g. is compact, if B = ∗ or B itself is compact); forinstance, if the closure V lies properly over B and g can be extended toV in such a way that no zeroes appear on the boundary, then we definethe degree of g as

(4.2) deg(g) = I(i− g′) ∈ π[M ]−[N ]G (B) ,

where g′ : V −→ B×K×M , g′(b, z, y) = (b, g(b, z, y)) and (i−g′)(b, z, y) =(b, (z, 0) − g(b, z, y)) ∈ B ×K ×M . This can be defined because i− g′

is properly fixed, since Fix(i− g′) = g−1(0).The degree is an invariant which detects solutions of the equation

(4.3) g(b, z, y) = 0 ,

which can be seen as a family of equations in the sense of [14, 15]parametrized, equivariantly, by the metric G-space B. It has the fol-lowing properties.

12 Carlos Prieto

4.4. deg(g) = 0 =⇒ (4.3) has a solution.

(This follows from the equivariant version of 2.6).

Excision 4.5. g−1(0) ⊂ W ⊂ V , W open in B ×K ×N =⇒ deg(g) =deg(g|W ).


Additivity 4.6. V = V1∪V2, V1, V2 open in B×K×N and g−1(0)∩V1∩V2 proper over B =⇒ deg(g) = deg(g|V1) + deg(g|V2)− deg(g|V1 ∩ V2).


Homotopy Invariance 4.7. If gt is a homotopy between g0 and g1 suchthat for every t, g−1

t (0) lies properly over B, then deg(g0) = deg(g1).

(It follows from the equivariant version of 2.7. In fact, the inverse isalso true, if we allow the domain Vt of gt to vary along with t).

Clearly, it is not necessary to assume in (4.1) that G acts triviallyon K. On the other hand, as described in 3.4, we may more generallyassume that K is a separable Banach space. The situation is as follows.Let

(4.8) B ×K ×N ⊃ Vg

−→K ×M

be equivariant and such that g−1(0) ⊂ V , as well as the closure of

(b, z, y) | (z, y) = (z′, 0)− g(b, z′, y′) ∈ K ×M, (b, z′, y′) ∈ V

in B ×K × N lie properly over B. Then i − g′ : (b, z, y) +→ (b, (z, 0) −(g(b, z, y)), (b, z, y) ∈ V , is strongly fixed and thus its Leray-Schaudertype index, I(i− g′) is defined. Hence we define the degree of g by

(4.9) deg(g) = I(i− g′) ∈ π[M ]−[N ]G (B)

as before. It has the same properties as the finite dimensional index.As in [14], K may have an action of G by isometries. There, the

authors consider the case B = ∗, in which our degree lies in the sta-

ble equivariant stem Π[M ]−[N ]G of stable homotopy classes of equivariant

maps between the G-spheres SM and SN , given by the one-point com-pactifications of the G-modules M and N , respectively.

As a last comment in this section it should be remarked that thedegree in [14, 15] is defined in a nonstable equivariant homotopy group ofG-spheres. After stabilizing, their degree becomes ours. This explains,in particular, that their additivity (property (e) of the degree in [14], p.445) only holds up to one suspension, whereas ours is plain.


A The Lefschetz-Hopf theorem

In this appendix, a short account of the results in [24] is given. There wegive a conceptual proof of a Lefschetz-Hopf trace formula for computingthe index of a globally defined fixed point situation. We prove thefollowing.

Theorem A.1. Let p : E −→ B be a proper G-ENRB such that h∗G(E)is a projective, finitely generated h∗G(B)-module, and let M and N beG-modules. Then, if f : E ×N −→ E ×M is an equivariant map overB such that Fix(f) −→ B is proper and f−1(E × 0) ⊂ E × B for someball B ⊂ N , then

I(f) = trace(f∗ : h∗G(E) −→ h∗G(E)) ∈ h[M ]−[N ]G (B) ,

where f∗ is, up to suspension, the endomorphism of degree [M ] − [N ]induced by (the stable map)

(A.2) f : E × (N,N − B) −→ E × (M,M − 0) .

Proof: It is an application of Proposition 4.4 in [10]. We sketch it.There is s category B-StabG, whose objects are triples (X,X ′; ρ),

where X is a G-space over B, X ′ is an invariant subspace and ρ is anelement of the real representation ring RO(G).

Its morphisms are the stable maps given by

(A.3) B-StabG((X,X ′; ρ), (Y, Y ′; σ)) =

= colimK

!

(X,X ′)× (K ⊕ ρ,K ⊕ ρ− 0), (Y, Y ′)× (K ⊕ σ,K ⊕ σ − 0)"

,

also called stable maps from (X,X ′) to (Y, Y ′) of degree σ−ρ ∈ RO(G),where [·] denotes G-homotopy classes of G-maps over B of pairs, and Kvaries in the category (made small) of unitary (complex) representationsof G, the direction given by K ≤ L ⇐⇒ ∃M such that K ⊕M ∼= L.(By taking K large enough, K ⊕ ρ and K ⊕ σ become G-modules).

This category can be endowed with the structure of a monoidalcategory, and inside it the proper G-ENRBs, E, are strongly dualizable,whose dual is (B × L,B × L− E), if E is a G-neighborhood retract inB × L.

Under the assumptions of A.1, the defining sequence (2.1) of theindex, defines the trace, (2.2), of the morphism (A.3) in the categoryG-StabB. Since the cohomology h∗G defines a functor from this categoryto the category of h∗G(B)-modules, which satisfies the hypothesis of [10,4.4], then it preserves traces, thus sending (2.2) to the trace we seek. ⊓-

For all details of the proof see [24].

14 Carlos Prieto

B The transfer

Given an equivariant fixed point situation as (2.0) there is anotherhomomorphism related to the index homomorphism If : h∗G(B) −→

h∗+[M ]−[N ]G (B), called the transfer homomorphism of f . To define it,

consider, as before, first the case E = B ×K and take the sequence ofequivariant maps of pairs(B.1)

(V, V − F )!"

!!

(id,i−f)"" V ×B (E ×M,E ×M − 0)

(E ×N,E ×N − Bρ)!"

!!

"" (E ×N,E ×N − F )

(E ×N,E ×N − 0)

B × (K ×N,K ×N − 0) ""❴❴❴❴❴❴❴❴❴❴ V × (K ×M,K ×M − 0) .

This, again, induces a stable map as (A.3)

τf : B −→ V

of degree [M ]− [N ] ∈ RO(G), or equivalently, a homomorphism

τVf : h∗G(V ) −→ h∗G(B)

for any RO(G)-graded G-equivariant cohomology theory h∗G, called atransfer homomorphism of f .

Since we may restrict f to any W ⊂ V , as to approach Fix(f), thenall transfers τWG fit together to pass to the limit and yield the minimaltransfer

τf : f∗

G(Fix(f)) −→ h∗+[M ]−[N ]G (B) ,

through which all other transfers factor. This shows, in particular, that

(B.2) Fix(f) = ø =⇒ τWf = 0 for every W .

These transfers have all properties, which generalize the ones in [8] ascan be seen in [26]. Thus they can provide applications of fixed pointtheory to algebraic topology. As an example of an application to thislast, in [25] it is proved that any equivariant stable map α : X −→ Ybetween pointed G-spaces of degree [M ]− [N ] ∈ RO(G), that is, a mapin the category StabG, factors through a transfer of some G-fixed pointsituation f over X and a nonstable map, namely, one has

α : Xτf

"" Vψ

"" Y .


Here V is an open invariant neighborhood of the fixed point set Fix(f)and ψ : V −→ Y is an equivariant (nonstable) map.

Carlos PrietoInstituto de Matematicas, UNAM,04510, Mexico, D.F., MEXICO,[email protected].

References

[1] Atiyah, M. F. and Segal, G. B., Equivariant K-theory,mimeographed notes, Warwick, 1965.

[2] Brouwer, LEJ, Uber Abbildung von Mannigfaltigkeiten, Math.Ann. 71 (1911), 97–115.

[3] tom Dieck, T., Transformation groups and representation theory,Lecture Notes in Math. 766, Springer-Verlag, Berlin-Heidelberg-New York, 1979.

[4] Dold, A., Fixed point index and fixed point theorem for Euclideanneighborhood retracts, Topology, 4 (1985), 1–8.

[5] Dold, A., Lectures on Algebraic Topology, Die Grundlehrender mathematischen Wissenschaften in Einzeldarstellungen, Band200, Springer-Verlag, Berlin-Heidelberg-New York, 1972.

[6] Dold, A., Eine geometrische Beschreibung des Fixpunktindexes,Arch. Math. 25 (1974), 297–302.

[7] Dold, A., The fixed point index of fibre-preserving maps, Invent.Math. 25 (1974), 281–297.

[8] Dold, A., The fixed point transfer for fibre-preserving maps, Math.Z. 148 (1976), 215–244.

[9] Dold, A., Fixed point theory and homotopy theory, ContemporaryMath. 12 (1982), 105–115.

[10] Dold, A., Puppe, D., Duality, trace and transfer, Proc. of theInternat. Conference on Geometric Topology PWN Warszawa(1980), 81–102.

[11] Granas, A., The Leray-Schauder index and the fixed point theoryfor arbitrary ANRs, Bull. Soc. Math. France, 100 (1972), 209–228.

16 Carlos Prieto

[12] Hopf, H., Eine Verallgemeinerung der Euler-PoincareschenFormel, Nachr. Akad. Wiss. Gottingen Math.-Phys. Kl II (1928),127–136.

[13] Hopf, H., Uber die algebraische Anzahl von Fixpunkten, Math. Z.29 (1929), 493–524.

[14] Ize, J., Massabo, I. and Vignoli, A., Degree theory for equivariantmaps, I, Trans. Amer. Math. Soc. 315 (1989), 433–510.

[15] Ize, J., Massabo, I., Vignoli, A., Degree theory for equivariantmaps, the general S1-action, Memoirs Amer. Math. Soc., 100,No. 481, (1992).

[16] Kosniowsky, C., Equivariant cohomology and stable cohomotopy,Math. Ann. 210 (1974), 83–104.

[17] Lefschetz, S., Intersections and transformations of complexes andmanifolds, Trans. Amer. Math. Soc. 28 (1926), 1–49.

[18] Leray, J. and Schauder, J., Topologie et equations fonctionelles,Ann. Scient. Ec. Norm. Sup. 51 (1934), 45–78.

[19] Lewis, G., May, P. and McClure, J., Ordinary RO(G)-graded co-homology, Bull. Am. Math. Soc. 4 (1981), 208–212.

[20] Prieto, C., Coincidence index for fiber-preserving maps. An ap-proach to stable cohomotopy, manuscripta math. 47 (1984), 233–249.

[21] Prieto, C., KO(B)-graded stable cohomotopy over B and RO(G)-graded G-equivariant stable cohomotopy: A fixed point theoreti-cal approach to the Segal Conjecture, Contemporary Math. 58, II(1987), 89–108.

[22] Prieto, C., Teorıa FIX de diagramas, en “Conferencias del Tallerde Topologıa Algebraica” IV Coloquio, Depto. de Matematicas,CINVESTAV, Mexico (1985).

[23] Prieto, C., FIX-Theory of diagrams, Comtemporary Math. 72

(1988), 207–224.

[24] Prieto, C., Una formula de Lefschetz-Hopf para el ındice de coin-cidencia equivariante parametrizado, Aportaciones Mat. 5 (1988),73–88.


[25] Prieto, C., Transfers generate the equivariant stable homotopy cat-egory, Topology & its Appl. 58 (1994), 181–191.

[26] Prieto, C. and Ulrich, H., Equivariant fixed point index and fixedpoint transfer in nonzero dimensions, Trans. Amer. Math. Soc.328 (1991), 731–745.

[27] Segal, G. B., Equivariant K-theory, Publ. Math. I.H.E.S. 34

(1968), 128–151.

[28] Segal, G. B., Equivariant stable homotopy theory, Proc. Internat.Congress Math. Nice 1970, 2 Parıs (1971), 59–63.

[29] Ulrich, H., Der Fixpunktindex fasernweiser Abbildungen, Diplo-marbeit, Heidelberg (1976).

[30] Ulrich, H., Der aquivariante Fixpunktindex vertikaler G-Abbil-dungen, Dissertation, Heidelberg (1983).

[31] Ulrich, H., Fixed Point Theory of Parametrized Equivariant Maps,Lect. Notes in Math., 1343, Springer-Verlag, Berlin-Heidelberg-New York (1988).

Morfismos, Vol. 5, No. 2, 2001, pp. 19–35

Existence of Nash equilibria in nonzero-sumergodic stochastic games in Borel spaces ∗

Rafael Benıtez-Medina

Abstract

In this paper we study nonzero-sum stochastic games with Borelstate and action spaces, and the average payoff criterion. Undersuitable assumptions we show the existence of Nash equlibria instationary strategies. Our hypotheses include ergodicity condi-tions and an ARAT (additive reward, additive transition) struc-ture.

2000 Mathematics Subject Classification: 91A15, 91A10.Keywords and phrases: nonzero-sum stochastic games, ARAT games,Nash equilibria, expected average payoff.

1 Introduction

This paper concerns nonzero-sum stochastic games with Borel state andaction spaces, and the average payoff criterion with possibly unboundedpayoffs. This class of games has many applications, for instance, inqueueing and economic theory (see [1], [2], [12], [27]).

The problem we are interested in is the existence of Nash equilibria instationary strategies. To do this we impose ergodicity conditions alreadyused by several authors for markov games and control problems (e.g.[1], [6], [9], [10], [14], [15], [19], [22]) together with a so-called ARAT(additive reward, additive transition law) structure. Similar results have

htiwsemagrof]41[elneuKdna]5[ihcgaBdnahsohGybdeniatboneebbounded payoffs. Other related works include [18], which deals with

∗Research partially supported by a CONACyT scholarship. This paper is a partof the author’s M. Sc. Thesis presented at the Department of Mathematics ofCINVESTAV-IPN.

19

20 Rafael Benıtez-Medina

Borel state space and bounded payoffs, and [27], in which the statespace is countable.

For stochastic games with a discounted payoff criterion there isa larger literature. For instance, for zero-sum problems in countablespaces, see [1], [17], [26]; for uncountable spaces, see [11], [13], [21], [25].On the other hand, for the nonzero-sum case in countable spaces, see[27], and for uncountable spaces, see [11], [23], [24].

