Morfismos, Vol 5, No 2, 2001

VOLUMEN 5 NÚMERO 2 JULIO A DICIEMBRE DE 2001 ISSN: 1870-6525

MORFISMOS Comunicaciones Estudiantiles Departamento de Matem´aticas Cinvestav Editores Responsables • Isidoro Gitler • Jes´ us Gonz´alez • Isa´ıas L´opez

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Editores Asociados • Ricardo Berlanga • Samuel Gitler • Emilio Lluis Puebla • Guillermo Pastor • V´ıctor P´erez Abreu • Carlos Prieto • Carlos Renter´ıa • Luis Verde

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VOLUMEN 5 NÚMERO 2 JULIO A DICIEMBRE DE 2001 ISSN: 1870-6525

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Morfismos

Editorial Guidelines “Morfismos” is the journal of the students of the Mathematics Department of CINVESTAV. One of its main objectives is for students to acquire experience in writing mathematics. MORFISMOS appears twice a year.

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If you have any suggestions about the journal, let the Editors know and we will gladly study the possibility of implementing them. We expect this journal to foster, as a preliminary experience, the development of a correct style of writing mathematics.

Morfismos

Contenido

Degree and fixed point index. An account Carlos Prieto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Existence of Nash equilibria in nonzero-sum ergodic stochastic games in Borel spaces Rafael Ben´ıtez-Medina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Monte Carlo approach to insurance ruin problems using conjugate processes Luis F. Hoyos-Reyes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Sobre la estrechez de un espacio topol´ ogico Alejandro Ram´ırez P´ aramo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Medida de colisin de un (a, d, b)-superproceso con su medida inicial Jos´e 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 degree and Lefschetz-Hopf fixed point index is discussed in the light of work done mainly by A. Dold, H. Ulrich and the author. Generalizations to certain coincidence situations including the equivariant cases are presented, as well as how to deal with the infinite dimensional cases. In two appendices a proof of the Lefschetz-Hopf theorem for these indices is referred to, as well as a generalization of Dold’s fixed point transfer is sketched.

2000 Mathematics Subject Classification: 55M20, 54H25.

Keywords and phrases: Generalized fixed point problems, Brouwer degree, Lefschetz-Hopf fixed point index.

1

Introduction

1.0 Consider a system of equations

(1.1)

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

where the unknown are restricted by some conditions. These restrictions can 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 in this boundary, Brouwer [2] defined in 1911 the concept of degree, which to such g assigns an integer, deg(g), such that if it is nonzero, then the equation 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 = Rk × Rl−k ,

V ⊂ Rk ,

such that g(x) = a if and only if f (x) = (x, 0); hence, we look for solutions x ∈ V for the equation f (x) = (x, 0) ∈ Rk × Rl−k .

This is a generalized fixed point problem. For the classical problem, k = l, Lefschetz [17] defined in 1926 an invariant, L(f ), with integral values, called the Lefschetz number, defined for V a polyhedron and f such that f (V ) ⊂ V . This number, which is easy to compute, has the property 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 defined another integral invariant for the case k = l, V open and bounded and such 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 as the Lefschetz number, namely, I(f ) ̸= 0 implies that f has fixed points.

This index deals with more general situations, but it is also more difficult to compute. Their relationship is given by the so-called Lefschetz-Hopf theorem which states that in the case that both L(f ) and I(f ) are defined, 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 ⊂ Rk = Rl × Rk−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 (1.3)

f : V −→ Rk × Rm ,

V ⊂ Rk × Rn

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 generality, fixed point indices which decide the existence of solutions for this 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 general spaces Eb , which include finite polyhedra and smooth manifolds, which in their time were considered by Lefschetz and Hopf. The problem is now the following. Let (1.4)

f : V −→ E × M ,

V ⊂ E × N open ,

where E is a euclidean neighborhood retract over B, an ENRB for short, namely a continuous family given by p : E −→ B, of retracts Eb = 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 × M if 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

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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 (generalized) 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], where he generalized the work of Lefschetz and Hopf as well as previous work of himself, [4] and [6]. In these last, he studied the case B = {∗} (see also [5]). The general case was studied by the author in [20]. This case will be discussed in section 2.

Very frequently the problem presents symmetries, that is, all the spaces E, B, M, N admit group actions for a group G, and p and f are compatible with those actions, i.e. they are equivariant. The solution of the problem in this case is sharper, and if M = N = 0 it has been given basically by Dold in [9] and by Ulrich in [30, 31], although tom Dieck has said something about it too [3]. Its generalization for real G-modules of finite dimension M and N was given by Ulrich and the author 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 several questions in nonlinear analysis. The development of this problem is as follows. Leray and Schauder [18] 1934 defined an index for the case B = {∗}, M = N = 0 and E a separable Banach space, requiring f to be such that the closure of the image of V under f , f (V ) ⊂ E is compact. Granas [11] generalized this to the case in which E is an absolute neighborhood retract (an ANR) and Ulrich [29] did it in the parametrized case (B ̸= {∗}). The general case (ANRB s and M and N finite 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 following commutative diagrams, called fixed point situations over B (2.1)

E ×◗N ⊃ V

◗◗◗ ◗◗◗ ◗◗◗ p◦proj1 ◗◗◗◗ ""

f

B,

!! E × M ✉✉ ✉✉ ✉ ✉✉ ##✉✉ p◦proj1

5

Degree and fixed point index

where p : E −→ B is an ENRB , i.e. a vertical retract (meaning that the retraction commutes with the projections p and projB ) of an open set in B × K, K a euclidean space (K = Rk ), M and N are euclidean spaces (M = Rm , N = Rn ) too and f is properly fixed over B, i.e. the restriction 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, for each 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 validity of a parametrized Heine-Borel theorem; namely, there exists a function ρ : 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 of balls in K × N = Rk+n of radius varying according to ρ).

Consider the following sequence of maps of pairs (V, V − ! F)

(2.2)

(1)

!

""

i−f

!! B × (K × M, K × M − 0)

"

# !!

(E × N, E × ! N − Bρ ) (E × N, E × N − F ) (2)

""

"

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

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) a homotopy equivalence (of the second spaces of the pairs), and all maps are over B (i.e., they preserve the fibers). Thus they induce a stable map (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) we get 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 times on the right side, gives hi (B) −→ hi+m−n (B),

i ∈ Z,

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Carlos Prieto

and thus (2.4). This homomorphism is called the index homomorphism of f . Important examples of h∗ are ordinary cohomology, K-theory, or stable cohomotopy. All these examples are multiplicative theories having an element 1 ∈ h0 (B); hence, for these theories, we may define the 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 ), it vanishes 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 see some special cases.

Let B = {∗} and m = n(= 0); the three cohomology theories mentioned above are such that h0 (∗) = Z. In this case, the index I(f ) is an integer, which is the same in all cases; this is the classical fixed point index, 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 some f.

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

Homotopy 2.7. Let f : V −→ E × M , V ⊂ E × N be properly fixed over B × I (I = [0, 1]). Then its restrictions f0 : V0 −→ E0 × M and f1 : V1 −→ E1 × M to bottom B × {0} ≈ B and top B × {1} ≈ B of the cylinder 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 fixed over 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, then so 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 over B. If V ′ ⊂ V is open and such that Fix(f ) ⊂ V ′ , then f ′ = f |V ′ is properly fixed and I(f ′ ) = I(f ) ∈ hm−n (B).

The next property allows us to define the index for general ENRB s.

It is this property which constitutes the main difference between index

Degree and fixed point index

7

and degree and shows the convenience to work with the index rather than with the degree, which, in general, can not be defined for arbitrary euclidean neighborhood retracts.

Commutativity 2.10. Let E = B × L −→ B and E ′ = B × L′ −→ B with 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 the composite (ψ × 1M )ϕ : ϕ−1 (U × M ) −→ E ′ × M ,

ϕ−1 (U × M ) ⊂ U ′ ⊂ E ′ × N

is properly fixed, then also the composite ϕ(ψ×1N )

(ψ × 1N )−1 (U ′ ) −−−−→ 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 the index: Proposition and Definition 2.11. If p : E −→ B is an ENRB , M and N are euclidean spaces and V ⊂ E × N is open, then every map over B, f : V −→ E × M admits a decomposition f: V

α×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 depends only on f and not on the factorization f = β(α × 1N ). Thus we define the fixed point index of f as I(f ) = I(g).

Proof: (sketch) Let r

E

##

W "" ⊂ B × K i

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

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Carlos Prieto

(r×1N )(i×1N ) = id : V −→ V ) and β = f (r×1N ) : U −→ E×M . Then β(α × 1N ) = f . The rest of the proof is a straightforward application of 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 index with 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 case m < 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 map f : B × S1 −→ B × S1 , f (b, z) = (b, b · z). This is a properly fixed map over B (for the projection B × S1 −→ B and M = N = 0).

If one takes stable cohomotopy as the cohomology theory, then 0 (B) = Πst = Z/2 which also is I(f ) is the nontrivial element of πst 1 ∗ (B × S1 ) −→ π ∗ (B × S1 ), seen as a the Lefschetz trace of f ∗ : πst st ∗ (B)-modules. homomorphism of πst (b) [20, 4.27] Let S2 = C ∪ {∞} be the Riemann sphere. If k ∈ Z, then the map S2 ⊃ C

f

!! S2 × C ,

f (z) = (z, z k )

is properly fixed over B = S2 (E = B, p = id, M = C = R2 ), and 2 (B) = Πst = Z. I(f ) = k ∈ πst 0

3

Generalizations of the index

3.0. Very often the situations one studies present some kind of symmetries; if these are given by the action of a compact Lie group G, there are cohomology theories which are fine enough to detect the symmetries.