The remainder of the paper is organized as follows. Section 2 in-troduces standard material on stochastic games and strategies, and theoptimality criteria. The core of the paper is contained in section 3: af-ter introducing some assumptions, we present our main result, Theorem3.10, on the existence of Nash equilibria. Finally, after some technicalpreliminaries in section 4, the proof of Theorem 3.10 is presented insection 5.

2 The game model

For notational ease, we shall consider a stochastic game with only twoplayers. For N > 2 players, the situation is completely analogous. Webegin with the following remark on terminology and notation.

2.1 Remark.

(a) A Borel subset X of a complete and separable metric space iscalled a Borel space, and its Borel σ-algebra is denoted by B(X).We only deal with Borel spaces, and so measurable always means“Borel measurable”. Given a Borel space X, we denote by IP(X)the family of probability measures on X, endowed with the weaktopology σ(IP(X), Cb(X)), where Cb(X) stands for the space ofcontinuous bounded functions on X. In this case, IP(X) is a Borelspace. Moreover, if X is compact, then so is IP(X).

(b) Let X and Y be Borel spaces. A measurable function φ : Y →IP(X) is called a transition probability from Y to X, and we denoteby IP(X|Y ) the family of all those transition probabilities. If φis in IP(X|Y ), then we write its values either as φ(y)(B) or asφ(B|y), for all y ∈ Y and B ∈ B(X). Finally, if X = Y then φ issaid to be a Markov transition probability on X.

The stochastic game model. We shall consider the two-personnonzero-sum stochastic game model

Nash equilibria in nonzero-sum stochastic games 21

GM := (X,A,B, IKA, IKB, Q, r1, r2),(1)

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 IKA ∈ B (X × A) and IKB ∈ B (X × B) are the constraint sets.That is, for each x ∈ X, the x-section in IKA, namely

A(x) := a ∈ A|(x, a) ∈ IKA,

represents the set of admissible actions for player 1 in the state x. Sim-ilarly, the x-section in IKB, i.e.

B(x) := b ∈ B|(x, b) ∈ IKB,

stands for the family of admissible actions for player 2 in the state x.Let

IK := (x, a, b)|x ∈ X, a ∈ A(x), b ∈ B(x),

which is a Borel subset of X × A × B. Then Q ∈ IP(X| IK) is thegame’s transition law, and, finally, ri : IK → IR is a measurable functionrepresenting the reward function for player i = 1, 2.

The game is played as follows. At each stage t = 0, 1, . . . , the play-ers 1 and 2 observe the current state x ∈ X of the system, and in-dependently choose actions a ∈ A(x) and b ∈ B(x), respectively. Asa consequence of this, the following happens: (1) player i receives animmediate reward ri(x, a, b), i = 1, 2; and (2) the system moves to anew state with distribution Q(·|x, a, b). The goal of each player is tomaximize, in the sense of Definition 2.2, below, his long-run expectedaverage reward (or payoff) per unit time.

2.2 Strategies

LetH0 := X andHt := IK×Ht−1 for t = 1, 2, . . . . For each t, an elementht = (x0, a0, b0, . . . , xt−1, at−1, bt−1, xt) of Ht represents a “history” ofthe game up to time t. A strategy for player 1 is then definedas a sequence π1 = π1

t , t = 0, 1, . . . of transition probabilities π1t in

IP(A|Ht) such that

π1t (A(xt)|ht) = 1 ∀ht ∈ Ht, t = 0, 1, . . . .

We denote by Π1 the family of all strategies for player 1.


Now define IA(x) := IP(A(x)) for each state x ∈ X, and let S1 bethe class of all transition probabilities φ ∈ IP(A|X) such that φ(x) isin IA(x) for all x ∈ X. Then a strategy π1 = π1

t ∈ Π1 is calledstationary if there exists φ ∈ S1 such that

π1t (·|ht) = φ(xt)(·) ∀ht ∈ Ht, t = 0, 1, . . . .

We will identify S1 with the family of stationary strategies for player 1.The sets of strategies Π2 and S2 for player 2 are defined similarly,

writing B(x) and IB(x) := IP(B(x)) in lieu of A(x) and IA(x), respec-tively.

Let (Ω,F) be the canonical measurable space that consists of thesample space Ω := (X × A × B)∞ and its product σ-algebra F . Thenfor each pair of strategies (π1,π2) ∈ Π1 × Π2 and each initial statex ∈ X there exists a probability measure P π1,π2

x and a stochastic process(xt, at, bt), t = 0, 1, . . . . defined on (Ω,F) in a canonical way, wherext, at and bt represent the state and the actions of players 1 and 2,respectively, at each stage t = 0, 1, . . .. The expectation operator withrespect to P π1,π2

x is denoted by Eπ1,π2

x .

2.3 Average payoff criteria

For each n = 1, 2, . . . and i = 1, 2, let

J in(π

1,π2, x) := Eπ1,π2

x [n−1!

t=0

ri(xt, at, bt)]

be the n-stage expected total payoff (or reward) of player i when theplayers use the strategies π1 ∈ Π1 and π2 ∈ Π2, given the initial statex0 = x.

The corresponding long-run expected average payoff (EAP) per unittime is then defined as

J i(π1,π2, x) := lim infn→∞

J in(π

1,π2, x)/n.(2)

The EAP is also known as the ergodic payoff (or ergodic reward) crite-rion.

2.2 Definition. A pair of strategies (π1∗,π2∗) is called a Nash equi-librium (for the EAP criterion) if

J1(π1∗,π2∗, x) ≥ J1(π1,π2∗, x) for all π1 ∈ Π1, x ∈ X,


and

J2(π1∗,π2∗, x) ≥ J2(π1∗,π2, x) for all π2 ∈ Π2, x ∈ X.

Our aim is to establish, under certain assumptions, the existence ofa Nash equilibrium (φ∗,ψ∗) in S1 × S2.

We introduce the following notation. For any given function f :IK → IR and probability measures φ ∈ IA(x) and ψ ∈ IB(x), we write

f(x,φ,ψ) :=!

A(x)

!

B(x)f(x, a, b)ψ(db)φ(da)

whenever the integrals are well defined. In particular, for ri and Q asin (1),

ri(x,φ,ψ) :=!

A(x)

!

B(x)ri(x, a, b)ψ(db)φ(da)

and

Q(·|x,φ,ψ) :=!

A(x)

!

B(x)Q(·|x, a, b)ψ(db)φ(da).

3 Main result

We first introduce our assumptions, and then present our main result.

3.1 Assumption. (a) For each state x ∈ X, the sets A(x) and B(x)of admissible actions are compact.

(b) For each (x, a, b) in IK, r1(x, ·, b) is upper semicontinuous (u.s.c)on A(x), and r2(x, a, ·) is u.s.c on B(x).

(c) For each (x, a, b) in IK and each bounded measurable function von X, the functions

!

Xv(y)Q(dy|x, ·, b) and

!

Xv(y)Q(dy|x, a, ·)

are continuous on A(x) and B(x), respectively.(d) There exists a constant r and a measurable function w(·) ≥ 1 on

X such that

|ri(x, a, b)| ≤ rw(x) ∀(x, a, b) ∈ IK, i = 1, 2,(3)

and, in addition, part (c) holds when v is replaced with w.

The next two assumptions are used to guarantee that the state pro-cess Xt is ergodic in a suitable sense.


3.2 Assumption. There exists a probability measure ν ∈ IP(X), apositive number α < 1, and a measurable function β : IK → [0, 1] forwhich the following holds for all (x, a, b) ∈ IK and D ∈ B(X):(a) Q(D|x, a, b) ≥ β(x, a, b)ν(D);(b)

!X w(y)Q(dy|x, a, b) ≤ αw(x) + β(x, a, b)||ν||w, where w(·) ≥ 1 is

the function in Assumption 3.1(d), and ||ν||w :=!wdν.

(c) inf!X β(x,φ(x),ψ(x))ν(dx) > 0, where the infimum is over all the

pairs (φ,ψ) in S1 × S2.

3.3 Assumption. There exists a σ-finite measure λ on X with re-spect to which, for each pair (φ,ψ) ∈ S1 × S2, the Markov transitionprobability Q(·|x,φ(x),ψ(x)) is λ-irreducible.

We next introduce some notation and then we mention some impor-tant consequences of the above assumptions.

3.4 Definition. IBw(X) denotes the linear space of real-valued mea-surable functions u on X with a finite w-norm, which is defined as

||u||w := supx∈X

|u(x)|/w(x),(4)

and IMw(X) stands for the normed linear space of finite signed measuresµ on X such that

||µ||w :="

Xwd|µ| < ∞.(5)

Note that the integral!udµ is finite for each u ∈ IBw(X) and µ in

IMw(X), because, by (4) and (5),

|"

udµ| ≤ ||u||w"

wd|µ| = ||u||w||µ||w < ∞.

3.5 Remark. Suppose that Assumptions 3.2 and 3.3 are satisfied.Then:(a) For each pair (φ,ψ) ∈ S1 × S2, the state (Markov) process Xtis positive Harris recurrent; hence, in particular, the Markov transi-tion probability Q(·|x,φ(x),ψ(x)) admits a unique invariant probabilitymeasure in IMw(X) which will be denoted by q(φ,ψ); thus

q(φ,ψ)(D) ="

XQ(D|x,φ(x),ψ(x))q(φ,ψ)(dx) ∀D ∈ B(X).


(b) Xt is w-geometrically ergodic, that is, there exist positive con-stants θ < 1 and M such that

|!X u(y)Qn(dy|x,φ(x),ψ(x)) −

!X u(y)q(φ,ψ)(dy)|

≤ w(x)||u||wMθn(6)

for every u ∈ IBw(X), x ∈ X, and n = 0, 1, . . . , where Qn denotes then-step Markov transition probability. This result follows from Lemmas3.3 and 3.4 in [6] where it was assumed the positive Harris recurrence inpart (a). However, as shown in Lemma 4.1 of [15], the latter recurrenceis a consequence of Assumptions 3.2 and 3.3.

3.6 Assumption. There exists a probability measure γ in IMw(X)(i.e.

!wdγ < ∞) and a strictly positive density function g(x, a, b, ·)

such that

Q(D|x, a, b) ="

Dg(x, a, b, y)γ(dy)

for all D ∈ B(X ) and (x, a, b) ∈ IK.

Note that Assumption 3.6 implies 3.3 with λ = γ.

3.7 Assumption. The transition density g(x, a, b, y) is such that

limn→∞

"

X|g(x, an, bn, y)− g(x, a, b, y)|w(y)γ(dy) = 0 ∀x ∈ X(7)

if an → a in A(x) and bn → b in B(x), where w(·) is the function inAssumption 3.1(d).

The next two assumptions require that the game model (1) has aso-called ARAT (additive reward, additive transition law) structure.

3.8 Assumption. There exist substochastic kernels Q1 ∈ IP(X| IKA)and Q2 ∈ IP(X| IKB) such that

Q(·|x, a, b) = Q1(·|x, a) +Q2(·|x, b)

for all x ∈ X, a ∈ A(x), b ∈ B(x). Further, Q1(D|x, ·) and Q2(D|x, ·)are continuous on A(x) and B(x), respectively, for each D ∈ B(X).


3.9 Assumption. For i = 1, 2 there exist measurable functions

ri1 : IKA → IR, ri2 : IKB → IR,

such that(a) ri(x, a, b) = ri1(x, a) + ri2(x, b) for all x ∈ X, a ∈ A, b ∈ B.Moreover, for each x ∈ X,(b) the functions ri1(x, ·) and ri2(x, ·) are continuous on A(x) and B(x),respectively, and(c) maxa∈A(x) |ri1(x, a)| ≤ w(x), and maxb∈B(x) |ri2(x, b)| ≤ w(x).

Observe that (c) and the condition γ ∈ IMw(X) in Assumption 3.6yield that

!

Xmaxa∈A(x)

|ri1(x, a)|γ(dx) < ∞,!

Xmaxb∈B(x)

|ri2(x, b)|γ(dx) < ∞.

3.10 Theorem. Under Assumptions 3.1, 3.2 and 3.6- 3.9, there is apair (φ∗,ψ∗) ∈ S1 × S2 that is a Nash equilibrium.

The remainder of this work is devoted to prove Theorem 3.10.

4 Preliminaries

Suppose that one of the players, say player 2, selects a fixed station-ary strategy ψ in S2. Then the game model GM in (1) reduces to aMarkov control model

MCM1(ψ) = (X,A, IKA, Qψ, r1,ψ)(8)

where X, A and IKA are as in (1), and the transition law Qψ inIP(X| IKA) and the reward function r1,ψ : IKA → IR are given by

Qψ(·|x, a) := Q(·|x, a,ψ(x)) and r1,ψ(x, a) := r1(x, a,ψ(x)),

respectively. Then from Corollary 5.12 in [10], for instance, we get thefollowing.

4.1 Lemma. Suppose that Assumptions 3.1, 3.2 and 3.3 are satisfied.Then for each fixed ψ ∈ S2, there exists a stationary strategy φ∗ ∈ S1

that is expected average reward (EAR) optimal for the Markov controlmodel in (8), i.e.,

J1(φ∗,ψ, x) = maxπ1∈Π1

J1(π1,ψ, x) =: ρ∗1(ψ) ∀x ∈ X.(9)


Moreover, there exists a function h1φ∗,ψ ∈ IBw(X) such that (ρ∗1(ψ), h1φ∗,ψ)

is the unique solution in IR× IBw(X) of the equation

ρ∗1(ψ) + h1φ∗,ψ(x) = r1(x,φ∗(x),ψ(x))+

!X h1φ∗,ψ(y)Q(dy|x,φ∗(x),ψ(x))(10)

= maxφ∈IA(x)[r1(x,φ,ψ(x))+

!X h1φ∗,ψ(y)Q(dy|x,φ,ψ(x))](11)

for all x ∈ X, and such that!X h1φ∗,ψ(y)q(φ

∗,ψ)(dy) = 0, with q(φ∗,ψ)as in the Remark 3.5(a).

In other words, (9) states that φ∗ ∈ S1 is an optimal response ofplayer 1, given that player 2 uses the fixed stationary strategy ψ ∈ S2.Similarly, we can obtain an optimal response ψ∗ ∈ S2 of player 2 ifplayer 1 uses a fixed strategy φ ∈ S1.

We now wish to express the optimal average reward ρ∗1(ψ) in (9),in a more convenient form. We will use the following fact, which isborrowed from Proposition 10.2.3 in [9].