More precisely we shall be concerned in first place with the equivariant index, which will be defined for situations like (2.1), but now assuming that G acts on all spaces involved and that every map in question commutes with the group action. To be precise, p : E −→ B will be a G-ENRB , i.e., G acts on both E and B, p is G-equivariant and E is a vertical equivariant retract of an open (invariant) set B × K, where now K is a G-module. In fact, K, M and N are now all G-modules,

Degree and fixed point index

9

that is, euclidean spaces with a linear action of G. Observe that in this case the fixed point set F of f is G-invariant and hence it is possible to choose 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 the sequence of maps (2.2) consists of equivariant maps (over B) and thus produces 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 group G, (see e.g. Appendix A).

As before, this sequence induces, equivalently to (2.1), an equivariant index homomorphism of f as (3.2)

∗+[M ]−[N ]

If : h∗G (B) −→ hG

(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 the right, 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]). All these theories are multiplicative and have an element 1 ∈ h0G (B). We define the equivariant fixed point index of f as the element (3.3)

[M ]−[N ]

IG (f ) = If (1) ∈ hG

(B) .

Once again, this index has the properties 2.5. through 2.10. and can thus be extended to general G-ENRB s exactly in the same way as before, (2.11).

The case B = {∗} and M = N (= 0) is interesting. Equivariant ∗ is such that π 0 (∗) ∼ A(G), the Burnside ring of stable cohomotopy πG = G

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 is an element of A(G). In [8] it is proved that every element of A(G) is the equivariant index of some equivariant f . In fact, more generally, in [M ]−[N ] [30, 31] and [21] it is proved that every element in πG (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(f H ) of the restrictions f h : V H −→ E H of f to the spaces whose points remain fixed under the action of the elements of the! closed" subgroups H ⊂ G. In [30], relationships between IG (f ) and I(f H ) are thoroughly studied.

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

An absolute neighborhood retract over B, an ANRB , p : E −→ B is defined as a vertical retract of an open set in B × K, where K now denotes a separable Banach space. We consider fixed point situations, that is, commutative diagrams (3.5)

E ×◗N ⊃ V

◗◗◗ ◗◗◗ ◗◗◗ ◗◗◗ ◗""

f

B

!! E × M ✇ ✇✇ ✇✇ ✇ ✇ ## ✇ ✇

where E −→ B is an ANRB , M and N are euclidean spaces (possibly with a group action, in whose case f has to be equivariant) and f is strongly fixed, which means that besides being properly fixed, the closure, f (V ), of its image (or at least to the image, f (W ), of some open neighborhood W of Fix(f )) lies properly over B. In the case that E = B × K −→ B, it is possible to approximate f by maps f ′ : V ′ −→ B × P × M , properly fixed over B, where V ′ is open in B × P × N and P is a finite polyhedron contained in K. Since then P 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 are homotopic (in the sense of 2.7; thus I(f ′ ) = I(f ′′ ). Therefore, we may define the index of f , I(f ), as I(f ′ ) for f ′ close enough to f .

Degree and fixed point index

11

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 of defining the index of the situation (3.4) for a general ANRB p : E −→ B holds.

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 refers to an important class of equations (cf. [14, 15]). We shall discuss the degree, which we shall define via the index, and using the properties of this last, we shall show its fundamental properties.

Let G be a compact Lie group and M , N (finite dimensional) real G-modules, and let K be a euclidean space. If B is a metric G-space and (4.1)

B×K ×N ⊃V

g

!! K × M

is an equivariant map, with V open and invariant in B × K × N , and is such that the set of solutions g −1 (0) of the equation g(b, z, y) = 0 lies properly over B (e.g. is compact, if B = {∗} or B itself is compact); for instance, if the closure V lies properly over B and g can be extended to V in such a way that no zeroes appear on the boundary, then we define the degree of g as (4.2)

[M ]−[N ]

deg(g) = I(i − g′ ) ∈ πG

(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 following 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 ).

(This follows from the equivariant version of 2.9).

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 ).

(This follows from the equivariant version of 2.8).

Homotopy Invariance 4.7. If gt is a homotopy between g0 and g1 such that for every t, gt−1 (0) lies properly over B, then deg(g0 ) = deg(g1 ).

(It follows from the equivariant version of 2.7. In fact, the inverse is also 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 trivially on K. On the other hand, as described in 3.4, we may more generally assume that K is a separable Banach space. The situation is as follows.

Let (4.8)

g

B × K × N ⊃ V −→ 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-Schauder type index, I(i − g′ ) is defined. Hence we define the degree of g by (4.9)

[M ]−[N ]

deg(g) = I(i − g′ ) ∈ πG

(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[M ]−[N ] ble equivariant stem ΠG of stable homotopy classes of equivariant maps between the G-spheres SM and SN , given by the one-point compactifications of the G-modules M and N , respectively.

As a last comment in this section it should be remarked that the degree in [14, 15] is defined in a nonstable equivariant homotopy group of G-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.

13

Degree and fixed point index

A

The Lefschetz-Hopf theorem

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

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 be G-modules. Then, if f : E × N −→ E × M is an equivariant map over B such that Fix(f ) −→ B is proper and f −1 (E × 0) ⊂ E × B for some ball B ⊂ N , then [M ]−[N ]

I(f ) = trace(f ∗ : h∗G (E) −→ h∗G (E)) ∈ hG

(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 an element of the real representation ring RO(G).

Its morphisms are the stable maps given by (A.3)

B-Stab G ((X, X ′ ; ρ), (Y, Y ′ ; σ)) = ! " = colim (X, X ′ ) × (K ⊕ ρ, K ⊕ ρ − 0), (Y, Y ′ ) × (K ⊕ σ, K ⊕ σ − 0) , K

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 K varies in the category (made small) of unitary (complex) representations of 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 monoidal category, and inside it the proper G-ENRB s, E, are strongly dualizable, whose dual is (B × L, B × L − E), if E is a G-neighborhood retract in B × L. Under the assumptions of A.1, the defining sequence (2.1) of the index, defines the trace, (2.2), of the morphism (A.3) in the category G-StabB . Since the cohomology h∗G defines a functor from this category to 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 another homomorphism related to the index homomorphism If : h∗G (B) −→ ∗+[M ]−[N ] hG (B), called the transfer homomorphism of f . To define it, consider, as before, first the case E = B × K and take the sequence of equivariant 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 τfV : h∗G (V ) −→ h∗G (B) for any RO(G)-graded G-equivariant cohomology theory h∗G , called a transfer homomorphism of f .

Since we may restrict f to any W ⊂ V , as to approach Fix(f ), then all transfers τGW fit together to pass to the limit and yield the minimal transfer ∗+[M ]−[N ] (B) , τˇf : fˇG∗ (Fix(f )) −→ hG through which all other transfers factor. This shows, in particular, that (B.2)

Fix(f ) = ø =⇒ τfW = 0

for every W .

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

τf

"" V

ψ

"" Y .

Degree and fixed point index

15

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

Carlos Prieto Instituto de Matem´ aticas, UNAM, 04510, M´exico, D.F., MEXICO, cprieto@matem.unam.mx.

References [1] Atiyah, M. F. and Segal, G. B., mimeographed notes, Warwick, 1965.

Equivariant K-theory,

¨ [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-HeidelbergNew York, 1979.

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

[5] Dold, A., Lectures on Algebraic Topology, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Band 200, 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, Contemporary Math. 12 (1982), 105–115.

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

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

16

Carlos Prieto

[12] Hopf, H., Eine Verallgemeinerung der Euler-Poincar´eschen Formel, Nachr. Akad. Wiss. G¨ ottingen 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., Massab´ o, I. and Vignoli, A., Degree theory for equivariant maps, I, Trans. Amer. Math. Soc. 315 (1989), 433–510.

[15] Ize, J., Massab´ o, I., Vignoli, A., Degree theory for equivariant maps, 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 and manifolds, Trans. Amer. Math. Soc. 28 (1926), 1–49.

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

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

[20] Prieto, C., Coincidence index for fiber-preserving maps. An approach 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 theoretical approach to the Segal Conjecture, Contemporary Math. 58, II (1987), 89–108.

[22] Prieto, C., Teor´ıa FIX de diagramas, en “Conferencias del Taller de Topolog´ıa Algebraica” IV Coloquio, Depto. de Matem´aticas, CINVESTAV, M´exico (1985).

[23] Prieto, C., FIX-Theory of diagrams, Comtemporary Math. 72 (1988), 207–224.

[24] Prieto, C., Una f´ ormula de Lefschetz-Hopf para el ´ındice de coincidencia equivariante parametrizado, Aportaciones Mat. 5 (1988), 73–88.

Degree and fixed point index

17

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

[26] Prieto, C. and Ulrich, H., Equivariant fixed point index and fixed point 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 Par´ıs (1971), 59–63.

[29] Ulrich, H., Der Fixpunktindex fasernweiser Abbildungen, Diplomarbeit, Heidelberg (1976).