4.2 Lemma. Suppose that Assumptions 3.1, 3.2 and 3.3 are satisfied,and let (φ,ψ) ∈ S1 × S2 be an arbitrary pair of stationary strategies.Then for i = 1, 2 we have:(a) The EAP in (2) satisfies that

J i(φ,ψ, x) = limn→∞

J in(φ,ψ, x)/n = ρi(φ,ψ),(12)

where

ρi(φ,ψ) :="

Xri(x,φ(x),ψ(x))q(φ,ψ)(dx)(13)

with q(φ,ψ) as in Remark 3.5(a).(b) The function hiφ,ψ defined on X as

hiφ,ψ(x) :=∞#

t=0

Eφ,ψx [ri(xt,φ(xt),ψ(xt))− ρi(φ,ψ)]

belongs to IBw(X), and, moreover, its w-norm is independent of (φ,ψ):

||hiφ,ψ||w ≤ rM/(1− θ),(14)


where r is the constant in (3), and M and θ are as in (6).(c) The pair (ρi(φ,ψ), hiφ,ψ) is the unique solution in IR× IBw(X) of theso-called Poisson equation

ρi(φ,ψ)+ hiφ,ψ(x) = ri(x,φ(x),ψ(x))+

!X hiφ,ψ(y)Q(dy|x,φ(x),ψ(x))(15)

that satisfies the condition"

Xhiφ,ψ(y)q(φ,ψ)(dx) = 0.

5 Proof of Theorem 3.10

From (12) and Corollary 5.12(a) in [10], we can write ρ∗1(ψ) in (9) as

ρ∗1(ψ) = ρ1(φ∗,ψ) = max

φ∈S1

ρ1(φ,ψ).(16)

Similarly, for each φ ∈ S1 there exists ψ∗ ∈ S2 such that

ρ∗2(φ) = ρ2(φ,ψ∗) = max

ψ∈S2

ρ2(φ,ψ)(17)

We next use (16) and (17) to introduce a multifunction τ fromS1 × S2 to 2S1×S2 as follows: for each pair (φ,ψ) in S1 × S2

τ(φ,ψ) := (φ∗,ψ∗)|ρ1(φ∗,ψ) = ρ∗1(ψ), ρ2(φ,ψ∗) = ρ∗2(φ).(18)

To complete the proof of Theorem 3.10 we shall proceed in twosteps, which is in fact a standard procedure (see Ghosh and Bagchi [5],Himmelberg et. al. [11], Parthasarathy [23], for instance).

Step 1. Introduce a topology on Si (i = 1, 2) with respect to whichSi is compact and metrizable.

Step 2. Show that the multifunction τ is upper semicontinuous(u.s.c.), that is , if (i)(φn,ψn) → (φ∞,ψ∞) in S1×S2, and (ii) (φ∗n,ψ

∗n) ∈

τ(φn,ψn) is such that (φ∗n,ψ∗n) → (φ∗∞,ψ∗

∞), then (φ∗∞,ψ∗∞) is in τ(φ∞,ψ∞).

From these two steps and Fan’s fixed point theorem (Theorem 1 in[4]), it will follow that the multifunction τ has a fixed point (φ∗,ψ∗) inS1 × S2, that is

(φ∗,ψ∗) ∈ τ(φ∗,ψ∗).(19)

Finally, from (16)− (18) and (19) we shall conclude that (φ∗,ψ∗) is aNash equilibrium.


In step 1 we shall use the topology introduced by Warga (see Theo-rem IV.3.1 in [28]): Let F1 be the Banach space of measurable functionsf : IKA → IR such that f(x, a) is continuous in a ∈ A(x) for each x ∈ Xand

||f || :=!

Xmaxa∈A(x)

|f(x, a)|γ(dx) < ∞,

with γ as in Assumption 3.6. We shall identify two stationary strategiesφ and φ

′in S1 if φ = φ

′γ-a.e. (almost everywhere), and , on the other

hand, φ ∈ S1 can be identified with the linear functional ∆φ ∈ F ∗1 given

by

∆φ(f) :=!

X

!

Af(x, a)φ(da|x)γ(dx).

Thus S1 can be identified with a subset of F ∗1 , and endowing S1 with

the weak∗ topology it can be shown that S1 is compact and metrizable[28]. The set S2 is topologized analogously.

To proceed with step 2, suppose that

(φn,ψn) → (φ∞,ψ∞) in S1 × S2,(20)

and that(φ∗n,ψ

∗n) ∈ τ(φn,ψn) ∀n(21)

is such that(φ∗n,ψ

∗n) → (φ∗∞,ψ∗

∞).(22)

By (21) and the definition (18) of τ , together with (10) (or (15)), forall x ∈ X we have

ρ∗1(ψn) + h1φ∗n,ψn(x) = r1(x,φ∗n(x),ψn(x))

+"X h1φ∗n,ψn

(y)Q(dy|x,φ∗n(x),ψn(x))(23)

and, similarly,

ρ∗2(φn) + h2φn,ψ∗n(x) = r2(x,φn(x),ψ∗

n(x))

+"X h2φn,ψ∗

n(y)Q(dy|x,φn(x),ψ∗

n(x)).(24)

Now observe that, by Assumptions 3.8 and 3.9, for each D ∈ B(X),the functions Q1(D|x, a) and ri1(x, a) are in F1, whereas Q2(D|x, b) andri2(x, b) are in F2. Therefore, by (20) and (22),

!

Xr1(x,φ

∗n(x),ψn(x))γ(dx) →

!

Xr1(x,φ

∗∞(x),ψ∞(x))γ(dx),(25)


and similarly for i = 2. Moreover, for any D ∈ B(X),

!X Q(D|x,φ∗n(x),ψn(x))γ(dx) →!

X Q(D|x,φ∗∞(x),ψ∞(x))γ(dx),(26)

and similarly for (φn,ψ∗n) → (φ∞,ψ∗

∞).

5.1 Lemma. There is a subsequence m of n and numbers ρ1 andρ2 such that

ρ∗1(ψm) = ρ1(φ∗m,ψm) → ρ1(27)

andρ∗2(φm) = ρ2(φm,ψ∗

m) → ρ2.(28)

Proof: Let ρi(φ,ψ) be as in (13). We next show that, for i = 1, 2,

|ρi(φ,ψ)| ≤ r||ν||w/(1− α) ∀(φ,ψ) ∈ S1 × S2,(29)

with r as in (3), and ν and α as in Assumption 3.2. Clearly, (29)implies (27) and (28) .

To prove (29), note that Assumption 3.2(b) yields

"

Xw(y)Q(dy|x, a, b) ≤ αw(x) + ||ν||w(30)

because β(x, a, b) ≤ 1. Now let (φ,ψ) be an arbitrary pair in S1 × S2.Integrating both sides of (30) with respect to φ(da|x) and ψ(db|x),and then integrating with respect to the invariant probability measureq(φ,ψ) yields

"

Xw(y)q(φ,ψ)(dy) ≤ α

"

Xw(y)q(φ,ψ)(dy) + ||ν||w,

and, therefore,

"

Xw(y)q(φ,ψ)(dy) ≤ ||ν||w/(1− α).

The latter inequality, together with (3) and (13), gives

|ρi(ψ,φ)| ≤!X |ri(x,φ(x),ψ(x))|q(φ,ψ)(dx)

≤ r!X w(y)q(φ,ψ)(dy)

≤ r||ν||w/(1− α),


i.e., (29) holds. This in turn gives that the sequences ρ1(φ∗n,ψn) andρ2(φn,ψ∗

n) are uniformly bounded , and so the lemma follows. For notational convenience, we shall write the subsequence m ⊂

n in (27) and (28) as the original sequence, n. Moreover, let

un(·) := h1φ∗n,ψn(·), and un(·) := un(·)/w(·).(31)

By (14), the constant m0 := rM/(1− θ) satisfies that

|un(x)| ≤ m0 ∀x, n.

Let U be the space of all γ-equivalence classes of real-valued mea-surable functions u on X such that |u(x)| ≤ m0 γ-a.e. By the Alaoglu(or Banach-Alaoglu) Theorem (see page 424 in [3], for instance), U is acompact and metrizable subset of L∞(γ) ≡ L∞(X,B(X), γ) equippedwith the relative weak* topology σ(L∞(γ), L1(γ)). Therefore, we canassume that un converges in the weak* topology to some functionu∗ in L∞(γ). Let u∗(x) := u∗(x)w(x) for all x ∈ X. Then, as in theproof of Theorem 4 in [19], using Assumption 3.7, one can show thatas n → ∞.

maxa∈A(x)

maxb∈B(x)

|!

X(un(y)− u∗(y))Q(dy|x, a, b)| → 0 ∀x ∈ X(32)

with un(·) as in (31). In turn, (32) and Assumption 3.8 yield that

maxφ∈IA(x)

maxψ∈IB(x)

|!

X(un(y)− u∗(y))Q(dy|x,φ,ψ)| → 0 ∀x ∈ X.(33)

We also have the following.

5.2 Lemma. If (φn,ψn) → (φ,ψ) in S1 × S2, then, as n → ∞,

"X

"X u(y) Q(dy|x,φn(x),ψn(x))γ(dx)

→"X

"X u(y)Q(dy|x,φ(x),ψ(x))γ(dx)(34)

for any function u ∈ IBw(X).

Proof: Choose an arbitrary function u ∈ IBw(X). By definition of theweak convergence of φn → φ and ψn → ψ in S1 and S2, respectively,and Assumption 3.8, to prove the lemma it suffices to show that thefunctions

(x, a) →!

Xu(y)Q1(dy|x, a)(35)


and

(x, b) →!

Xu(y)Q2(dy|x, b)(36)

are in F1 and F2, respectively. With this in mind, first note that"X u(y)Qi(dy|x, ·) is continuous in a ∈ A(x) and b ∈ B(x), for i = 1and i = 2, respectively, (see Lemma 8.3.7(a) in [9]). Moreover, by (4)and Assumption 3.2(b) (using that β(x, a, b) ≤ 1),

maxa∈A(x)

|!

Xu(y)Q1(dy|x, a)| ≤ ||u||w(αw(x) + ||ν||w) ∀x ∈ X.

Hence, as"wdγ < ∞ (by Assumption 3.6), the function in (35) is in

F1. Similarly, the function in (36) is in F2. By Lemmas 5.1 and 5.2, together with (25), (26) and (33), letting

n → ∞ in (23) we obtain γ-a.e.

ρ1 + u∗(x) = r1(x,φ∗∞(x),ψ∞(x))+

"X u∗(y)Q(dy|x,φ∗∞(x),ψ∞(x))

(37)

= maxφ∈IA(x)[r1(x,φ,ψ∞(x))+

"X u∗(y)Q(dy|x,φ,ψ∞(x))],

where the second equality comes from (10)- (11) replacing (φ∗,ψ) with(φ∗n,ψn).

Finally, arguing as in the last part of the proof of Theorem 5.8 in[10], let D ∈ B(X) be the set with γ(D) = 1 on which (37) holds, andlet h∗ : X → IR be such that h∗(x) := u∗(x) for x ∈ D, and

h∗(x) := maxφ∈IA(x)

[r1(x,φ,ψ∞(x)) +!

Xu∗(y)Q(dy|x,φ,ψ∞(x))]− ρ1

for all x in the complement Dc of D. As γ(Dc) = 0, by Lemma 6.3 in[10], we have Q(Dc|x, a, b) = 0 for all (x, a, b) ∈ IK. Therefore, (37)holds for all x ∈ X when u∗(·) is replaced with h∗(·). This implies (byLemma 4.1) that

ρ1 = ρ∗1(ψ∞) = ρ1(φ∗∞,ψ∞).(38)

An analogous argument using (21), (22) and (24) with obvious changes,shows that

ρ2 = ρ∗2(φ∞) = ρ2(φ∞,ψ∗∞).(39)


In other words, (38) and (39) state that, under (20)- (22), the pair(φ∗∞,ψ∗

∞) is in τ(φ∞,ψ∞), and so the set-valued map defined by (18)is u.s.c. Thus, as was already noted, it follows from Fan’s fixed pointtheorem that τ has a fixed point (as in (19), say), which completes theproof of Theorem 3.10.

AcknowledgementThe author wishes to thank Dr. Onesimo Hernandez-Lerma for his

very valuable comments and suggestions.

Rafael Benıtez-MedinaDepartamento de Matematicas,CINVESTAV-IPN,A. Postal 14-740,07000, Mexico D.F., MEXICO,[email protected].

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Monte Carlo approach to insurance ruinproblems using conjugate processes ∗

Luis F. Hoyos-Reyes 1

Abstract

In this paper is discussed a simulation method developed by S.Asmussen called conjugate processes which is based on a versionof Wald’s fundamental identity. With this method it is possibleto simulate within finite time risk reserve processes with infinitetime horizons. This allows us to construct Monte Carlo estimatorsfor the ruin probability, which is one of the main problems ininsurance risk theory. Some examples of the Poisson/Exponentialand Poisson/Uniform cases are presented.

2000 Mathematics Subject Classification: 60K30, 65C05.Keywords and phrases: Conjugate processes, Monte Carlo estimators,risk reserve process, ruin probability, Wald’s fundamental identity.

1 Introduction

One of the main problems in insurance risk theory is to estimate theruin probability [1-8,11]. It can be roughly described as follows.

The risk reserve process over (0, t] is the difference between a pre-mium deterministic process u + ct and the accumulated claims Zt (acompound Poisson process), for some given initial capital u ≥ 0 . Thepremium income rate c is fixed by the insurance company and is in-dependent of t. The idea is to study the behavior of the risk reserveprocess that models the accumulated capital over finite or infinite time

∗Research partially supported by a CONACyT scholarship. This paper is part ofthe author’s M. Sc. Thesis presented at the Division de Ciencias Basicas e Ingenierıa,UAM-Iztapalapa.

1Professor at Departamento de Sistemas, UAM-Azcapotzalco.

37

38 Luis F. Hoyos-Reyes

horizons, in particular the probability that exists a moment τ when therisk process is negative. This is called the ruin probability.

The main purpose of this paper is to introduce Monte Carlo esti-mators (MCEs) for the ruin probability in infinite time horizon usingconjugate processes. This approach allows us to simulate within finitetime a risk reserve process with infinite time horizon. Here, we constructa MCE for the ruin probability using the empirical distribution of theruin events after a sufficiently large number of simulations. Some ex-amples of the Poisson/Exponential (P/E) and Poisson/Uniform (P/U)cases are presented.