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

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

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

Existence of Nash equilibria in nonzero-sum ergodic stochastic games in Borel spaces ∗ Rafael Ben´ıtez-Medina

Abstract In this paper we study nonzero-sum stochastic games with Borel state and action spaces, and the average payoff criterion. Under suitable assumptions we show the existence of Nash equlibria in stationary strategies. Our hypotheses include ergodicity conditions and an ARAT (additive reward, additive transition) structure.

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 and action spaces, and the average payoff criterion with possibly unbounded payoffs. This class of games has many applications, for instance, in queueing and economic theory (see [1], [2], [12], [27]).

The problem we are interested in is the existence of Nash equilibria in stationary strategies. To do this we impose ergodicity conditions already used 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 been obtained by Ghosh and Bagchi [5] and Ku ¨enle [14] for games with bounded payoffs. Other related works include [18], which deals with ∗

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

19

20

Rafael Ben´ıtez-Medina

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

For stochastic games with a discounted payoff criterion there is a larger literature. For instance, for zero-sum problems in countable spaces, 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 introduces standard material on stochastic games and strategies, and the optimality criteria. The core of the paper is contained in section 3: after introducing some assumptions, we present our main result, Theorem 3.10, on the existence of Nash equilibria. Finally, after some technical preliminaries in section 4, the proof of Theorem 3.10 is presented in section 5.

2

The game model

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

2.1 Remark. (a) A Borel subset X of a complete and separable metric space is called 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 weak topology σ(IP(X), Cb (X)), where Cb (X) stands for the space of continuous bounded functions on X. In this case, IP(X) is a Borel space. 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 denote

by IP(X|Y ) the family of all those transition probabilities. If φ is in IP(X|Y ), then we write its values either as φ(y)(B) or as φ(B|y), for all y ∈ Y and B ∈ B(X). Finally, if X = Y then φ is said to be a Markov transition probability on X. The stochastic game model. We shall consider the two-person nonzero-sum stochastic game model

Nash equilibria in nonzero-sum stochastic games

(1)

21

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

where X is the state space, and A and B are the action spaces for players 1 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. Similarly, 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 the game’s transition law, and, finally, ri : IK → IR is a measurable function representing the reward function for player i = 1, 2.

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

2.2

Strategies

Let H0 := X and Ht := IK ×Ht−1 for t = 1, 2, . . . . For each t, an element ht = (x0 , a0 , b0 , . . . , xt−1 , at−1 , bt−1 , xt ) of Ht represents a “history” of the game up to time t. A strategy f or player 1 is then defined as a sequence π 1 = {πt1 , t = 0, 1, . . .} of transition probabilities πt1 in IP(A|Ht ) such that πt1 (A(xt )|ht ) = 1 ∀ht ∈ Ht , t = 0, 1, . . . .

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

22

Rafael Ben´ıtez-Medina

Now define IA(x) := IP(A(x)) for each state x ∈ X, and let S1 be the class of all transition probabilities φ ∈ IP(A|X) such that φ(x) is in IA(x) for all x ∈ X. Then a strategy π 1 = {πt1 } ∈ Π1 is called stationary if there exists φ ∈ S1 such that πt1 (·|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), respectively.

Let (Ω, F ) be the canonical measurable space that consists of the sample space Ω := (X × A × B)∞ and its product σ-algebra F. Then for each pair of strategies (π 1 , π 2 ) ∈ Π1 × Π2 and each initial state 1 2 x ∈ X there exists a probability measure Pxπ ,π and a stochastic process {(xt , at , bt ), t = 0, 1, . . . .} defined on (Ω, F) in a canonical way, where xt , at and bt represent the state and the actions of players 1 and 2, respectively, at each stage t = 0, 1, . . .. The expectation operator with 1 2 1 2 respect to Pxπ ,π is denoted by Exπ ,π .

2.3

Average payoff criteria

For each n = 1, 2, . . . and i = 1, 2, let Jni (π 1 , π 2 , x)

:=

1 2 Exπ ,π [

n−1 !

ri (xt , at , bt )]

t=0

be the n-stage expected total payoff (or reward) of player i when the players use the strategies π 1 ∈ Π1 and π 2 ∈ Π2 , given the initial state x0 = x.

The corresponding long-run expected average payoff (EAP) per unit time is then defined as (2)

J i (π 1 , π 2 , x) := lim inf Jni (π 1 , π 2 , x)/n. n→∞

The EAP is also known as the ergodic payoff (or ergodic reward) criterion.

2.2 Definition. A pair of strategies (π 1∗ , π 2∗ ) is called a Nash equilibrium (for the EAP criterion) if J 1 (π 1∗ , π 2∗ , x) ≥ J 1 (π 1 , π 2∗ , x) for all π 1 ∈ Π1 , x ∈ X,

Nash equilibria in nonzero-sum stochastic games

23

and J 2 (π 1∗ , π 2∗ , x) ≥ J 2 (π 1∗ , π 2 , x) for all π 2 ∈ Π2 , x ∈ X. Our aim is to establish, under certain assumptions, the existence of a 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) :=

!

!

f (x, a, b)ψ(db)φ(da)

A(x) B(x)

whenever the integrals are well defined. In particular, for ri and Q as in (1), ! ! ri (x, φ, ψ) := ri (x, a, b)ψ(db)φ(da) A(x) B(x)

and Q(·|x, φ, ψ) :=

3

!

!

Q(·|x, a, b)ψ(db)φ(da).

A(x) B(x)

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 v on X, the functions !

X

v(y)Q(dy|x, ·, b) and

!

X

v(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 (3)

|ri (x, a, b)| ≤ r¯w(x)

∀(x, a, b) ∈ IK, i = 1, 2,

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

The next two assumptions are used to guarantee that the state process {Xt } is ergodic in a suitable sense.

24

Rafael Ben´ıtez-Medina

3.2 Assumption. There exists a probability measure ν ∈ IP(X), a positive number α < 1, and a measurable function β : IK → [0, 1] for which 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 respect to which, for each pair (φ, ψ) ∈ S1 × S2 , the Markov transition probability Q(·|x, φ(x), ψ(x)) is λ-irreducible.

We next introduce some notation and then we mention some important consequences of the above assumptions.

3.4 Definition. IBw (X) denotes the linear space of real-valued measurable functions u on X with a finite w-norm, which is defined as ||u||w := sup |u(x)|/w(x),

(4)

x∈X

and IMw (X) stands for the normed linear space of finite signed measures µ on X such that ||µ||w :=

(5)

!

"

X

wd|µ| < ∞.

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 {Xt } is positive Harris recurrent; hence, in particular, the Markov transition probability Q(·|x, φ(x), ψ(x)) admits a unique invariant probability measure in IMw (X) which will be denoted by q(φ, ψ); thus q(φ, ψ)(D) =

"

Q(D|x, φ(x), ψ(x))q(φ, ψ)(dx) X

∀D ∈ B(X).

25

Nash equilibria in nonzero-sum stochastic games

(b) {Xt } is w-geometrically ergodic, that is, there exist positive constants θ < 1 and M such that (6)

|

!

X

u(y)Qn (dy|x, φ(x), ψ(x)) −

!

X

u(y)q(φ, ψ)(dy)|

≤ w(x)||u||w M θn

for every u ∈ IBw (X), x ∈ X, and n = 0, 1, . . . , where Qn denotes the n-step Markov transition probability. This result follows from Lemmas 3.3 and 3.4 in [6] where it was assumed the positive Harris recurrence in part (a). However, as shown in Lemma 4.1 of [15], the latter recurrence is 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) = g(x, a, b, y)γ(dy) D

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 (7)

lim

"

n→∞ X

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

∀x ∈ X

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

The next two assumptions require that the game model (1) has a so-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).

26

Rafael Ben´ıtez-Medina

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 3.6 yield that !

max |ri1 (x, a)|γ(dx) < ∞,

X a∈A(x)

!

max |ri2 (x, b)|γ(dx) < ∞.

X b∈B(x)

3.10 Theorem. Under Assumptions 3.1, 3.2 and 3.6- 3.9, there is a pair (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 f ixed stationary strategy ψ in S2 . Then the game model GM in (1) reduces to a M arkov control model (8)

M CM1 (ψ) = (X, A, IKA , Qψ , r1,ψ )

where X, A and IKA are as in (1), and the transition law Qψ in IP(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 the following.

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 control model in (8), i.e., (9)

J 1 (φ∗ , ψ, x) = max J 1 (π 1 , ψ, x) =: ρ∗1 (ψ) ∀x ∈ X. π 1 ∈Π1

Nash equilibria in nonzero-sum stochastic games

27

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 (10)

ρ∗1 (ψ) + h1φ∗ ,ψ (x) = r1 (x, φ∗ (x), ψ(x)) ! + X h1φ∗ ,ψ (y)Q(dy|x, φ∗ (x), ψ(x)) = maxφ∈IA(x) [r (x, φ, ψ(x)) ! 1 + X h1φ∗ ,ψ (y)Q(dy|x, φ, ψ(x))]

(11)

for all x ∈ X, and such that as in the Remark 3.5(a).

!

X

h1φ∗ ,ψ (y)q(φ∗ , ψ)(dy) = 0, with q(φ∗ , ψ)

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

Similarly, we can obtain an optimal response ψ ∗ ∈ S2 of player 2 if player 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 is borrowed 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 (12)

J i (φ, ψ, x) = lim Jni (φ, ψ, x)/n = ρi (φ, ψ),

where (13)

ρi (φ, ψ) :=

n→∞

"

X

ri (x, φ(x), ψ(x))q(φ, ψ)(dx)

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

∞ # t=0

Exφ,ψ [ri (xt , φ(xt ), ψ(xt )) − ρi (φ, ψ)]

belongs to IBw (X), and, moreover, its w-norm is independent of (φ, ψ): (14)

||hiφ,ψ ||w ≤ r¯M/(1 − θ),

28

Rafael Ben´ıtez-Medina

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 the so-called Poisson equation (15)

ρi (φ, ψ)+ hiφ,ψ (x) = ri (x, φ(x), ψ(x)) ! + X hiφ,ψ (y)Q(dy|x, φ(x), ψ(x))

that satisfies the condition "

5

X

hiφ,ψ (y)q(φ, ψ)(dx) = 0.

Proof of Theorem 3.10

From (12) and Corollary 5.12(a) in [10], we can write ρ∗1 (ψ) in (9) as (16)

ρ∗1 (ψ) = ρ1 (φ∗ , ψ) = max ρ1 (φ, ψ). φ∈S1

Similarly, for each φ ∈ S1 there exists ψ ∗ ∈ S2 such that (17)

ρ∗2 (φ) = ρ2 (φ, ψ ∗ ) = max ρ2 (φ, ψ) ψ∈S2

We next use (16) and (17) to introduce a multifunction τ from S1 × S2 to 2S1 ×S2 as follows: for each pair (φ, ψ) in S1 × S2 (18)

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

To complete the proof of Theorem 3.10 we shall proceed in two steps, 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 which Si 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 , ψn∗ ) ∈ ∗ ), then (φ∗ , ψ ∗ ) is in τ (φ , ψ ). τ (φn , ψn ) is such that (φ∗n , ψn∗ ) → (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 ∗ ) in S1 × S2 , that is (φ∗ , ψ ∗ ) ∈ τ (φ∗ , ψ ∗ ).

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

Nash equilibria in nonzero-sum stochastic games

29

In step 1 we shall use the topology introduced by Warga (see Theorem IV.3.1 in [28]): Let F1 be the Banach space of measurable functions f : IKA → IR such that f (x, a) is continuous in a ∈ A(x) for each x ∈ X and ! max |f (x, a)|γ(dx) < ∞, ||f || := X a∈A(x)

with γ as in Assumption 3.6. We shall identify two stationary strategies ′ ′ φ 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 ∈ F1∗ given by ! ! ∆φ (f ) :=

f (x, a)φ(da|x)γ(dx).

X

A

Thus S1 can be identified with a subset of F1∗ , 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 (21) is such that (22)

(φ∗n , ψn∗ ) ∈ τ (φn , ψn ) ∀n ∗ (φ∗n , ψn∗ ) → (φ∗∞ , ψ∞ ).

By (21) and the definition (18) of τ , together with (10) (or (15)), for all 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,

(24)

ρ∗2 (φn ) + h2φn ,ψn∗ (x) = r2 (x, φn (x), ψn∗ (x)) " + X h2φn ,ψn∗ (y)Q(dy|x, φn (x), ψn∗ (x)).

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) and ri2 (x, b) are in F2 . Therefore, by (20) and (22), (25)

!

X

r1 (x, φ∗n (x), ψn (x))γ(dx) →

!

X

r1 (x, φ∗∞ (x), ψ∞ (x))γ(dx),

30

Rafael Ben´ıtez-Medina

and similarly for i = 2. Moreover, for any D ∈ B(X), (26)

!

Q(D|x, φ∗n (x), ψn (x))γ(dx) → ! ∗ X Q(D|x, φ∞ (x), ψ∞ (x))γ(dx),

X

∗ ). and similarly for (φn , ψn∗ ) → (φ∞ , ψ∞

5.1 Lemma. There is a subsequence {m} of {n} and numbers ρˆ1 and ρˆ2 such that ρ∗1 (ψm ) = ρ1 (φ∗m , ψm ) → ρˆ1 (27) and (28)

∗ ρ∗2 (φm ) = ρ2 (φm , ψm ) → ρˆ2 .

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 "

(30)

X

w(y)Q(dy|x, a, b) ≤ αw(x) + ||ν||w

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 measure q(φ, ψ) yields "

X

w(y)q(φ, ψ)(dy) ≤ α

"

X

w(y)q(φ, ψ)(dy) + ||ν||w ,

and, therefore, "

X

w(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),

Nash equilibria in nonzero-sum stochastic games

31

i.e., (29) holds. This in turn gives that the sequences {ρ1 (φ∗n , ψn )} and {ρ2 (φn , ψn∗ )} 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 u ˜n (·) := un (·)/w(·).

(31)

By (14), the constant m0 := r¯M/(1 − θ) satisfies that |˜ un (x)| ≤ m0 ∀x, n.

Let U be the space of all γ-equivalence classes of real-valued measurable 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 a compact and metrizable subset of L∞ (γ) ≡ L∞ (X, B(X), γ) equipped with the relative weak* topology σ(L∞ (γ), L1 (γ)). Therefore, we can assume that {˜ un } converges in the weak* topology to some function ∞ u ˜∗ in L (γ). Let u∗ (x) := u ˜∗ (x)w(x) for all x ∈ X. Then, as in the proof of Theorem 4 in [19], using Assumption 3.7, one can show that as n → ∞.

(32)

max max |

a∈A(x) b∈B(x)

!

X

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

with un (·) as in (31). In turn, (32) and Assumption 3.8 yield that (33) max

max |

φ∈IA(x) ψ∈IB(x)

!

X

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

We also have the following.

5.2 Lemma. If (φn , ψn ) → (φ, ψ) in S1 × S2 , then, as n → ∞, (34)

" "

X X

u(y) Q(dy|x, φn (x), ψ (x))γ(dx) " "n → X X u(y)Q(dy|x, φ(x), ψ(x))γ(dx)

for any function u ∈ IBw (X).

Proof: Choose an arbitrary function u ∈ IBw (X). By definition of the weak 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 the functions !

(35) u(y)Q1 (dy|x, a) (x, a) → X

32

Rafael Ben´ıtez-Medina

and (36)

(x, b) →

!

X

u(y)Q2 (dy|x, b)

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 = 1 and 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),

"

max |

a∈A(x)

"

!

X

u(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.

(37)

ρˆ1 + u∗ (x) = r1 (x, φ∗∞ (x), ψ∞"(x)) + X u∗ (y)Q(dy|x, φ∗∞ (x), ψ∞ (x)) ψ (x)) = maxφ∈IA(x) [r1 (x, φ, " ∞ + 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, and let h∗ : X → IR be such that h∗ (x) := u∗ (x) for x ∈ D, and h∗ (x) := max [r1 (x, φ, ψ∞ (x)) + φ∈IA(x)

!

X

u∗ (y)Q(dy|x, φ, ψ∞ (x))] − ρˆ1

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 (by Lemma 4.1) that (38)

ρˆ1 = ρ∗1 (ψ∞ ) = ρ1 (φ∗∞ , ψ∞ ).

An analogous argument using (21), (22) and (24) with obvious changes, shows that ∗ ρˆ2 = ρ∗2 (φ∞ ) = ρ2 (φ∞ , ψ∞ (39) ).

33

Nash equilibria in nonzero-sum stochastic games

In other words, (38) and (39) state that, under (20)- (22), the pair ∗ ) is in τ (φ , ψ ), and so the set-valued map defined by (18) (φ∗∞ , ψ∞ ∞ ∞ is u.s.c. Thus, as was already noted, it follows from Fan’s fixed point theorem that τ has a fixed point (as in (19), say), which completes the proof of Theorem 3.10. ✷

Acknowledgement The author wishes to thank Dr. On´esimo Hern´andez-Lerma for his very valuable comments and suggestions.

Rafael Ben´ıtez-Medina Departamento de Matem´ aticas, CINVESTAV-IPN, A. Postal 14-740, 07000, M´exico D.F., MEXICO, rbenitez@math.cinvestav.mx.

References [1] Altman E., Hordijk A. and Spieksma F. M., Contraction conditions for average and α-discount optimality in countable state Markov games with unbounded rewards, Math. Oper. Res., 22 (1997), 588-618.

[2] Curtat L. O., Markov equilibria of stochastic games with complementarities, Games and Economic Behavior, 17 (1996), 177-199.

[3] Dunford N. and Schwartz J. T., Linear Operators, Vol. 1, Interscience Publishers, New York, 1958.

[4] Fan K., Fixed point and minimax theorems in locally convex topological spaces, Proc. Nat. Acad. Sci., U. S. A., 38 (1952), 539-560.

[5] Ghosh M. K. and Bagchi A., Stochastic games with average payoff criterion, Appl. Math. Optim., 38 (1998), 283-301.

[6] Gordienko E. and Hern´andez-Lerma O., Average cost Markov control processes with weighted norms: existence of canonical policies, Appl. Math. (Warsaw), 23 (1995), 199-218.

[7] Hern´andez-Lerma O., Adaptive Springer-Verlag, New York, 1989.

Markov

Control

Process,

34

Rafael Ben´ıtez-Medina

[8] Hern´andez-Lerma O. and Lasserre J. B., Discrete-Time Markov Control Processes: Basic Optimality Criteria, Springer-Verlag, New York, 1996.