This paper is organized as follows. We begin in §2 by introducingbasic terminology and notation. In §3 we show a formulation for theconjugate process and construct the MCEs. In §4 we compute exam-ples of the P/E and P/U cases. Finally, §5 presents some concludingremarks.

2 Preliminaries

Assumption 2.1

(a) The claims arrive according to a Poisson process Ntt≥0 withintensity λ and interclaim times Ttt≥1.

(b) The claim sizes X1, X2, . . . are i.i.d nonnegative random variableswith a finite mean µ.

(c) Xi and Ntt≥0 are independent.

Definition 2.2 The accumulated claim process is Zt :=!Nt

n=0Xn fort ≥ 0, with X0 := 0.

We next recall the classical risk reserve process [2,7,8].

Definition 2.3 Let u be the initial capital and c > 0 be the premiumincome rate.

We define the risk reserve process

Yt := Zt − ct, t ∈ (0,∞),

and the time to ruin

τ := inf t > 0 : Yt > u .

Insurance ruin problems 39

Definition 2.4 A family (Fθ)θ∈Θ of distributions on R is called a con-jugate family if the Fθ are mutually equivalent with densities of theform

(1)dFθ

dFθ0(x) = exp (θ − θ0)x− hθ0(θ)

and if for some fixed θ0 ∈ Θ the parameter set Θ contains all θ ∈ R forwhich (1) defines a probability density for some hθ0(θ).

Then, by definition, Pθ0 := P is the probability law of the process Yt.In addition, θ0 < 0 is the solution of

φ′X(−θ0) = c/λ,

where φX(β) := E!eβX

"is the moment generating function of X. This

definition of θ0 allows us to choose the sign of EθYt as we prove below(Proposition 2.7).

Also note that φθ0(β) = Eθ0

!eβX

"= φX(β).

Equation (1) implies that hθ0(θ) is given in terms of the cumulantgenerating function of Fθ0 by

hθ0(θ) := logEθ0e(θ−θ0)X .

The accumulated claim process Zt is a compound Poisson process, soits moment generating function [2] is

(2) φZt(β) = eλt(φX(β)−1).

Proposition 2.5 Let θ, θ0 ∈ Θ with θ = θ0. Then

φθ(β) =φθ0(β + θ − θ0)

φθ0(θ − θ0).

Proof: Using (1)


φθ(β) =! ∞

−∞eβxdFθ(x)

=! ∞

−∞eβx

e(θ−θ0)x

Eθ0e(θ−θ0)X

dFθ0(x)

=φθ0(β + θ − θ0)

φθ0(θ − θ0). !

Proposition 2.6 logEeβYtt = λ(φX(β)− 1)− βc.

Proof: By definition of Yt, we can see that

EeβYt = φZt(β)/eβct.

Hence, from (2) we have

EeβYt = eλt(φX(β)−1)−βct.

Applying the log function to both sides of the latter equation and di-viding by t, completes the proof. !

Now from Proposition 2.5

EθeβZt = Eθ0e

(β+θ−θ0)Zt/Eθ0e(θ−θ0)Zt .

Using (2)

EθeβZt = eλtφθ0

(θ−θ0)(φθ(β)−1),

which implies that under Pθ, Zt is also a compound Poisson process witharrival rate λθ = λφθ0(θ − θ0) and claims distribution Fθ. Therefore,replacing E with Eθ in Proposition 2.6 we obtain

(3)logEθeβYt

t= λθ(φθ(β)− 1)− βc = λφθ0(θ − θ0)(φ0(β)− 1)− βc.

Proposition 2.7 If φ′′X exists in an interval I that contains −θ0, then

µθ := EθYt > 0 when θ > 0

and

µθ < 0 when θ > 0.


Proof: Let χθ(β) :=logEθeβYt

t , so that from (3)

χθ(β) = λθ(φθ(β)− 1)− βc.

In particular, if θ = θ0,

χθ0(β) = λθ0(φθ0(β)− 1)− βc,

and taking β = −θ0 we have

χ′θ0(θ0) = λφ′

θ0(−θ0)− c.

On the other hand, recalling that φθ0(β) = φX(β), we get

φ′X(−θ0) = c/λ = φ′

θ0(−θ0),

which yields

χ′θ0(−θ0) = 0.

Also, for all β ∈ I

χ′′θ0(β) = λφ′′

X(β) = λEθ0(X2eβX) > 0,

so that −θ0 is a local minimum, and χθ0(·) is convex on I.Now let θ ∈ Θ. Then Proposition 2.5 implies

χθ(β) = χθ0(β + θ − θ0)− χθ0(θ − θ0),

and, therefore,

(4) χ′θ(0) = χ′

θ0(θ − θ0),

and, moreover,

χ′θ(β) = λφ′

X(β + θ − θ0).

On the other hand,

χ′θ(β) = (EθYte

βYt)/EθeβYt ,

and so

χ′θ(0) = EθYt = µθ.


Using (4)

µθ = χ′θ0(θ − θ0),

and

µ0 = χ′θ0(−θ0) = 0.

The last two equalities and the convexity of χθ0(·) yield the desiredconclusion. !

3 Monte Carlo estimators

Our main purpose in this section is to estimate the ruin probabilityconsidering an infinite time horizon for the risk reserve process. Wefirst introduce some definitions.

Definition 3.1 Let u and τ be as in Definition 2.3. The ruin probabil-ity in finite time is

Ψ(u, T ) := P (τ < T ),

and the ruin probability in infinite time is

Ψ(u) := P (τ < ∞).

The premium income rate c is usually taken as

c = (1 + ρ)EZt/t.

As Zt is a compound Poisson process, c is independent of t. The numberρ is called the safety loading, and is related to the capital expectedgrowth as follows.

Proposition 3.2 If θ > 0, then Pθ(τ < ∞) = 1.

Proof: Under the law Pθ, Zt is a compound Poisson process withNt ∼ Poisson(λθ) and claim sizes Xi ∼ Fθ. Therefore

Eθ(Zt) = λθtEθX.


Applying the strong law of large numbers to the accumulated claimprocess yields

(5) limt→∞

1

t(Zt − λθtEθX) = 0 a.s.

Now, from Proposition 2.7 we have Eθ(Zt−ct) > 0 and, therefore, ρ < 0,and using (5)

limt→∞

1

t(Zt − λθtEθX − ρλθtEθX) = −ρλθEθX > 0 a.s.

This implies

(Zt − λθt(1 + ρ)EθX) → +∞ a.s.,

which completes the proof. !Consider a conjugate family (Fθ)θ∈Θ governing a random walk Stt≥0

in discrete or continuous time. Define FT := σ(St; t ≤ T ), with the usualextension to stopping times.

Next, we present the version of the Wald’s fundamental identity usedby Asmussen [1,2]. The proof can be seen in [3].

Theorem 3.3 Let τ be a stopping time for Stt≥0 and G ∈ Fτ , G ⊆τ < ∞. Then for each θ0, θ ∈ Θ

(6) Pθ0G = Eθ [exp (θ0 − θ)Sτ − τχθ(θ0 − θ) ;G] .

From Definition 2.3 and (6)

(7)dPθ0

dPθ= exp (θ0 − θ)Yτ − τχθ(θ − θ0) ,

and integrating (7) over τ < ∞ we can express the ruin probabilityin infinite time as

Ψ(u) = Eθ [(exp (θ0 − θ)Yτ − τχθ(θ − θ0)) · I τ < ∞] .


Proposition 3.4 Let θ > 0. If we compute n simulations of the con-jugate process

Rθ := exp (θ0 − θ)Yτ − τχθ(θ − θ0) ,

then with probability 1

1

n

n!

i=1

Riθ → Ψ(u) as n → ∞,

where Riθ is the final value of the realization of the conjugate process

after simulation i (i = 1, 2, . . . ).

Proof: By Proposition 3.2 the ruin occurs almost surely, and so eachof the n simulations of Rθ can be performed in a finite number of steps.Moreover, as

EθRθ = Ψ(u),

by the strong law of large numbers it follows that, with probability 1,

1

n

n!

i=1

Riθ → Ψ(u) as n → ∞. !

We call 1n

"ni=1R

iθ a Monte Carlo estimator (MCE) for Ψ(u).

Observe that integrating (7) over τ < T we can write the ruinprobability in a finite time T as

Ψ(u, T ) = Eθ [(exp (θ0 − θ)Yτ − τχθ(θ − θ0)) · I τ < T] .

Then, in this case, the corresponding conjugate process is

RTθ := exp (θ0 − θ)Yτ − τχθ(θ − θ0) · I τ < T ,

and so we could construct an analogous MCE for Ψ(u, T ).

Remark 3.5 (a) Observe that if θ = θ0, then RTθ0

= I τ < T. Thusto simulate RT

θ0is equivalent to simulate the original process Yt,

which, in insurance terminology is called a crude simulation [7,8].


(b) We can simplify RTθ taking θ as the Lundberg value θ1 := γ +

θ0, where γ > 0 is the unique solution of Lundberg’s equationχθ0(γ) = 0. In this case, RT

θ1is called the Lundberg process.

Using Proposition 2.7 one can see that χθ1(θ0 − θ1) = 0, whichimplies that

RTθ1 = exp(−γYτ ) · I τ < T .

Therefore, taking ∆ > 0, θ = (1 +∆) · θ1 and using Theorem 3.3we obtain the following expression:

(8) RTθ1(1+∆) = exp −(γ + θ1∆)Yτ + τχθ1(θ1∆) · I τ < T .

(c) In (b), the corresponding variance σ2θ = VarθRT

θ is

σ2θ1(1+∆) = Eθ1 [exp −2(γ + θ1)Yτ + τχθ1(θ1∆) · I τ < T]−Ψ2(u, T ),

and for the infinite horizon case is

σ2θ1(1+∆) = Eθ1 exp −2(γ + θ1)Yτ + τχθ1(θ1∆)−Ψ2(u).

(d) The overshot B(u) of the risk process, defined as B(u) := Yτ −u isuseful to calculate σ2

θ1. It is known [1,2] that when the claims are

exponentially distributed and the arrival process is Poisson (P/Ecase), B(u) is exponentially distributed:

(9) Pθ(B(u) > b) = exp(−b/EθX).

4 P/E and P/U examples

4.1 Example P/E

The Poisson/Exponential case has been extensively researched [1-5] be-cause it is easy to calculate the ruin probability for the infinite time


horizon. It is a well known fact [5] that if the safety loading ρ is posi-tive, then

(10) Ψ(u) =1

1 + ρexp

!− ρu

µ(1 + ρ)

".

Let us consider the P/E case with µ := EX = 1, λ = 0.8, ρ = 0.1and T = ∞. The right-hand side of (10) depends on the initial capitalu. Let Ψ(u) = 0.05. Then the initial capital is u = 31.904, and thepremium income rate is c = (1 + ρ)λµ (remember that we deal with acompound Poisson Process Zt).

We solve the Lundberg equation for γ using Proposition 2.6:

χθ0(γ) = λ(φX(γ)− 1)− cγ = 0.

Then γ = 0 is the trivial solution, and the other solution is

γ = (c− λ)/λ = 0.0909.

Now we calculate the variances:

σ2θ0 = Eθ0I

2τ < ∞− E2θ0Iτ < ∞ = Ψ(u)−Ψ2(u) = 0.0475

σ2θ1 = Varθ1Rθ1 = Varθ1e

−γTτ = Varθ1e−γ(u+B(u)) = e−2γVarθ1e

−γB(u).

From (1), Eθ1X = (1 − γ)−1, which together with (9) implies thatB(u) ∼ exp(1− γ). Thus

Eθ1e−2γB(u) = (1− γ)/(1 + γ) and Eθ1e

−γB(u) = 1− γ.

Hence,

σ2θ1 = e−2γu

!1− γ

1 + γ− (1− γ)2

"= 2.08× 10−5 < σ2

θ0 .

Note that the difference between the variances is significant, which isan statistical advantage [9] to construct confidence intervals for Ψ(u).

To show some numerical results, let µ = 1, λ = 0.8, ρ = 0.1, c = 0.88.¿From Proposition 2.6, we can see that Nθ

t is a Poisson process witharrival rate c, and Xθ is exponentially distributed with parameter λ/c.


Moreover, γ = 0.0909, θ0 = −0.0488 and θ1 = 0.0421.One can compare the theoretical results versus the Monte Carlo

estimators (MCEs) in Table 1, where

Table 1: Infinite Time Horizon P/E

u n Ψ(u) Ψ(u) σMCE SMCE εR31.9 100 0.05 0.0498 4.5× 10−4 5.00× 10−4 4.0× 10−3

31.9 1000 0.05 0.0499 1.4× 10−4 1.41× 10−4 2.0× 10−3

16.7 1000 0.20 0.1997 5.7× 10−4 5.90× 10−4 1.5× 10−3

σMCE :=!σ2θ/n

"1/2is the standard error of the MCE, SMCE is the

corresponding estimator, and the relative error is

εR :=| 1− Ψ(u)/Ψ(u) | .

Notice the good fitness between the standard error σMCE and its esti-mator SMCE . Obviously, we have better aproximations to Ψ(u) takinglarger samples because the MCEs are consistent.

4.2 Example P/U

Let us assume that the claims size distribution is uniform over (0, 1).First of all, we need to find the distibution Fθ, and then we have toshow an expression for the conjugate process RT

θ1(1−∆). From (1)

Fθ(x) =e(θ−θ0)x − 1

eθ−θ0 − 1, 0 < x < 1.

Recall that under Pθ, Zt is also a compound Poisson process with pa-rameter

λθ =#eγ+θ1∆ − 1

$/(γ + θ1∆).

From Proposition 2.6 and (8) we obtain that in the finite horizon case,and letting γ0 := γ + θ1∆,

(11) RTθ1(1+∆) = exp

%

−γ0Yτ + τ

%eγ+θ1 − 1

γ0

&

− 1− γ0c

&

· Iτ < T,

whereas in the infinite time horizon


(12) Rθ1(1+∆) = exp

!

−γ0Yτ + τ

!eγ+θ1 − 1

γ0

"

− 1− γ0c

"

.

Let γ = 0.05. Then from Lundberg’s equation

eγ − 1

γ− 1− cγ = 0,

we get c = 0.508439. Moreover, simple computations show that −θ0 =0.025078 and θ1 = γ + θ0 = 0.024922.