[9] Hern´andez-Lerma O. and Lasserre J. B., Further Topics on Discrete-Time Markov Control Processes, Springer-Verlag, New York, 1999.

[10] Hern´andez-Lerma O. and Lasserre J. B., Zero-sum stochastic games in Borel spaces:average payoff criteria, SIAM J. Control Optim., 39 (2001), 1520-1539.

[11] Himmelberg C., Parthasarathy T., Raghavan T. E. S. and Van Vleck F., Existence of ρ-equilibrium and optimal stationary strategies in stochastic games, Proc. Amer. Math. Soc., 60 (1976), 245251.

[12] Karatzas I., Shubik M. and Sudderth W. D., Construction of stationary Markov equilibria in a strategic market game, Math. Oper. Res., 19 (1994), 975-1006.

[13] Kumar P. R. and Shiau T. H., Existence of value and randomized strategies in zero-sum discrete time stochastic dynamic games, SIAM J. Control Optim., 19 (1981), 561-585.

[14] K¨ uenle H. U., Equilibrium strategies in stochastic games with additive cost and transition structure and Borel state and action spaces, International Game Theory Review, 1 (1999), 131-147.

[15] Luque-V´azquez F. and Hern´andez-Lerma O., Semi-Markov control models with average costs, Appl. Math. (Warsaw), 26 (1999), 315331.

[16] Maitra A. and Parthasarathy T., On stochastic games, J. Optim. Theory Appl., 5 (1970), 289-300.

[17] Mohan S. R., Neogy S. K., Parthasarathy T. and Sinha S., Vertical linear complementarity and discounted zero-sum stochastics games with ARAT structure, Math. Program., Ser. A, 86 (1999), 637-648.

[18] Nowak A. S., Stationary equilibria for nonzero-sum average payoff ergodic stochastic games with general space state, Annals of the International Society of Dynamic Games, 1 (1993), 232-246.

Nash equilibria in nonzero-sum stochastic games

35

[19] Nowak A. S., Optimal strategies in a class of zero-sum ergodic stochastic games, Math. Meth. Oper. Res., 50 (1999), 399-419.

[20] Nowak, A. S., Zero-sum average payoff stochastic games with general state space, Games and Economic Behavior , 7 (1994), 221232.

[21] Nowak, A. S., On zero-sum stochastic games with general state space, I. Prob. Math. Stat., 4 (1984), 13-32.

[22] Nowak A. S. and Altman E., ϵ-Nash equilibria for stochastic games with uncountable state space and unbounded costs, (1998).

Preprint.

[23] Parthasarathy T., Existence of equilibrium stationary strategies in discounted stochastic games, Shankhya. The Indian Journal of Statistics, 44 (1982), Series A, 114-127.

[24] Parthasarathy T. and Sinha S., Existence of stationary equilibrium strategies in nonzero-sum discounted stochastic games with uncountable state space, and state independent transitions, International Journal of Game Theory, 18 (1989), 189-194.

[25] Ramirez-Reyes F., Existence of optimal strategies for zero-sum stochastic games with discounted payoff, Morfismos, 5 (2001), 6383.

[26] Sennot L. I., Zero-sum stochastic games with unbounded cost: discounted and average cost cases, Math. Meth. Oper. Res., 39 (1994), 209-225.

[27] Sennot L. I., Nonzero-sum stochastic games with unbounded cost: discounted and average cost cases, Math. Meth. Oper. Res., 40 (1994), 145-162.

[28] Warga J., Optimal Control of Differential and Funtional Equations, Academic Press, New York, 1972.

Morfismos, Vol. 5, No. 2, 2001, pp. 37–50

Monte Carlo approach to insurance ruin problems 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 version of Wald’s fundamental identity. With this method it is possible to simulate within finite time risk reserve processes with infinite time horizons. This allows us to construct Monte Carlo estimators for the ruin probability, which is one of the main problems in insurance risk theory. Some examples of the Poisson/Exponential and 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 the ruin probability [1-8,11]. It can be roughly described as follows.

The risk reserve process over (0, t] is the difference between a premium deterministic process u + ct and the accumulated claims Zt (a compound Poisson process), for some given initial capital u ≥ 0 . The premium income rate c is fixed by the insurance company and is independent of t. The idea is to study the behavior of the risk reserve process that models the accumulated capital over finite or infinite time ∗ Research partially supported by a CONACyT scholarship. This paper is part of the author’s M. Sc. Thesis presented at the Divisi´ on de Ciencias B´ asicas e Ingenier´ıa, UAM-Iztapalapa.

1 Professor at Departamento de Sistemas, UAM-Azcapotzalco.

37

38

Luis F. Hoyos-Reyes

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

The main purpose of this paper is to introduce Monte Carlo estimators (MCEs) for the ruin probability in infinite time horizon using conjugate processes. This approach allows us to simulate within finite time a risk reserve process with infinite time horizon. Here, we construct a MCE for the ruin probability using the empirical distribution of the ruin events after a sufficiently large number of simulations. Some examples 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 introducing basic terminology and notation. In §3 we show a formulation for the conjugate process and construct the MCEs. In §4 we compute examples of the P/E and P/U cases. Finally, §5 presents some concluding remarks.

2

Preliminaries

Assumption 2.1 (a) The claims arrive according to a Poisson process {Nt }t≥0 with intensity λ and interclaim times {Tt }t≥1 . (b) The claim sizes X1 , X2 , . . . are i.i.d nonnegative random variables with a finite mean µ. (c) Xi and {Nt }t≥0 are independent.

Definition 2.2 The accumulated claim process is Zt := t ≥ 0, with X0 := 0.

! Nt

n=0 Xn

for

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 premium income 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 conjugate family if the Fθ are mutually equivalent with densities of the form dFθ (x) = exp {(θ − θ0 )x − hθ0 (θ)} dFθ0

(1)

and if for some fixed θ0 ∈ Θ the parameter set Θ contains all θ ∈ R for which (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 Eθ Yt 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 cumulant generating function of Fθ0 by hθ0 (θ) := log Eθ0 e(θ−θ0 )X .

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

φZt (β) = eλt(φX (β)−1) .

(2)

Proposition 2.5 Let θ, θ0 ∈ Θ with θ ̸= θ0 . Then φθ (β) = Proof:

Using (1)

φθ0 (β + θ − θ0 ) . φθ0 (θ − θ0 )

40

Luis F. Hoyos-Reyes

φθ (β) =

!

∞

−∞ ! ∞

eβx dFθ (x)

e(θ−θ0 )x dFθ0 (x) Eθ0 e(θ−θ0 )X −∞ φθ (β + θ − θ0 ) . ! = 0 φθ0 (θ − θ0 ) =

Proposition 2.6 Proof:

logEeβYt t

eβx

= λ(φX (β) − 1) − βc.

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 dividing by t, completes the proof. !

Now from Proposition 2.5 Eθ eβZt = Eθ0 e(β+θ−θ0 )Zt /Eθ0 e(θ−θ0 )Zt .

Using (2) Eθ eβZt = eλtφθ0 (θ−θ0 )(φθ (β)−1) , which implies that under Pθ , Zt is also a compound Poisson process with arrival rate λθ = λφθ0 (θ − θ0 ) and claims distribution Fθ . Therefore, replacing E with Eθ in Proposition 2.6 we obtain

(3)

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

Proposition 2.7 If φ′′X exists in an interval I that contains −θ0 , then µθ := Eθ Yt > 0

when

θ>0

and µθ < 0

when

θ > 0.

Insurance ruin problems

Proof:

Let χθ (β) :=

log Eθ eβYt , 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 (X 2 eβX ) > 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) = (Eθ Yt eβYt )/Eθ eβYt , and so χ′θ (0) = Eθ Yt = µθ .

41

42

Luis F. Hoyos-Reyes

Using (4) µθ = χ′θ0 (θ − θ0 ), and µ0 = χ′θ0 (−θ0 ) = 0.

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

3

Monte Carlo estimators

Our main purpose in this section is to estimate the ruin probability considering an infinite time horizon for the risk reserve process. We first introduce some definitions.

Definition 3.1 Let u and τ be as in Definition 2.3. The ruin probability 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 number ρ is called the safety loading, and is related to the capital expected growth as follows.

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

Proof: Under the law Pθ , Zt is a compound Poisson process with Nt ∼ Poisson(λθ ) and claim sizes Xi ∼ Fθ . Therefore Eθ (Zt ) = λθ tEθ X.

43

Insurance ruin problems

Applying the strong law of large numbers to the accumulated claim process yields 1 (Zt − λθ tEθ X) = 0 t→∞ t

(5)

lim

a.s.

Now, from Proposition 2.7 we have Eθ (Zt −ct) > 0 and, therefore, ρ < 0, and using (5) 1 (Zt − λθ tEθ X − ρλθ tEθ X) = −ρλθ Eθ X > 0 t→∞ t lim

a.s.

This implies (Zt − λθ t(1 + ρ)Eθ X) → +∞

a.s.,

which completes the proof. !

Consider a conjugate family (Fθ )θ∈Θ governing a random walk {St }t≥0 in discrete or continuous time. Define FT := σ(St ; t ≤ T ), with the usual extension to stopping times.

Next, we present the version of the Wald’s fundamental identity used

by Asmussen [1,2]. The proof can be seen in [3].

Theorem 3.3 Let τ be a stopping time for {St }t≥0 and G ∈ Fτ , G ⊆ {τ < ∞}. Then for each θ0 , θ ∈ Θ (6)

Pθ0 G = Eθ [exp {(θ0 − θ)Sτ − τ χθ (θ0 − θ)} ; G] .

From Definition 2.3 and (6)

(7)

dPθ0 = exp {(θ0 − θ)Yτ − τ χθ (θ − θ0 )} , dPθ

and integrating (7) over {τ < ∞} we can express the ruin probability in infinite time as

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

44

Luis F. Hoyos-Reyes

Proposition 3.4 Let θ > 0. If we compute n simulations of the conjugate process Rθ := exp {(θ0 − θ)Yτ − τ χθ (θ − θ0 )} , then with probability 1 n 1!

Ri → Ψ(u) n i=1 θ

n → ∞,

as

where Rθi 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 each of the n simulations of Rθ can be performed in a finite number of steps.

Moreover, as Eθ Rθ = Ψ(u), by the strong law of large numbers it follows that, with probability 1,

"

n 1!

Ri → Ψ(u) n i=1 θ

as

n → ∞.

!

We call n1 ni=1 Rθi a Monte Carlo estimator (MCE) for Ψ(u).

Observe that integrating (7) over {τ < T } we can write the ruin probability in a finite time T as

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

Then, in this case, the corresponding conjugate process is RθT := exp {(θ0 − θ)Yτ − τ χθ (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 RθT0 = I {τ < T }. Thus to simulate RθT0 is equivalent to simulate the original process Yt , which, in insurance terminology is called a crude simulation [7,8].

Insurance ruin problems

45

(b) We can simplify RθT taking θ as the Lundberg value θ1 := γ + θ0 , where γ > 0 is the unique solution of Lundberg’s equation χθ0 (γ) = 0. In this case, RθT1 is called the Lundberg process.

Using Proposition 2.7 one can see that χθ1 (θ0 − θ1 ) = 0, which implies that RθT1 = exp(−γYτ ) · I {τ < T } .

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

(8)

RθT1 (1+∆) = exp {−(γ + θ1 ∆)Yτ + τ χθ1 (θ1 ∆)} · I {τ < T } .

(c) In (b), the corresponding variance σθ2 = Varθ RθT is σθ21 (1+∆) = Eθ1 [exp {−2(γ + θ1 )Yτ + τ χθ1 (θ1 ∆)} · I {τ < T }]−Ψ2 (u, T ), and for the infinite horizon case is

σθ21 (1+∆) = Eθ1 exp {−2(γ + θ1 )Yτ + τ χθ1 (θ1 ∆)} − Ψ2 (u). (d) The overshot B(u) of the risk process, defined as B(u) := Yτ − u is useful to calculate σθ21 . It is known [1,2] that when the claims are exponentially distributed and the arrival process is Poisson (P/E case), B(u) is exponentially distributed:

(9)

4 4.1

Pθ (B(u) > b) = exp(−b/Eθ X).

P/E and P/U examples Example P/E

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

46

Luis F. Hoyos-Reyes

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

(10)

Ψ(u) =

"

!

1 ρu exp − .

1+ρ µ(1 + ρ)

Let us consider the P/E case with µ := EX = 1, λ = 0.8, ρ = 0.1 and T = ∞. The right-hand side of (10) depends on the initial capital u. Let Ψ(u) = 0.05. Then the initial capital is u = 31.904, and the premium income rate is c = (1 + ρ)λµ (remember that we deal with a compound 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: σθ20 = Eθ0 I 2 {τ < ∞} − Eθ20 I{τ < ∞} = Ψ(u) − Ψ2 (u) = 0.0475 σθ21 = Varθ1 Rθ1 = Varθ1 e−γTτ = Varθ1 e−γ(u+B(u)) = e−2γ Varθ1 e−γB(u) .

From (1), Eθ1 X = (1 − γ)−1 , which together with (9) implies that B(u) ∼ exp(1 − γ). Thus Eθ1 e−2γB(u) = (1 − γ)/(1 + γ)

and

Eθ1 e−γB(u) = 1 − γ.

Hence, σθ21

=e

−2γu

!

"

1−γ − (1 − γ)2 = 2.08 × 10−5 < σθ20 .

1+γ

Note that the difference between the variances is significant, which is an 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 Ntθ is a Poisson process with arrival rate c, and Xθ is exponentially distributed with parameter λ/c.

47

Insurance ruin problems

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 31.9 31.9 16.7

n 100 1000 1000 !

Ψ(u) 0.05 0.05 0.20

ˆ Ψ(u) 0.0498 0.0499 0.1997

σM CE 4.5 × 10−4 1.4 × 10−4 5.7 × 10−4

SM CE 5.00 × 10−4 1.41 × 10−4 5.90 × 10−4

εR 4.0 × 10−3 2.0 × 10−3 1.5 × 10−3

"1/2

σM CE := σθ2 /n is the standard error of the MCE, SM CE is the corresponding estimator, and the relative error is ˆ εR :=| 1 − Ψ(u)/Ψ(u) |.

Notice the good fitness between the standard error σM CE and its estimator SM CE . Obviously, we have better aproximations to Ψ(u) taking larger 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 to show an expression for the conjugate process RθT1 (1−∆) . From (1) e(θ−θ0 )x − 1 , 0 < x < 1. eθ−θ0 − 1 Recall that under Pθ , Zt is also a compound Poisson process with parameter Fθ (x) =

#

$

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)

RθT1 (1+∆)

%

= exp −γ0 Yτ + τ

%

eγ+θ1 − 1 γ0

whereas in the infinite time horizon

&

&

− 1 − γ0 c · I{τ < T },

48

(12)

Luis F. Hoyos-Reyes

!

Rθ1 (1+∆) = exp −γ0 Yτ + τ

!

eγ+θ1 − 1 γ0

"

"

− 1 − γ0 c .

Let γ = 0.05. Then from Lundberg’s equation eγ − 1 − 1 − cγ = 0, γ 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, so we cannot compare the real and the estimated values like we did under the P/E assumptions. However, it is possible to estimate the variance of the conjugate process and to compute the estimator SM CE of the standard error; see Table 2.

Table 2: Infinite Time Horizon P/U ˆ ∆ Ψ(u) σ ˆθ2 SM CE −2 1.00 0.199 1.5 × 10 1.2 × 10−2 −4 0.10 0.223 6.2 × 10 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 consistent with the asymptotic optimality proved by Asmussen [2].

5

Concluding remarks

In the previous sections we have introduced MCEs for the ruin probability using conjugate processes. In particular, we have shown formulations for 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, and so it is suitable for the simulation approach.

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

Insurance ruin problems

49

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

Acknowledgement The author is grateful to Dr. On´esimo Hern´andez-Lerma for his valuable comments and useful suggestions.

Luis F. Hoyos-Reyes Departamento de Sistemas, UAM - Azcapotzalco, Av. San Pablo No. 180, 02200 M´exico D.F., MEXICO, hrlf@correo.azc.uam.mx.

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

[2] Asmussen, S., Conjugate processes and the simulation of ruin problems, 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 theory: A matrix-algorithmic approach, Insurance: Mathematics and Economics 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-IV Proceedings, Mexico City 1990.

[8] Embrechts, P. and Wouters, P., Simulating risk solvency, Insurance: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.

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Luis F. Hoyos-Reyes

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

Morfismos, Vol. 5, No. 2, 2001, pp. 51–61 Morfismos, Vol. 5, No. 2, 2001, pp. 51–61

Sobre la estrechez de un espacio topol´ogico ∗ Alejandro Ram´ırez P´aramo

1

Resumen En este trabajo se muestran algunos resultados sobre la estrechez en 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| · sup{t(Xs ) : s ∈ S}, cuando X y Xs pertenecen a C2 para toda s ∈ S.

2000 Mathematics Subject Classification: 54A25.

Keywords and phrases: Estrechez, clase C2 .

1

Introducci´ on

En este trabajo se presentan algunos resultados conocidos y otros (posiblemente) no tan conocidos sobre la funci´on cardinal estrechez; cabe se˜ nalar que los resultados proposici´on 3.2, el corolario 3.4 y el lema 4.10, no se encuentran en la bibliograf´ıa consultada por el autor.

Una funci´on cardinal topol´ogica (llamada, tambi´en, invariante cardinal topol´ogico), es una funci´on φ que va de la clase de los espacios topol´ogicos (algunas veces de una subclase de ´estos) a la clase de los n´ umeros cardinales infinitos de tal forma que φ(X) = φ(Y ) para espacios X y Y homeomorfos (para un estudio detallado sobre funciones cardinales recomendamos al lector [3] y [4]).

Una cuesti´on inmediata sobre funciones cardinales es c´omo determinar el cardinal que debe asociarse al espacio X; lo cual, en muchas ∗

El contenido de este trabajo representa parte de la tesis de grado presentada por el autor dentro del programa de maestr´ıa de la Facultad de Ciencias F´ısicoMatem´ aticas de la Universidad Aut´ onoma de Puebla.

1 Estudiante inscrito en el programa de doctorado de la Facultad de Ciencias F´ısico-Matem´ aticas de la Universidad Aut´ onoma de Puebla.