Unfortunately, there are no theoretical results for the P/U case, sowe cannot compare the real and the estimated values like we did underthe P/E assumptions. However, it is possible to estimate the varianceof the conjugate process and to compute the estimator SMCE of thestandard error; see Table 2.

Table 2: Infinite Time Horizon P/U∆ Ψ(u) σ2

θ SMCE

1.00 0.199 1.5× 10−2 1.2× 10−2

0.10 0.223 6.2× 10−4 2.5× 10−3

0.05 0.220 9.9× 10−5 9.9× 10−4

0.00 0.220 4.2× 10−6 2.0× 10−4

The computations were made with n = 100, and u = 30.Observe that the best estimation occurs when ∆ = 0, which is con-

sistent with the asymptotic optimality proved by Asmussen [2].

5 Concluding remarks

In the previous sections we have introduced MCEs for the ruin probabil-ity using conjugate processes. In particular, we have shown formulationsfor the conjugate process under the P/U assumptions for both finite (11)and infinite (12) time horizons.

The P/U case has not been discussed enough in the literature, andso it is suitable for the simulation approach.

Finally it is important to mention two main advantages of the MCEsusing conjugate processes: (i) the relative simplicity of the formulation,


and (ii) the minimum computational resources needed compared withthe diffusion approach [1,2], and the martingale approach [6].

AcknowledgementThe author is grateful to Dr. Onesimo Hernandez-Lerma for his

valuable comments and useful suggestions.

Luis F. Hoyos-ReyesDepartamento de Sistemas,UAM - Azcapotzalco,Av. San Pablo No. 180,02200 Mexico D.F., MEXICO,[email protected].

References

[1] Asmussen, S., Approximations for the probability of ruin withinfinite time, Scandinavian Actuarial J. 20 (1984), 31-57.

[2] Asmussen, S., Conjugate processes and the simulation of ruinproblems, Stochastic Processes and their Applications 20 (1985),213-229.

[3] Asmussen, S., Applied Probability and Queues, Wiley, Chichester,U.K., 1987.

[4] Asmussen, S. and Rolski, T., Computational methods in risk the-ory: A matrix-algorithmic approach, Insurance: Mathematics andEconomics 10 (1991), 259-274.

[5] Beard, R.E., Pentikainen, T. and Pessonen, E., Risk Theory,Chapman and Hall, New York, 1984.

[6] Dassios, A. and Embrechts, P., Martingales and insurance risk,Commun. Statist.-Stochastic Models 5 (1989), 181-217.

[7] Embrechts, P., Stochastic Modelling in insurance, CLAPEM-IVProceedings, Mexico City 1990.

[8] Embrechts, P. and Wouters, P., Simulating risk solvency, Insur-ance:Mathematics and Economics 9 (1990), 141-148.

[9] Ross, S.M., Stochastic Processes, Wiley, New York, 1983.

[10] Ross, S.M., A Course in Simulation, Macmillan, New York, 1990.


[11] Tijms, H.C., Stochastic Models, An Algorithmic Approach, Wiley,Chichester, U.K., 1998.


Sobre la estrechez de un espacio topologico ∗

Alejandro Ramırez Paramo 1

Resumen

En este trabajo se muestran algunos resultados sobre la estrechezen la clase C2 de los espacios Hausdorff y compactos; en particular,se demuestran las igualdades t(X) = hπχ(X) y t(

!Xs : s ∈

S) = |S| · supt(Xs) : s ∈ S, cuando X y Xs pertenecen a C2para toda s ∈ S.

2000 Mathematics Subject Classification: 54A25.Keywords and phrases: Estrechez, clase C2.

1 Introduccion

En este trabajo se presentan algunos resultados conocidos y otros (posi-blemente) no tan conocidos sobre la funcion cardinal estrechez; cabesenalar que los resultados proposicion 3.2, el corolario 3.4 y el lema4.10, no se encuentran en la bibliografıa consultada por el autor.

Una funcion cardinal topologica (llamada, tambien, invariante car-dinal topologico), es una funcion φ que va de la clase de los espaciostopologicos (algunas veces de una subclase de estos) a la clase de losnumeros cardinales infinitos de tal forma que φ(X) = φ(Y ) para es-pacios X y Y homeomorfos (para un estudio detallado sobre funcionescardinales recomendamos al lector [3] y [4]).

Una cuestion inmediata sobre funciones cardinales es como deter-minar el cardinal que debe asociarse al espacio X; lo cual, en muchas

∗El contenido de este trabajo representa parte de la tesis de grado presentadapor el autor dentro del programa de maestrıa de la Facultad de Ciencias Fısico-Matematicas de la Universidad Autonoma de Puebla.

1Estudiante inscrito en el programa de doctorado de la Facultad de CienciasFısico-Matematicas de la Universidad Autonoma de Puebla.

51


52 Alejandro Ramırez Paramo

ocasiones no es sencillo de responder. De aquı la necesidad de tenerresultados que permitan obtener cotas para las funciones cardinales.En la tercera seccion de este trabajo, se relaciona el concepto de con-junto κ−cerrado con la estrechez para abordar el problema anterior.Se muestra ademas, en esta seccion, que las funciones t (estrechez) yhπχ (π−caracter hereditario) satisfacen la desigualdad t(X) ≤ hπχ(X)para cualquier espacio topologico. De manera natural surge la pregunta:¿para que espacios X se da la igualdad t(X) ≤ hπχ(X)?; este problemaes abordado en la cuarta seccion; en esta misma se muestra que si Xs :s ∈ S es una familia de espacios topologicos, tales que cadaXs es Haus-dorff y compacto, entonces t(

!Xs : s ∈ S) = |S| · tS(

!Xs : s ∈ S),

en donde tS(!Xs : s ∈ S) = supt(Xs) : s ∈ S.

2 Notaciones y definiciones

Aquı, ω representa ambos, el primer ordinal y cardinal infinito, ademas,κ denota un cardinal el cual siempre sera ≥ ω y κ+ es el sucesor de κ.Usamos P(X) para denotar al conjunto potencia de X y |X| para lacardinalidad de X. Si S es un conjunto y κ un cardinal, [S]κ denota lacoleccion de subconjuntos de S con cardinalidad κ; [S]≤κ se usa para lacoleccion de subconjuntos de S con cardinalidad ≤ κ y [S]<κ denotaraa la coleccion de subconjuntos de S con cardinalidad menor que κ.

Sea X un espacio topologico, x ∈ X y A un subconjunto de X. Laclausura de A en X se denota clX(A). Con Vx denotamos al conjuntode abiertos en X que contienen a x. Por otro lado, x es punto deacumulacion completo de A, si para todo U ∈ Vx, se cumple que|A| = |A ∩ U |. Es posible demostrar que en un espacio Hausdorff ycompacto, cualquier conjunto infinito tiene un punto de acumulacioncompleto. Se usa C2 para designar a la clase de los espacios Hausdorffy compactos. Si Xs : s ∈ S es una familia, no vacıa, de espaciostopologicos, denotamos con X su producto cartesiano con la topologıaproducto. Si S0 ⊆ S es no vacıo, XS0 denota al espacio producto

!Xs :

s ∈ S0. Ademas, bajo estas condiciones, prS0 : X → XS0 es la funcionproyeccion.

Las funciones cardinales, estrechez y π−caracter se definen a travesde los siguientes numeros cardinales: la estrechez y el π−caracter delpunto x ∈ X son, respectivamente: t(x,X) = minβ : ∀C ⊆ X conx ∈ clXC, existe B ⊆ C tal que |B| ≤ β y x ∈ clX(B) y πχ(x,X) =min|V| : V es π−base local de x (donde V es una π−base local de

Estrechez y compacidad 53

x ∈ X si cada V ∈ V es abierto no vacıo en X y para cada U ∈ Vx, existeV ∈ V tal que V ⊆ U). La estrechez y el π−caracter del espacio X sedefinen, respectivamente, de la siguiente manera: t(X) = supt(x,X) :x ∈ X + ω y πχ(X) = supπχ(x,X) : x ∈ X + ω. Otra funcioncardinal que usaremos es la amplitud cuya definicion es como sigues(X) = min|A| : A es discreto en X+ ω.

Se dice que una funcion cardinal φ es monotona si para todo Ysubespacio de X, φ(Y ) ≤ φ(X). No es difıcil demostrar que una funcioncardinal φ es monotona si hφ(X) = supφ(Y ) : Y es subespacio deX = φ(X).

3 La estrechez

Un concepto de gran utilidad al trabajar con la estrechez es la κ−clausurade un conjuto: Sea X un espacio topologico, A un subconjunto de Xy κ un numero cardinal; la κ−clausura de A es el conjunto [A]κ =!clX(B) : B ⊆ A y |B| ≤ κ.Un conjunto A con la propiedad de que para todo B ⊆ A con |B| ≤

κ, clX(B) ⊆ A se dice que es un conjunto κ−cerrado.

Proposicion 3.1 Sea X un espacio topologico y κ un numero cardinal.Para todo C ∈ P(X)\∅, C es κ−cerrado si y solo si C = [C]κ.

Demostracion: La necesidad es inmediata. Para demostrar la sufi-ciencia, sea C ⊆ X no vacıo, y sea A ⊆ [C]κ con |A| ≤ κ; por de-mostrar que clX(A) ⊆ [C]κ. Para cada x ∈ A, existe Bx ⊆ C talque x ∈ clX(Bx) y |Bx| ≤ κ. Sea B =

!x∈ABx, entonces B ⊆ C y

|B| ≤"

x∈A |Bx| ≤ κ · κ = κ, de donde clX(B) es uno de los uniendosen [C]κ. Por otra parte, dado que A ⊆

!x∈A clX(Bx) ⊆ clX(B) se tiene

que clX(A) ⊆ clX(B) ⊆ [C]κ; por tanto clX(A) ⊆ [C]κ. Ası [C]κ esκ−cerrado en X.

Proposicion 3.2 Sea f : X → Y una funcion cerrada y κ un numerocardinal. Si C ∈ P(X)\∅ es κ−cerrado en X entonces, f(C) esκ−cerrado en Y .

Demostracion: Sea C ∈ P(X)\∅ κ−cerrado en X y A ⊆ f(C), novacıo, tal que |A| ≤ κ. Queremos demostrar que clY (A) ⊆ f(C). Porcada a ∈ A, elija xa ∈ C tal que f(xa) = a; sea C ′ el conjunto formadopor tales puntos. Entonces C ′ ⊆ C tal que |C ′| ≤ κ; ası, puesto que C


es κ−cerrado en X, clX(C ′) ⊆ C; de donde f(clX(C ′)) ⊆ f(C). Clara-mente A ⊆ f(clX(C ′)); luego, clY (A) ⊆ clY f(clX(C ′)) = f(clX(C ′))(pues f es cerrada). Por tanto clY (A) ⊆ f(clX(C)). Por tanto f(C) esκ−cerrado en Y .

El siguiente resultado es de gran utilidad, permite acotar por arribaa la estrechez de un espacio.

Lema 3.3 Sea X un espacio topologico arbitrario y κ un cardinal.Entonces t(X) ≤ κ si y solo si para todo C ∈ P(X)\∅, clX(C) = [C]κ.

Demostracion: Para probar la necesidad, sea C ∈ P(X)\∅. Puestoque [C]κ ⊆ clX(C); es suficiente con demostrar que clX(C) ⊆ [C]κ.Si x ∈ clX(C), existe B ⊆ C tal que |B| ≤ t(x,X) ≤ t(X) ≤ κ yx ∈ clX(B) ⊆ [C]κ. Por tanto clX(C) ⊆ [C]κ.

Ahora probaremos la suficiencia. Sea C ⊆ X y x ∈ clX(C), porhipotesis, clX(C) = [C]κ, luego, existe B ⊆ C tal que x ∈ clX(B) y|B| ≤ κ. Por tanto t(x,X) ≤ κ, dado que x ∈ X fue arbitrario, se sigueque t(X) ≤ κ.

Corolario 3.4 t(X) ≤ κ si y solo si [C]κ es cerrado en X para todo C ∈P(X)\∅. (Equivalentemente, t(X) ≤ κ si y solo si todo subconjuntoκ−cerrado en X es cerrado en X.)

Teorema 3.5 t(X) es igual al menor cardinal κ con la propiedad deque para cualquier subconjunto no cerrado C de X existe B ⊆ C talque |B| ≤ κ y clX(B)\C = ∅.

Demostracion: Sea β = t(X). Veamos primero que κ ≤ β. Sea C ⊆ Xno cerrado, entonces podemos tomar x ∈ clX(C)\C; luego, existe B ⊆ Ctal que x ∈ clX(B) y |B| ≤ t(x,X) ≤ β. Evidentemente clX(B)\C = ∅.De aquı que β satisface la misma propiedad que κ; por tanto κ ≤ β.

Para verificar que β ≤ κ, suponemos que [C]κ no es cerrado, porhipotesis existe A ⊆ [C]κ tal que |A| ≤ κ y clX(A)\[C]κ = ∅, lo cualcontradice la proposicion 3.1. Por tanto [C]κ es cerrado; ası β ≤ κ.

La estrechez es una funcion monotona y se preserva bajo mapeoscerrados.

Proposicion 3.6 (i) La funcion t es monotona.(ii) Si f : X → Y es continua y cerrada de X sobre Y ,entonces t(Y ) ≤ t(X).


Demostracion: (i) Observe que si C es un subconjunto del subespacioY de X, y κ = t(X), entonces [C]Yκ = Y ∩ [C]κ; donde [C]Yκ es laκ−clausura de C en Y .

(ii) Sea F ∈ P(Y )\∅ y κ = t(X). Demostraremos que [F ]κ escerrado en Y . Denotemos por E al conjunto [F ]κ.

Afirmacion: f−1(E) = [f−1(E)]κ. En efecto, como κ = t(X), dellema 3.3 se tiene que [f−1(E)]κ = clX(f−1(E)), de donde, trivialmentef−1(E) ⊆ [f−1(E)]κ.

Por otro lado, si x ∈ [f−1(E)]κ, entonces existe C ⊆ f−1(E) talque x ∈ clXC y |C| ≤ κ. Puesto que f es continua, f(x) ∈ clY f(C).Ahora bien, f(C) ⊆ E y |f(C)| ≤ κ; luego clY f(C) ⊆ E. Por tantof(x) ∈ E; ası, x ∈ f−1(E) y por tanto [f−1(E)]κ ⊆ f−1(E). De donde,f−1(E) = [f−1(E)]κ. Puesto que [f−1(E)]κ es cerrado en X, de lasobreyectividad de f y el hecho de que f es cerrada se tiene que E escerrado en Y , i.e. [F ]κ es cerrado en Y ; por tanto t(Y ) ≤ κ = t(X).