51

52

Alejandro Ram´ırez P´ aramo

ocasiones no es sencillo de responder. De aqu´ı la necesidad de tener resultados que permitan obtener cotas para las funciones cardinales.

En la tercera secci´on de este trabajo, se relaciona el concepto de conjunto κ−cerrado con la estrechez para abordar el problema anterior.

Se muestra adem´as, en esta secci´on, que las funciones t (estrechez) y hπχ (π−caracter hereditario) satisfacen la desigualdad t(X) ≤ hπχ(X) para cualquier espacio topol´ogico. De manera natural surge la pregunta: ¿para qu´e espacios X se da la igualdad t(X) ≤ hπχ(X)?; este problema es abordado en la cuarta secci´on; en esta misma se muestra que si {Xs : s ∈ S} es una familia de espacios topol´ogicos, tales que cada Xs es Haus! ! dorff y compacto, entonces t( {Xs : s ∈ S}) = |S| · tS ( {Xs : s ∈ S}), ! en donde tS ( {Xs : s ∈ S}) = sup{t(Xs ) : s ∈ S}.

2

Notaciones y definiciones

Aqu´ı, ω representa ambos, el primer ordinal y cardinal infinito, adem´as, κ denota un cardinal el cual siempre ser´a ≥ ω y κ+ es el sucesor de κ.

Usamos P(X) para denotar al conjunto potencia de X y |X| para la cardinalidad de X. Si S es un conjunto y κ un cardinal, [S]κ denota la colecci´on de subconjuntos de S con cardinalidad κ; [S]≤κ se usa para la colecci´on de subconjuntos de S con cardinalidad ≤ κ y [S]<κ denotar´a a la colecci´on de subconjuntos de S con cardinalidad menor que κ.

Sea X un espacio topol´ogico, x ∈ X y A un subconjunto de X. La clausura de A en X se denota clX (A). Con Vx denotamos al conjunto de abiertos en X que contienen a x. Por otro lado, x es punto de acumulaci´on completo de A, si para todo U ∈ Vx , se cumple que |A| = |A ∩ U |. Es posible demostrar que en un espacio Hausdorff y compacto, cualquier conjunto infinito tiene un punto de acumulaci´on completo. Se usa C2 para designar a la clase de los espacios Hausdorff y compactos. Si {Xs : s ∈ S} es una familia, no vac´ıa, de espacios topol´ogicos, denotamos con X su producto cartesiano con la topolog´ıa ! producto. Si S0 ⊆ S es no vac´ıo, XS0 denota al espacio producto {Xs : s ∈ S0 }. Adem´as, bajo estas condiciones, prS0 : X → XS0 es la funci´on proyecci´on.

Las funciones cardinales, estrechez y π−caracter se definen a trav´es de los siguientes n´ umeros cardinales: la estrechez y el π−caracter del punto x ∈ X son, respectivamente: t(x, X) = min{β : ∀C ⊆ X con x ∈ clX C, 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 vac´ıo en X y para cada U ∈ Vx , existe V ∈ V tal que V ⊆ U ). La estrechez y el π−caracter del espacio X se definen, respectivamente, de la siguiente manera: t(X) = sup{t(x, X) : x ∈ X} + ω y πχ(X) = sup{πχ(x, X) : x ∈ X} + ω. Otra funci´on cardinal que usaremos es la amplitud cuya definici´on es como sigue s(X) = min{|A| : A es discreto en X} + ω.

Se dice que una funci´on cardinal φ es mon´otona si para todo Y subespacio de X, φ(Y ) ≤ φ(X). No es dif´ıcil demostrar que una funci´on cardinal φ es mon´otona si hφ(X) = sup{φ(Y ) : Y es subespacio de X} = φ(X).

3

La estrechez

Un concepto de gran utilidad al trabajar con la estrechez es la κ−clausura de un conjuto: Sea X un espacio topol´ogico, A un subconjunto de X y κ un n´ umero 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.

Proposici´ on 3.1 Sea X un espacio topol´ ogico y κ un n´ umero cardinal.

Para todo C ∈ P(X)\{∅}, C es κ−cerrado si y s´ olo si C = [C]κ .

Demostraci´ on: La necesidad es inmediata. Para demostrar la suficiencia, sea C ⊆ X no vac´ıo, y sea A ⊆ [C]κ con |A| ≤ κ; por demostrar que clX (A) ⊆ [C]κ . Para cada x ∈ A, existe Bx ⊆ C tal ! que x ∈ clX (Bx ) y |Bx | ≤ κ. Sea B = x∈A Bx , entonces B ⊆ C y " |B| ≤ x∈A |Bx | ≤ κ · κ = κ, de donde clX (B) es uno de los uniendos ! en [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 . As´ı [C]κ es κ−cerrado en X. ✷ Proposici´ on 3.2 Sea f : X → Y una funci´ on cerrada y κ un n´ umero cardinal. Si C ∈ P(X)\{∅} es κ−cerrado en X entonces, f (C) es κ−cerrado en Y .

Demostraci´ on: Sea C ∈ P(X)\{∅} κ−cerrado en X y A ⊆ f (C), no vac´ıo, tal que |A| ≤ κ. Queremos demostrar que clY (A) ⊆ f (C). Por cada a ∈ A, elija xa ∈ C tal que f (xa ) = a; sea C ′ el conjunto formado por tales puntos. Entonces C ′ ⊆ C tal que |C ′ | ≤ κ; as´ı, puesto que C

54

Alejandro Ram´ırez P´ aramo

es κ−cerrado en X, clX (C ′ ) ⊆ C; de donde f (clX (C ′ )) ⊆ f (C). Claramente 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) es κ−cerrado en Y . ✷ El siguiente resultado es de gran utilidad, permite acotar por arriba a la estrechez de un espacio.

Lema 3.3 Sea X un espacio topol´ ogico arbitrario y κ un cardinal.

Entonces t(X) ≤ κ si y s´ olo si para todo C ∈ P(X)\{∅}, clX (C) = [C]κ .

Demostraci´ on: Para probar la necesidad, sea C ∈ P(X)\{∅}. Puesto que [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 y x ∈ clX (B) ⊆ [C]κ . Por tanto clX (C) ⊆ [C]κ .

Ahora probaremos la suficiencia. Sea C ⊆ X y x ∈ clX (C), por hip´otesis, 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 sigue que t(X) ≤ κ. ✷ Corolario 3.4 t(X) ≤ κ si y s´ olo si [C]κ es cerrado en X para todo C ∈ P(X)\{∅}. (Equivalentemente, t(X) ≤ κ si y s´ olo si todo subconjunto κ−cerrado en X es cerrado en X.) Teorema 3.5 t(X) es igual al menor cardinal κ con la propiedad de que para cualquier subconjunto no cerrado C de X existe B ⊆ C tal que |B| ≤ κ y clX (B)\C ̸= ∅.

Demostraci´ on: Sea β = t(X). Veamos primero que κ ≤ β. Sea C ⊆ X no cerrado, entonces podemos tomar x ∈ clX (C)\C; luego, existe B ⊆ C tal que x ∈ clX (B) y |B| ≤ t(x, X) ≤ β. Evidentemente clX (B)\C ̸= ∅.

De aqu´ı que β satisface la misma propiedad que κ; por tanto κ ≤ β.

Para verificar que β ≤ κ, suponemos que [C]κ no es cerrado, por hip´otesis existe A ⊆ [C]κ tal que |A| ≤ κ y clX (A)\[C]κ ̸= ∅, lo cual contradice la proposici´on 3.1. Por tanto [C]κ es cerrado; as´ı β ≤ κ. ✷ La estrechez es una funci´on mon´otona y se preserva bajo mapeos cerrados.

Proposici´ on 3.6 (i) La funci´ on t es mon´ otona. (ii) Si f : X → Y es continua y cerrada de X sobre Y , entonces t(Y ) ≤ t(X).

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55

Demostraci´ on: (i) Observe que si C es un subconjunto del subespacio Y de X, y κ = t(X), entonces [C]Yκ = Y ∩ [C]κ ; donde [C]Yκ es la κ−clausura de C en Y . (ii) Sea F ∈ P(Y )\{∅} y κ = t(X). Demostraremos que [F ]κ es cerrado en Y . Denotemos por E al conjunto [F ]κ .

Afirmaci´on: f −1 (E) = [f −1 (E)]κ . En efecto, como κ = t(X), del lema 3.3 se tiene que [f −1 (E)]κ = clX (f −1 (E)), de donde, trivialmente f −1 (E) ⊆ [f −1 (E)]κ .

Por otro lado, si x ∈ [f −1 (E)]κ , entonces existe C ⊆ f −1 (E) tal que x ∈ clX C 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 tanto f (x) ∈ E; as´ı, 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 la sobreyectividad de f y el hecho de que f es cerrada se tiene que E es cerrado en Y , i.e. [F ]κ es cerrado en Y ; por tanto t(Y ) ≤ κ = t(X). ✷ Proposici´ on 3.7 Para cualquier espacio X, t(X) ≤ hπχ(X); donde hπχ(X) = sup{πχ(Y ) : Y es subespacio de X}.

Demostraci´ on: Sea x0 ∈ X y C un subconjunto de X tal que x0 ∈ clX (C); sea adem´as κ = hπχ(X). Como πχ(clX (C)) ≤ κ, existe una π−base local B de x0 en clX (C) tal que |B| ≤ κ.