Proposicion 3.7 Para cualquier espacio X, t(X) ≤ hπχ(X); dondehπχ(X) = supπχ(Y ) : Y es subespacio de X.

Demostracion: Sea x0 ∈ X y C un subconjunto de X tal que x0 ∈clX(C); sea ademas κ = hπχ(X). Como πχ(clX(C)) ≤ κ, existe unaπ−base local B de x0 en clX(C) tal que |B| ≤ κ.

Note que si B ∈ B, entonces B ∩ C = ∅. En efecto, para talB ∈ B, existe UB abierto en X tal que B = UB ∩ clX(C). Ası, siy ∈ B, entonces y ∈ clX(C) y y ∈ UB, por tanto UB ∩ C = ∅; peroUB ∩C ⊆ UB ∩ clX(C) = B. Por tanto B ∩C = ∅. De aquı resulta que,para cada B ∈ B, es posible tomar yB ∈ B ∩C; sea M = yB : B ∈ B.No es difıcil verificar que M ⊆ C, |M | ≤ κ y x0 ∈ clX(M).

La igualdad en t(X) ≤ hπχ(X) se tiene cuando X es Hausdorff ycompacto, lo cual se demostrara en la seccion 4.

Terminamos esta seccion con un resultado que nos dice como encon-trar una cota por arriba para la estrechez de un espacio producto (nonecesariamente formado con espacios Hausdorff y compactos).

Teorema 3.8 Sea Xs : s ∈ S una familia no vacıa de espaciostopologicos (no vacıos) y κ un cardinal. Si para todo S0 ∈ [S]<ω,t(XS0) ≤ κ, y |S| ≤ κ, entonces t(X) ≤ κ.

Demostracion: Sea C ⊆ X y x ∈ clX(C). Queremos demostrar queexiste C0 ⊆ C tal que |C0| ≤ κ y x ∈ clX(C0).


Denotemos por F la coleccion [S]<ω. Entonces |F| ≤ κ. Ahora bien,para cada F ∈ F , sea XF =

!s∈F Xs y sea prF : X → XF el mapeo

proyeccion sobre la cara XF .Puesto que t(XF ) ≤ κ y prF (x) ∈ prF (clXC) ⊆ clXF (prF (C)) =

[prF (C)]κ, entonces, para todo F ∈ F , existe MF ⊆ prF (C) tal que|MF | ≤ κ y prF (x) ∈ clXF (MF ); ahora bien, para cada MF , cons-truyase DF ⊆ C como sigue: por cada (xs)s∈F ∈ MF , elıjase un puntoen pr−1

F ((xs)s∈F ) ∩ C. Claramente, |DF | ≤ κ para todo F ∈ F .Considerese ahora C0 =

"F∈F DF . Puesto que DF ⊆ C para todo

F ∈ F , se tiene que C0 ⊆ C. Mas aun, como |F| ≤ κ y |DF | ≤ κ paratodo F ∈ F , se tiene que |C0| ≤ κ.

Por ultimo, veamos que, x ∈ clX(C0).Efectivamente, sea U un abierto canonico en X que contiene a x,

digamos U =!

s∈S Us, donde, para s ∈ S, Us = Xs salvo un numerofinito de ındices s1,...,sn. Entonces A = s1, ..., sn ∈ F ; luego, dadoque prA(x) ∈ clXA(MA), existe (xs1 , ..., xsn) ∈ MA ∩ (prA(C) ∩ (Us1 ×· · ·×Usn)). Para tal (xs1 , ..., xsn) existe y ∈ DA, y por tanto en C0, talque prA(y) = (xs1 , ..., xsn). De aquı, ya es claro que U ∩ C0 = ∅. Portanto x ∈ clX(C0). Ası t(X) ≤ κ.

4 Estrechez y compacidad

En la presente seccion daremos algunos resultados de la estrechez sobrela clase C2 de espacios Hausdorff y compactos; en particular, probaremoslas igualdades senaladas en el resumen.

Definicion 4.1 Una sucesion xα : 0 ≤ α < κ en el espacio X esuna sucesion libre de longitud κ si para todo β < κ: clXxα : α <β ∩ clXxα : α ≥ β = ∅.

Observe que si xα : 0 ≤ α < κ es una sucesion libre de longitud ken X, entonces xα : 0 ≤ α < κ es un subconjunto discreto en X.

Teorema 4.2 Si X ∈ C2 y t(X) ≤ κ, entonces X no tiene una sucesionlibre de longitud κ+.

Demostracion: Supongamos que xα : 0 ≤ α < κ+ es una sucesionlibre de longitud κ+. Puesto que X es compacto, el conjunto xα : 0 ≤α < κ+ tiene un punto de acumulacion completo, digamos z0. Por otraparte, dado que t(X) ≤ κ existe β0 < κ+ tal que z0 ∈ clXxα : 0 ≤


α < β0; luego, dado que clXxα : α < β0 ∩ clXxα : α ≥ β0 = ∅,existe una vecindad abierta U de z0 tal que U ∩ xα : α ≥ β0 = ∅.De aquı y el hecho de que U ∩ xα : α < β0 = ∅, se tiene que|U ∩ xα : α < β0| < κ+, lo cual contradice que z0 es punto de acumu-lacion completo del conjunto xα : 0 ≤ α < κ+.

El lector puede consultar la prueba del siguiente resultado en [3].

Lema 4.3 Sea X ∈ T3, K un subconjunto compacto de X y p ∈ X\K.Entonces existen conjuntos cerrados y Gδ, A y B en X tales que p ∈ A,K ⊆ B y A ∩B = ∅.

Teorema 4.4 Si X ∈ C2 y hπχ(X) > κ, entonces X tiene una sucesionlibre de longitud κ+.

Demostracion: Puesto que πχ(Y ) ≤ πχ(Y ), para demostrar el resultadoes suficiente suponer que πχ(X) > κ. Sea p ∈ X tal que πχ(p,X) ≥ κ+.Sea G la coleccion de todos los conjuntos cerrados, no vacıos y Gδ enX. La coleccion G tiene la siguiente propiedad:

(∗) si H ⊆ G y |H| ≤ κ, entonces existe una vecindad abierta R dep tal que H\R = ∅ para todo H ∈ H.

En efecto, supongamos que la coleccion G no satisface (∗); i.e., existeH ⊆ G con |H| ≤ κ tal que para toda vecindad abierta R de p existeHR ∈ H de tal forma que HR ⊆ R = ∅. Ahora, puesto que X escompacto, para cada HR existe UHR abierto en X de tal forma queHR ⊆ UHR ⊆ R. Entonces la coleccion UHR : HR ∈ H es una π−basede p con |UHR : HR ∈ H| ≤ κ. Lo cual contradice el hecho de queπχ(p,X) > κ.

Para continuar, construiremos subcolecciones Aα : 0 ≤ α < κ+ yBα : 0 ≤ α < κ+ de G tales que:

(1) p ∈ Aα y Aα ∩Bα = ∅, 0 ≤ α < κ+;(2) Si H = ∅ es interseccion finita de elementos de Aβ : 0 ≤ β <

α ∪ Bβ : 0 ≤ β < α, entonces H ∩Bα = ∅, 0 < α < κ+.La construccion es por induccion transfinita. Para obtener A0 y B0,

sea K compacto en X tal que p /∈ K; por el lema 4.3, existen A0, B0 ∈ Gtales que p ∈ A0, K ⊆ B0 y A0 ∩B0 = ∅. Ahora sea α fijo, 0 < α < κ+,y supongamos que Aβ : β < α y Bβ : β < α se tienen construidosde tal modo que (1) y (2) se verifican. Por construir Aα y Bα. Sea Hla coleccion de todas las intersecciones finitas no vacıas de elementos deAβ : β < α ∪ Bβ : β < α. Entonces H ⊆ G, H = ∅ y |H| ≤ κ,


ası que por (∗), existe una vecindad abierta R de p tal que H\R = ∅para todo H ∈ H. Como X\R es cerrado en el compacto X, entoncesX\R es compacto y p /∈ X\R; luego, por el lema 4.3, existen Aα, Bα

en G tales que p ∈ Aα, (X\R) ⊆ Bα y Aα ∩ Bα = ∅. Ahora H\R = ∅implica que H ∩Bα = ∅, de donde (1) y (2) se verifican. Ası termina laconstruccion.

Ahora bien, usando las partes (1) y (2) e induccion finita, se puededemostrar que para cada α fijo, 0 < α < κ+, la coleccion Aβ : β ≤α+ ∪ Bβ : β > α+ satisface la propiedad de la interseccion finita(i.e. cualquier subconjunto finito de dicha coleccion tiene interseccionno vacıa).

De lo anterior y la compacidad de X, tenemos que para cada α,existe xα ∈ (

!β≤αAβ) ∩ (

!β>αBβ). Entonces xα : 0 ≤ α < κ+ es

una sucesion libre en X de longitud κ+. Por supuesto, sea ε < κ+. Essuficiente demostrar que xα : α < ε ⊆ Bε y xα : α ≥ ε ⊆ Aε; puespor la segunda parte de (1) y el hecho de que Aα y Bα son cerrados yajenos, para todo α < κ+, tendrıamos que clX(xα : α < ε)∩clX(xα :α ≥ ε) = ∅. Para ver la primera contencion, note que si α < ε,entonces xα ∈ (

!β≤αAβ) ∩ (

!β>αBβ) ⊆ Bε. Para la segunda, tenemos

que xα ∈ (!

β≤αAβ) ∩ (!

β>αBβ), xα ∈ Aβ para β ≤ α, de donde, siα ≥ ε, xα ∈ Aε.

Corolario 4.5 Si X ∈ C2 y t(X) > κ, entonces X tiene una sucesionlibre de longitud κ+.

Demostracion: Como κ < t(X) y t(X) ≤ hπχ(X), entonces hπχ(X)> κ y, por tanto, X tiene una sucesion libre de longitud κ+.

Como se observa en el teorema 4.2 y el corolario 4.5, existe unarelacion entre la estrechez de un espacio Hausdorff y compacto y lalongitud de las sucesiones libres en el, a saber:

Teorema 4.6 (Arkhangel’skiı) Para X ∈ C2, t(X) = F (X), dondeF (X) = supλ : X tiene una sucesion libre de longitud λ+ ω.

Demostracion: Sea κ = t(X) y β = F (X). Si κ < β entonces X tieneuna sucesion libre de longitud κ+. Pero esto contradice 4.2. Por tantoκ ≥ β. Si κ > β, entonces, por el corolario 4.5, X tiene una sucesionlibre de longitud β+; lo cual contradice la definicion de β. Por tantoκ = β.


Teorema 4.7 (Sapirovskı) Para X ∈ C2, t(X) = hπχ(X). En particu-lar, todo subespacio de un espacio Hausdorff y compacto con estrecheznumerable tiene π−caracter numerable.

Demostracion: Puesto que t(X) ≤ hπχ(X), es suficiente demostrar queno ocurre que t(X) < hπχ(X). Si t(X) < hπχ(X), entonces, por el teo-rema 4.4, X tiene una sucesion libre de longitud t(X)+, contradiciendoel teorema anterior.

Teorema 4.8 (Arkhangel’skiı) Para X ∈ C2, t(X) ≤ s(X)+. En par-ticular, todo espacio compacto con amplitud numerable tiene estrecheznumerable.

Demostracion: Sea κ = s(X). Si t(X) > κ, entonces por el coro-lario 4.5, X tiene una sucesion libre de longitud κ+. Puesto que todasucesion libre es un conjunto discreto, entonces X tiene un subconjuntodiscreto de cardinalidad > s(X); lo cual es una contradiccion. Por tantot(X) ≤ s(X).

Suponga que Xs : s ∈ S es una coleccion, no vacıa, de espaciostopologicos. Supongase ademas que X es el espacio producto de dichosespacios. Bajo el supuesto de que D(2)|S| → X (donde D(2)|S| es elcubo de Cantor de peso |S|, vea [2], pg. 84) o F (2)|S| → X (dondeF (2)|S| es el cubo de Alexandroff de peso |S|, vea [2], pg. 84), es posibledemostrar que t(X) ≥ |S| · tS(X), donde tS(X) = t(Xs) : s ∈ S. Engeneral, la igualdad no siempre se da, aun en el caso finito:

Ejemplo 4.9 Considere κ un cardinal arbitrario con la topologıa dis-creta y el espacio compacto Z = 1

n : n ∈ ω∪0. En el producto κ×Z,identificamos los puntos de la forma (α, 0), para α ∈ κ; y denotamospor Vκ al espacio cociente resultante. Entonces, para todo cardinal κ,t(Vκ) = ω. Por otro lado, t(Vω × Vc) > ω (consulte [1]).

Para el caso en que cada Xs ∈ C2, la estrechez verifica la igualdaden t(X) ≥ |S| · tS(X). La prueba requiere de algunos resultados que acontinuacion se dan.

El siguiente resultado es una generalizacion del lema de la pagina113 de la referencia [4].

Lema 4.10 Si X ∈ T1 y Y es Hausdorff y localmente compacto,entonces t(X × Y ) ≤ t(X) · t(Y ).


Demostracion: Sea κ = t(X) · t(Y ) y A un subconjunto κ−cerrado enX × Y . Sea (x, y) ∈ clX×Y (A). Demostraremos que (x, y) ∈ A.

Sea B = (x× Y ) ∩A. Note que para demostrar que (x, y) ∈ A essuficiente con probar que y ∈ prY (B).

Afirmamos que B es cerrado en x× Y . En efecto, si B = ∅, nadaque probar.

Supongamos, pues, que B no es vacıo. Entonces B es κ−cerrado enx×Y . Por supuesto, si C ⊆ B tal que |C| ≤ κ, entonces C ⊆ x×Y ,C ⊆ A y |B| ≤ κ. Claramente, clx×Y C ⊆ (x × Y ); por otro lado,puesto que C ⊆ A y |C| ≤ κ, se tiene que clX×Y (C) ⊆ A; de dondeclx×Y C = clX×Y (C) ∩ (x × Y ) ⊆ A ∩ (x × Y ) = B. Ası, B esκ−cerrado en x× Y . Ahora bien, puesto que t(x× Y ) = t(Y ) ≤ κ,se tiene que B es cerrado en x× Y (y por tanto, cerrado en X × Y ).

Veamos, pues, que y ∈ prY (B). Supongamos, por el contrario, quey /∈ prY (B). Como B es cerrado en x × Y y prY : x × Y → Yes homeomorfismo, entonces prY (B) es un subespacio cerrado de Y .Sea V una vecindad compacta de y en Y que no intersecta a prY (B).Entonces X×V es una vecindad cerrada de (x, y) en X×Y ; por lo cual(x, y) ∈ clX×Y ((X×V )∩A). En particular, (x, y) ∈ clX×V [(X×V )∩A].

Ahora bien, como X × V es cerrado en X × Y y A es κ−cerrado enX × Y , entonces (X × V ) ∩ A es κ−cerrado en X × Y . En particular,(X×V )∩A es κ−cerrado enX×V . Como V es compacto en Y , entoncesprX : X × V → X, es cerrada, y por lo tanto prX((X × V ) ∩ A) es κcerrado en X. Ası, por la continuidad de prX : X ×V → X, obtenemosque x ∈ prX(clX×V [(X×V )∩A]) ⊆ clX [prX((X×V )∩A)] = prX((X×V ) ∩A). Por tanto, existe r ∈ V tal que (x, r) ∈ (x× V ) ∩A ⊆ B; loque significa que r ∈ V ∩ prY B; lo cual es una contradiccion. Por tantoy ∈ prY B. La prueba esta completa.

La siguiente proposicion es consecuencia inmediata del lema anterior.

Proposicion 4.11 Si para cada i ∈ 1, ..., n, Xi ∈ C2, entoncest(!n

i=1Xi) ≤ maxt(Xi) : i ∈ 1, ..., n.

Ahora daremos la prueba del caso general (en C2, por supuesto).

Teorema 4.12 Sea Xs : s ∈ S una familia no vacıa de espaciostopologicos (no vacıos). Si cada Xs ∈ C2, entonces t(X) = |S| · tS(X).

Demostracion: Sea κ = |S|·tS(X). Sea A = [C]κ, donde C ∈ P(X)\∅,y sea x ∈ clX(A). Puesto que cada Xs es compacto se tiene que,


para cualquier S0 ∈ [S]<ω, prS0 es una funcion cerrada; luego, porla proposicion 3.2, para cualquier S0 ∈ [S]<ω, prS0(A) es κ− cerradoen XS0 . Ası, dado que tS0(X) ≤ κ, se tiene que prS0(A) es cerrado enXS0 . Consecuentemente, para cada S0 ∈ [S]<ω prS0(x) ∈ prS0(A), dedonde se deduce que existe un punto a ∈ A tal que x |S0= a. Sea Bel conjunto formado por estos puntos, entonces B ∈ [A]≤κ, puesto que|[S]<ω| = |S| ≤ κ. Luego, claramente se tiene que x ∈ clXB ⊆ A. Porla definicion de B y el hecho de que A es κ−cerrado.

AgradecimientosA mi novia Homaira Athenea Ramırez Gutierrez, a quien ademas,

dedico este trabajo.Quiero agradecer a Jesus F. Tenorio Arvide su revision y sugerencias aeste trabajo.

Alejandro Ramırez ParamoFacultad de Ciencias Fısico Matematicas,Benemerita Universidad Autonoma de Puebla,Av. San Claudio y Rio Verde s/n,Puebla Pue., MEXICO,[email protected].

Referencias

[1] Arkhangel’skiı, A. V., Structure and Classification of TopologicalSpaces and Invariants, Russian Math Survey, 33 (1978), 33-96.

[2] Engelking, R., General Topology, Heldermann Verlag Berlin 1989.

[3] Hodel, R., Cardinal Functions in Topology I, Handbook of line-break Set-Theoretic Topology, K. Kunen and J. E. Vaughan, eds.,linebreak Amsterdam, 1984.

[4] Juhasz, I., Cardinal Functions in Topology (ten years later), Hel-dermann Verlag Berlin, 1989.



Medida de colision de un (α, d, β)-superprocesocon su medida inicial ∗

Jose Villa Morales 1

Resumen

Sea X = Xt : t ≥ 0 un (α, d,β)-Superproceso. Demostraremosque la medida de colision, M(X), de X con su valor inicial X0,definida heurısticamente por Mt(X)(ϕ) := ⟨ϕ((y + x)/2)δ(y −x), X0(dy)Xt(dx)⟩ existe. Mas precisamente si Jε, con ε > 0, es unmolificador, entonces limε→0⟨ϕ((y+x)/2)Jε(y−x), X0(dy)Xt(dx)⟩existe en probabilidad para cada ϕ ∈ Cb(Rd)+.

1991 Mathematics Subject Clasification: 60J55; 60G17.Keywords and phrases: (α, d,β)-superprocesos, medidas de colision.

1 Introduccion

En este artıculo demostraremos la existencia de la medida de colisionde un (α, d,β)-superproceso con su valor inicial. Intuitivamente estosignifica que al tiempo t el soporte de Xt no esta muy alejado del soportede X0. Nuestro problema es mas sencillo en relacion con lo hecho porMytnik en [9] y se utilizan las mismas ideas empleadas por este autor (elcual a su vez se basa en los trabajos [3] y [4]). Mytnik demuestra que siX(i), con parametros (αi, d,βi), i = 1, 2, son dos (α, d,β)-superprocesosindependientes, entonces la medida de colision y el tiempo local de coli-sion existen si d < α1/β1 + α2/β2 y d < α1/β1 + α2/β2 +maxα1,α2,respectivamente. El caso en que α = 2 y β = 1 es estudiado porEvans y Perkins en [2], por ejemplo. Es de hacer notar que cuando hay

∗Apoyo parcial del proyecto PIM99-ln de la UAA. El contenido de este trabajorepresenta parte de la tesis de doctorado que el autor se encuentra desarrollandodentro del programa de doctorado del CIMAT.

1Estudiante inscrito en el programa de doctorado del CIMAT.

63

64 Jose Villa Morales

independencia la existencia del lımite anterior depende de α, d y β, sinembargo veremos que en nuestro caso no es ası.

1.1 Notacion

• Sea E un espacio metrico y B(E) la σ-algebra de Borel de E.DE [0,∞) es el conjunto de las funciones E-valuadas definidas en[0,∞) que son continuas por la derecha y tienen lımite por laizquierda (funciones cadlag), con la topologıa de Skorohod.

• λ denota la media de Lebesgue en Rd.

• MF (Rd) denota el espacio de las medidas no negativas finitas enRd, con la topologıa debil.

• Sea B(Rd) (respectivamente Cb(Rd)) el conjunto de las funcionesBorel medibles acotadas en Rd (respectivamente continuas y aco-tadas). B(Rd)+ ⊂ B(Rd) (resp. Cb(Rd)+ ⊂ Cb(Rd)) denota alconjunto de las funciones no negativas en B(Rd) (resp. Cb(Rd)).

• ⟨f, µ⟩ =!f(x)µ(dx), ∥f∥∞ = supx |f(x)| y ||f ||1 = ⟨|f | ,λ⟩.

1.2 (α, d, β)-superprocesos

Un proceso X lo llamaremos (α, d,β)-superproceso, 0 < β ≤ 1 y 0 <α ≤ 1, si X es un proceso de Markov MF (Rd)-valuado homogeneo enel tiempo con trayectorias en DMF (Rd) [0,∞), tal que para cada f ∈B(Rd)+

Eµ

"e−⟨f,Xt⟩

#= e−⟨Vt(f),µ⟩,(1)

donde µ ∈ MF (Rd) y Vt (f) es la solucion de la ecuacion

Vt (f) = St(f)−$ t

0St−s(Vs(f)

1+β)ds, t ≥ 0.(2)

Aquı St : t ≥ 0 denota el semigrupo de un proceso de Markov enRd, estable y simetrico con exponente α. pt(x, y) = pt(y − x) es ladensidad de transicion de probabilidad correspondiente a St, es decir,para cada f ∈ L1

+(Rd),

Stf(x) =$

Rdf(y)pt(y − x)dy, t > 0.

Medida de Colision 65

St : t ≥ 0 es un semigrupo de contraccion, tal que Stf ≥ Vtf (ver[6], p. 93).

Es conocido (ver, por ejemplo, [6]) que la forma diferencial!

∂Vt∂t = − (−∆)α/2 − V 1+β

t ,V0 = f,

∆ ="d

i=1 ∂2/∂x2i , de la ecuacion (2) tiene una unica solucion. Por lo

tanto (2) tambien tiene una unica solucion.

Una manera de probar la existencia de un (α, d,β)-superproceso escomo lımite debil de un sistema de partıculas con ramificacion. En estecontexto St : t ≥ 0 es el semigrupo del proceso de Markov que deter-mina el movimiento de las partıculas. Por otra parte, la ramificacion delas partıculas, es decir, el numero de hijos que tiene cada partıcula, estadado por una distribucion crıtica que pertenece al dominio de atraccionde una ley estable de exponente 1 + β (ver [8]).

1.3 Medida de colision de Xt con X0 y resultado principal

Formalmente la medida de colision de Xt con X0 es

⟨ϕ ((y + x)/2) δ (y − x) , X0(dy)Xt(dx)⟩ , t > 0,(3)

donde δ es la delta de Dirac. (3) mide intuitivamente la colision del(α, d,β)-superproceso X en el tiempo t con la medida inicial X0.

Para hacer preciso (3) introduzcamos la siguiente notacion. Sea Juna funcion continua, no negativa, simetrica, con soporte contenido enla bola unitaria x ∈ Rd : ∥x∥ < 1 y tal que

#Rd J(x)dx = 1. La

funcionJε(x) := ε−dJ(ε−1x), ε > 0,

se llama molificador.

Definicion 1.3.1 Sea X un proceso cadlag MF (Rd)-valuado. Para ε >0 definimos M ε : DMF (Rd) [0,∞) → DMF (Rd) [0,∞) por

M εt (X)(ϕ) = ⟨ϕ ((y + x)/2) Jε (y − x) , X0(dy)Xt(dx)⟩

=$

Rd

$

Rdϕ ((y + x)/2) Jε (y − x)X0(dy)Xt(dx),


para cada ϕ ∈ B(Rd)+. La medida de colision de X con su valor inicialX0 es un proceso M(X) progresivamente medible MF (Rd)-valuado talque M ε

t (X)(ϕ) converge a Mt(X)(ϕ) en probabilidad, cuando ε → 0,para cada t > 0 y cada ϕ ∈ Cb(Rd)+. El lımite no depende de laeleccion del molificador Jε.

El resultado principal es el:

Teorema 1.3.2 Sea X un (α, d,β)-superproceso tal que la medida ini-cial µ es absolutamente continua con respecto a λ y con densidad aco-tada, g := dµ/dλ ∈ B(Rd)+. Entonces la medida de colision M(X)existe y

E!e−Mt(X)(ϕ)

"= e−⟨Vt(ϕg),µ⟩,

para cada t > 0 y cada ϕ ∈ Cb(Rd)+.

2 Demostracion del Teorema

Comenzaremos por demostrar una serie de lemas antes de dar la pruebadel resultado principal. Un resultado que sera esencial es el siguiente.

Lema 2.0.3 Sea ϕ ∈ B(Rd), entonces

limε→0

|| ⟨ϕ ((y + ·)/2) Jε (y − ·) , µ(dy)⟩ − ϕ(·)g(·)||1 = 0

Demostracion: Por el teorema de cambio de variable tenemos que

⟨ϕ ((y + x)/2) Jε (y − x) , µ(dy)⟩ =#

Rd

1

εdJ ((y − x)/ε)ϕ ((y + x)/2)

g(y)dy

=#

Rdϕ(x+ εy/2)g(x+ εy)J(y)dy.

Ya que µ ∈ MF (Rd), entonces g ∈ L1(Rd), ası por el Teorema de Fubiniresulta

∥⟨ϕ ((y + ·)/2) Jε (y − ·) , µ(dy)⟩ − ϕ(·)g(·)∥1=

#

Rd|#

Rdϕ(x+ εy/2)g(x+ εy)J(y)dy −

#

Rdϕ (x) g(x)J(y)dy|dx

≤#

Rd

#

Rd|ϕ(x+ εy/2)g(x+ εy)− ϕ (x) g(x)|J(y)dydx

=#

Rd||ϕ(·+ εy/2)g(·+ εy)− ϕ(·)g(·)||1J(y) dy.


Para mostrar la convergencia usaremos el teorema de la convergenciadominada de Lebesgue. Para esto comencemos notando que g ∈ L1 yϕ ∈ B(Rd) implican que ϕg ∈ L1, pues ∥ϕg∥1 ≤ ∥ϕ∥∞∥g∥1. Entonceslimz→0 ∥g(·+ z)− g(·)∥1 = 0 y limz→0 ∥ϕ(·+ z)g(·+ z)−ϕ(·)g(·)∥1 = 0(ver, por ejemplo, el Teorema 0.13 de [5]). Ası de la estimacion

||ϕ(·+ εy/2)g(·+ εy)− ϕ(·)g(·)||1≤ ||ϕ(·+ εy/2)(g(·+ εy)− g(·))||1

+||ϕ(·+ εy/2)(g(·)− g(·+ εy/2))||1+||ϕ(·+ εy/2)g(·+ εy/2)− ϕ(·)g(·)||1

≤ ∥ϕ∥∞ ∥g(·+ εy)− g(·)∥1 + ∥ϕ∥∞ ||g(·+ εy/2)− g(·)||1+||ϕ(·+ εy/2)g(·+ εy/2)− ϕ(·)g(·)||1,

vemos que el integrando · converge a cero. Y ademas

||ϕ(·+ εy/2)g(·+ εy)− ϕ(·)g(·)||1≤ ||ϕ(·+ εy/2)g(·+ εy)||1 + ∥ϕ(·)g(·)∥1≤ 2 ∥ϕ∥∞ ∥g∥1 .

Lo que demuestra la afirmacion.

Lema 2.0.4 Sea 1 ≤ k ≤ N y τ = T/N . Sean ϕi ∈ B(Rd)+, i = 1, 2,entonces

! kτ

(k−1)τ||Vt (ϕ1)

1+β − Vt (ϕ2)1+β ||1dt

≤ 2! kτ

(k−1)τ||St (ϕ1 + ϕ2) ||β∞||St (ϕ1 − ϕ2) ||1dt

+2! kτ

(k−1)τ||St (ϕ1 + ϕ2) ||β∞dt

! kτ

0||Vs (ϕ1)

1+β − Vs (ϕ2)1+β ||1ds.

Demostracion: De la desigualdad |x1+β − y1+β | ≤ 2|x − y|(x + y)β ,x, y ≥ 0, obtenemos

! kτ

(k−1)τ||Vt (ϕ1)

1+β − Vt (ϕ2)1+β ||1dt

≤! kτ

(k−1)τ

!

Rd2| (Vt (ϕ1)− Vt (ϕ2)) (x)| (Vt(ϕ1) + Vt(ϕ2))

β (x)dxdt


(Usando el Lema 2.1 de [6] y el hecho de que St es un semigrupo lineal,resulta)

≤ 2! kτ

(k−1)τ

!

Rd| (Vt (ϕ1)− Vt (ϕ2)) (x)|St(ϕ1 + ϕ2)

β(x)dxdt

≤ 2! kτ

(k−1)τ||St(ϕ1 + ϕ2)||β∞

!

Rd| (Vt (ϕ1)− Vt (ϕ2)) (x)|dxdt.

Por otra parte, de la relacion (2) vemos que

|Vt(ϕ1)−Vt(ϕ2)| ≤ |St(ϕ1 −ϕ2)|+! t

0|St−s(Vs(ϕ1)

1+β −Vs(ϕ2)1+β)|ds.

Ası

! kτ

(k−1)τ||Vt(ϕ1)

1+β − Vt(ϕ2)1+β ||1dt

≤ 2! kτ

(k−1)τ||St(ϕ1 + ϕ2)||β∞

!

Rd(|St(ϕ1 − ϕ2)(x)|

+! t

0|St−s(Vs(ϕ1)

1+β − Vs(ϕ2)1+β)(x)|ds)dxdt.

Por ser St una contraccion y t ≤ kτ , implica que

! kτ

(k−1)τ||Vt(ϕ1)

1+β − Vt(ϕ2)1+β ||1dt

≤ 2! kτ

(k−1)τ||St(ϕ1 + ϕ2)||β∞(||St(ϕ1 − ϕ2)||1

+! kτ

0||Vs(ϕ1)

1+β − Vs(ϕ2)1+β||1ds)dt.

Con lo cual la afirmacion queda demostrada.

Una consecuencia de los lemas anteriores es el siguiente resultado.

Lema 2.0.5 Para toda T > 0

limε→0

! T

0∥Vt (⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩)1+β − Vt(ϕ(·)g(·))1+β∥1dt

= 0.


Demostracion: Sea N =!2β+1∥ϕ∥β∞∥g∥β∞T

"+ 1 ([·] funcion mayor

entero) y sea τ = T/N . Por el Lema 2.0.4 obtenemos

# T

0∥Vt (⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩)1+β − Vt(ϕ(·)g(·))1+β∥1dt(4)

=N$

k=1

# kτ

(k−1)τ∥Vt (⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩)1+β

−Vt(ϕ(·)g(·))1+β∥1dt

≤N$

k=1

2# kτ

(k−1)τ∥St (⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩+ ϕ(·)g(·)) ∥β∞

·∥St (⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩ − ϕ(·)g(·)) ∥1dt

+N$

k=1

2# kτ

(k−1)τ∥St (⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩+ ϕ(·)g(·)) ∥β∞

·# kτ

0∥Vs (⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩)1+β

−Vt(ϕ(·)g(·))1+β∥1dt.

Ahora notese que

∥St (⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩+ ϕ(·)g(·)) ∥∞(5)

= supx

|St (⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩+ ϕ(·)g(·)) (x)|

= supx

#

Rdpt(z − x) (⟨ϕ ((y + z)/2) Jε(y − z), µ(dy)⟩+ ϕ(z)g(z)) dz

≤ ∥ϕ∥∞∥g∥∞ supx

#

Rdpt(z − x)(

#

RdJε(y − z)dy + 1)dz

= 2∥ϕ∥∞∥g∥∞.

Usando (4) y el hecho que St es una contraccion nos queda

lim supε→0

# kτ

(k−1)τ∥St(⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩(6)

+ϕ(·)g(·))∥β∞∥St(⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩−ϕ(·)g(·))∥1dt

≤ lim supε→0

2β∥ϕ∥β∞∥g∥β∞# kτ

(k−1)τ∥ ⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩

−ϕ(·)g(·)∥1dt


= 2β∥ϕ∥β∞∥g∥β∞τ lim supε→0

∥ ⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩

−ϕ(·)g(·)∥1= 0.

La ultima igualdad es debido al Lema 2.0.3.De esta forma sea demostrado que cuando tomamos el lim supε→0

en (4) la primera serie converge a cero. Ahora consideremos la segundaserie. Para k ≥ 1 definamos

bk = lim supε→0

! kτ

0∥Vs (⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩)1+β(7)

−Vt(ϕ(·)g(·))1+β"""1ds,

y b0 = 0. Supongamos que bk = 0 y veamos que bk+1 = 0. Del Lema2.0.4 y (6) resulta que

bk+1 = bk + lim supε→0

! (k+1)τ

kτ∥Vs (⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩)1+β

−Vs(ϕ(·)g(·))1+β∥1ds

≤ lim supε→0

2! (k+1)τ

kτ∥St(⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩

+ϕ(·)g(·))∥β∞∥St(⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩−ϕ(·)g(·))∥1dt

+ lim supε→0

2! (k+1)τ

kτ∥St(⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩

+ϕ(·)g(·))∥β∞! (k+1)τ

0∥Vs (⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩)1+β

−Vs(ϕ(·)g(·))1+β∥1ds

≤ 0 + 2β+1∥ϕ∥β∞∥g∥β∞τ lim supε→0

! (k+1)τ

0∥Vs(⟨ϕ ((y + ·)/2)

·Jε(y − ·), µ(dy)⟩)1+β − Vs(ϕ(·)g(·))1+β∥1ds.

En la ultima desigualdad hemos usado (5). Ası es que bk+1 ≤ 2β+1∥ϕ∥β∞·∥g∥β∞τ bk+1. Pero hemos elegido N de modo que 2β+1∥ϕ∥β∞∥g∥β∞τ < 1,entonces bk+1 = 0.

Para terminar notemos que al tomar lim supε→0 en la segunda suma-toria de (4) y usando (5), como antes, y (7) obtenemos la prueba delresultado.


Lema 2.0.6 Para toda t > 0

limε→0

∥Vt (⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩)− Vt(ϕ(·)g(·))∥1 = 0.

Demostracion: Usando (2) y que St es una contraccion nos da

limε→0

∥Vt (⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩)− Vt(ϕ(·)g(·))∥1≤ lim

ε→0∥St (⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩ − ϕ(·)g(·)) ∥1

+ limε→0

! t

0∥St−s(Vs (⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩)1+β

−Vs(ϕ(·)g(·))1+β)∥1ds≤ lim

ε→0∥⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩ − ϕ(·)g(·)∥1

+ limε→0

! t

0∥Vs (⟨ϕ ((y + ·)/2) Jε(y − ·), µ(dy)⟩)1+β

−Vs(ϕ(·)g(·))1+β∥1ds.

Por los Lemas 2.0.3 y 2.0.5 tenemos que los dos sumandos de la derechade la desigualdad son cero. En consecuencia el lımite es cero.

Finalmente veamos el siguiente resultado.

Lema 2.0.7 Sea Zε : 0 ≤ ε ≤ 1 una familia de procesos MF (Rd)-va-luados progresivamente medibles tal que

limε,ε′→0

E

"#e−Zε

t (ϕ) − e−Zε′t (ϕ)

$2%

= 0,

para toda t > 0 y cada ϕ ∈ Cb(Rd)+. Entonces existe un proceso Zprogresivamente medible MF (Rd)-valuado tal que Zε

t converge en proba-bilidad a Zt, cuando ε → 0, para toda t > 0.

Demostracion: Ya que&e−Zε

t (ϕ)'

es una sucesion de Cauchy en L2,

entonces existe una variable aleatoria Yt(ϕ) tal que e−Zεt (ϕ)−→Yt(ϕ)

en L2, cuando ε → 0. Por lo tanto e−Zεt (ϕ)−→Yt(ϕ) en probabilidad,

cuando ε → 0. Ası Zεt (ϕ) → Zt(ϕ) := − log Yt(ϕ), ε → 0, pues − log(·)

es una funcion continua. Es decir, para cada, t > 0 Zt es una variablealeatoria MF (Rd)-valuada tal que

limε→0

Pω : d(Zεt (ω), Zt(ω)) > δ = 0,


para cada δ > 0, donde d es una metrica enMF (Rd), la cual correspondea la topologıa debil (ver Dawson (1991), p. 42).

Ahora veamos que Z = Zt : t > 0 es progresivamente medible.Para cada T > 0 consideremos el espacio de medida ([0, T ]×Ω,B([0, T ])×FT ,λ × P ). Sea δ > 0, entonces por el teorema de Fubini y el teoremade la convergencia dominada de Lebesgue tenemos que

limε→0

(λ× P )d(Zεt , Zt) > δ = lim

ε→0

!

[0,T ]×Ω1d(Zε

t ,Zt)>δ(ω, t)dPdt

= limε→0

! T

0Pd(Zε

t , Zt) > δdt

=! T

0limε→0

Pd(Zεt , Zt) > δdt = 0.

Es decir Zε→Z en λ× P , cuando ε → 0. Ya que Zε es B([0, T ])× FT -medible, entonces tambien lo sera Z (ver Teorema 5.2.9 de [7]). Ası elproceso Z es progresivamente medible.

Demostracion: (del Teorema 1.3.2) Comencemos por observar queM ε

t (X)(ϕ) es igual a ⟨⟨ϕ ((y + x)/2) Jε(y − x), X0(dy)⟩ , Xt(dx)⟩, en-tonces

E"e−Mε

t (X)(ϕ)#= E

"e−⟨⟨ϕ((y+x)/2)Jε(y−x),X0(dy)⟩,Xt(dx)⟩

#.

Ahora tomando f(x) = ⟨ϕ ((y + x)/2) Jε(y − x), X0(dy)⟩ ∈ B(Rd)+(mas aun ∥f∥∞ ≤ ∥ϕ∥∞∥g∥∞µ(Rd)) y µ = X0 en (1) nos da

E"e−Mε

t (X)(ϕ)#= e−⟨Vt(⟨ϕ((y+x)/2)Jε(y−x),X0(dy)⟩),X0(dx)⟩.(8)

Analogamente vemos que

E$e−(Mε

t (X)(ϕ)+Mε′t (X)(ϕ))

%

= e−⟨Vt(⟨ϕ((y+x)/2)(Jε(y−x)+Jε′ (y−x)),X0(dy)⟩),X0(dx)⟩.

En consecuencia

E

&'e−Mε

t (X)(ϕ) − e−Mε′t (X)(ϕ)

(2)

= e−⟨Vt(2⟨ϕ((y+x)/2)Jε(y−x),µ(dy)⟩),µ(dx)⟩

+e−⟨Vt(2⟨ϕ((y+x)/2)Jε′ (y−x),µ(dy)⟩),µ(dx)⟩

−2e−⟨Vt(⟨ϕ((y+x)/2)(Jε(y−x)+Jε′ (y−x)),µ(dy)⟩),µ(dx)⟩.


Usando la desigualdad |e−a − e−b| ≤ |a − b|, a, b ∈ R, tenemos por elLema 2.0.6 que

limε,ε→0

E

!"e−Mε

t (X)(ϕ) − e−Mε′t (X)(ϕ)

#2$

= 0.

Entonces por el Lema 2.0.7 existe un proceso M(X) progresivamentemedible yMF (Rd)-valuado, tal queM ε

t (X)(ϕ) converge en probabilidada Mt(X)(ϕ) cuando ε → 0, para cada t > 0. Y ademas de (8) y el Lema2.0.6 resulta que

E%e−Mt(X)(ϕ)

&= lim

ε→0E

%e−Mε

t (X)(ϕ)&

= e−⟨Vt(ϕ(x)g(x)⟩),µ(dx)⟩.

Ası queda demostrado el resultado.

AgradecimientosLe agradezco a Jose Alfredo Lopez-Mimbela el haber revisado la

presente nota.

Jose Villa MoralesDepartamento de Matematicas y FısicaUniversidad de Autonoma de AguascalientesCIMATApartado Postal 40236000, Guanajuato, Gto. MEXICO,[email protected].

Referencias

[1] Dawson, D., Measure-valued Markov processes, Ecole d’Ete deProbabilites de Saint-Flour XXI, Lecture Notes in Math. 1541,Springer, Berlin, 1991.

[2] Evans, S. and Perkins, E., Collision local times, historical stochas-tic calculus and competing suprocesses, Elec. J. of Probab. 3(1998), 1-120.

[3] Fleischmann, K., Critical behavior of some measure-valued pro-cesses, Math. Nachr. 135 (1988), 131-147.

[4] Fleischmann, K and Gartner, J., Occupation time processes at acritical point, Math. Nachr. 125 (1986), 275-290.


[5] Folland, G., Introduction to partial differential equations, Secondedition Princeton University Press, Princeton, NJ, 1997.

[6] Iscoe, I., A weighted ocupation time for a class of measure-valuedbranching processes, Probab. Theory Relat. Fields 71 (1986), 85-116.

[7] Malliavin, P., Integration and Probability, Springer-Verlag, NewYork, 1995.

[8] Meleard, S. and Roelly, S., Discontinuous measure-valued branch-ing processes and generalized stochastic equations, Math. Nachr.154 (1991), 141-156.

[9] Mytnik, L., Collision measure and collision local time for (α, d,β)-Superprocesses, J. Theoret. Probab. 11 (1998), 733-763.

MORFISMOS, Comunicaciones Estudiantiles del Departamento de Matema-ticas del CINVESTAV, se termino de imprimir en el mes de enero de 2002 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

Degree and fixed point index. An account

Carlos Prieto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Existence of Nash equilibria in nonzero-sum ergodic stochastic games in Borelspaces

Rafael Benıtez-Medina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Monte Carlo approach to insurance ruin problems using conjugate processes

Luis F. Hoyos-Reyes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

ocigolopotoicapsenuedzehcertsealerboS

Alejandro Ramırez Paramo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Medida de colisin de un (a, d, b)-superproceso con su medida inicial

Jose Villa Morales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Morfismos, Vol 5, No 2, 2001

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