Note que si B ∈ B, entonces B ∩ C ̸= ∅. En efecto, para tal B ∈ B, existe UB abierto en X tal que B = UB ∩ clX (C). As´ı, si y ∈ B, entonces y ∈ clX (C) y y ∈ UB , por tanto UB ∩ C ̸= ∅; pero UB ∩ C ⊆ UB ∩ clX (C) = B. Por tanto B ∩ C ̸= ∅. De aqu´ı resulta que, para cada B ∈ B, es posible tomar yB ∈ B ∩ C; sea M = {yB : B ∈ B}.

No es dif´ıcil verificar que M ⊆ C, |M | ≤ κ y x0 ∈ clX (M ). ✷ La igualdad en t(X) ≤ hπχ(X) se tiene cuando X es Hausdorff y compacto, lo cual se demostrar´a en la secci´on 4.

Terminamos esta secci´on con un resultado que nos dice c´omo encontrar una cota por arriba para la estrechez de un espacio producto (no necesariamente formado con espacios Hausdorff y compactos).

Teorema 3.8 Sea {Xs : s ∈ S} una familia no vac´ıa de espacios topol´ ogicos (no vac´ıos) y κ un cardinal. Si para todo S0 ∈ [S]<ω , t(XS0 ) ≤ κ, y |S| ≤ κ, entonces t(X) ≤ κ.

Demostraci´ on: Sea C ⊆ X y x ∈ clX (C). Queremos demostrar que existe C0 ⊆ C tal que |C0 | ≤ κ y x ∈ clX (C0 ).

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Denotemos por F la colecci´on [S]<ω . Entonces |F| ≤ κ. Ahora bien, ! para cada F ∈ F , sea XF = s∈F Xs y sea prF : X → XF el mapeo proyecci´on sobre la cara XF .

Puesto que t(XF ) ≤ κ y prF (x) ∈ prF (clX C) ⊆ 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 , constr´ uyase DF ⊆ C como sigue: por cada (xs )s∈F ∈ MF , el´ıjase un punto en prF−1 ({(xs )s∈F }) ∩ C. Claramente, |DF | ≤ κ para todo F ∈ F. " Consid´erese ahora C0 = F ∈F DF . Puesto que DF ⊆ C para todo F ∈ F, se tiene que C0 ⊆ C. M´as a´ un, como |F| ≤ κ y |DF | ≤ κ para todo F ∈ F, se tiene que |C0 | ≤ κ.

Por u ´ltimo, veamos que, x ∈ clX (C0 ).

Efectivamente, sea U un abierto can´onico en X que contiene a x, ! digamos U = s∈S Us , donde, para s ∈ S, Us = Xs salvo un n´ umero finito de ´ındices s1 ,...,sn . Entonces A = {s1 , ..., sn } ∈ F; luego, dado que prA (x) ∈ clXA (MA ), existe (xs1 , ..., xsn ) ∈ MA ∩ (prA (C) ∩ (Us1 × · · · × Usn )). Para tal (xs1 , ..., xsn ) existe y ∈ DA , y por tanto en C0 , tal que prA (y) = (xs1 , ..., xsn ). De aqu´ı, ya es claro que U ∩ C0 ̸= ∅. Por tanto x ∈ clX (C0 ). As´ı t(X) ≤ κ. ✷

4

Estrechez y compacidad

En la presente secci´on daremos algunos resultados de la estrechez sobre la clase C2 de espacios Hausdorff y compactos; en particular, probaremos las igualdades se˜ naladas en el resumen.

Definici´ on 4.1 Una sucesi´on {xα : 0 ≤ α < κ} en el espacio X es una sucesi´on libre de longitud κ si para todo β < κ: clX {xα : α < β} ∩ clX {xα : α ≥ β} = ∅.

Observe que si {xα : 0 ≤ α < κ} es una sucesi´on libre de longitud k en X, entonces {xα : 0 ≤ α < κ} es un subconjunto discreto en X. on Teorema 4.2 Si X ∈ C2 y t(X) ≤ κ, entonces X no tiene una sucesi´ + libre de longitud κ .

Demostraci´ on: Supongamos que {xα : 0 ≤ α < κ+ } es una sucesi´on libre de longitud κ+ . Puesto que X es compacto, el conjunto {xα : 0 ≤ α < κ+ } tiene un punto de acumulaci´ on completo, digamos z0 . Por otra parte, dado que t(X) ≤ κ existe β0 < κ+ tal que z0 ∈ clX {xα : 0 ≤

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α < β0 }; luego, dado que clX {xα : α < β0 } ∩ clX {xα : α ≥ β0 } = ∅, existe una vecindad abierta U de z0 tal que U ∩ {xα : α ≥ β0 } = ∅.

De aqu´ı 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 acumulaci´on 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 = ∅. on Teorema 4.4 Si X ∈ C2 y hπχ(X) > κ, entonces X tiene una sucesi´ libre de longitud κ+ .

Demostraci´ on: Puesto que πχ(Y ) ≤ πχ(Y ), para demostrar el resultado es suficiente suponer que πχ(X) > κ. Sea p ∈ X tal que πχ(p, X) ≥ κ+ .

Sea G la colecci´on de todos los conjuntos cerrados, no vac´ıos y Gδ en X. La colecci´on G tiene la siguiente propiedad: (∗) si H ⊆ G y |H| ≤ κ, entonces existe una vecindad abierta R de p tal que H\R ̸= ∅ para todo H ∈ H. En efecto, supongamos que la colecci´on G no satisface (∗); i.e., existe H ⊆ G con |H| ≤ κ tal que para toda vecindad abierta R de p existe HR ∈ H de tal forma que HR ⊆ R = ∅. Ahora, puesto que X es compacto, para cada HR existe UHR abierto en X de tal forma que HR ⊆ U HR ⊆ R. Entonces la colecci´on {UHR : HR ∈ H} es una π−base de p con |{UHR : HR ∈ H}| ≤ κ. Lo cual contradice el hecho de que πχ(p, X) > κ.

Para continuar, construiremos subcolecciones {Aα : 0 ≤ α < κ+ } y {Bα : 0 ≤ α < κ+ } de G tales que: (1) p ∈ Aα y Aα ∩ Bα = ∅, 0 ≤ α < κ+ ; (2) Si H ̸= ∅ es intersecci´on finita de elementos de {Aβ : 0 ≤ β < α} ∪ {Bβ : 0 ≤ β < α}, entonces H ∩ Bα ̸= ∅, 0 < α < κ+ .

La construcci´on es por inducci´on transfinita. Para obtener A0 y B0 , sea K compacto en X tal que p ∈ / K; por el lema 4.3, existen A0 , B0 ∈ G tales 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 construidos de tal modo que (1) y (2) se verifican. Por construir Aα y Bα . Sea H la colecci´on de todas las intersecciones finitas no vac´ıas de elementos de {Aβ : β < α} ∪ {Bβ : β < α}. Entonces H ⊆ G, H ̸= ∅ y |H| ≤ κ,

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as´ı 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, entonces X\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. As´ı termina la construcci´on.

Ahora bien, usando las partes (1) y (2) e inducci´on finita, se puede demostrar que para cada α fijo, 0 < α < κ+ , la colecci´on {Aβ : β ≤ α+ } ∪ {Bβ : β > α+ } satisface la propiedad de la intersecci´ on finita (i.e. cualquier subconjunto finito de dicha colecci´on tiene intersecci´on no vac´ıa).

De lo anterior y la compacidad de X, tenemos que para cada α, ! ! existe xα ∈ ( β≤α Aβ ) ∩ ( β>α Bβ ). Entonces {xα : 0 ≤ α < κ+ } es una sucesi´on libre en X de longitud κ+ . Por supuesto, sea ε < κ+ . Es suficiente demostrar que {xα : α < ε} ⊆ Bε y {xα : α ≥ ε} ⊆ Aε ; pues por la segunda parte de (1) y el hecho de que Aα y Bα son cerrados y ajenos, para todo α < κ+ , tendr´ıamos que clX ({xα : α < ε}) ∩ clX ({xα : α ≥ ε}) = ∅. Para ver la primera contenci´on, note que si α < ε, ! ! entonces xα ∈ ( β≤α Aβ ) ∩ ( β>α Bβ ) ⊆ Bε . Para la segunda, tenemos ! ! que xα ∈ ( β≤α Aβ ) ∩ ( β>α Bβ ), xα ∈ Aβ para β ≤ α, de donde, si α ≥ ε, xα ∈ Aε . ✷ Corolario 4.5 Si X ∈ C2 y t(X) > κ, entonces X tiene una sucesi´ on libre de longitud κ+ .

Demostraci´ on: Como κ < t(X) y t(X) ≤ hπχ(X), entonces hπχ(X) > κ y, por tanto, X tiene una sucesi´on libre de longitud κ+ . ✷ Como se observa en el teorema 4.2 y el corolario 4.5, existe una relaci´on entre la estrechez de un espacio Hausdorff y compacto y la longitud de las sucesiones libres en ´el, a saber: Teorema 4.6 (Arkhangel’ski˘ı) Para X ∈ C2 , t(X) = F (X), donde F (X) = sup{λ : X tiene una sucesi´ on libre de longitud λ} + ω.

Demostraci´ on: Sea κ = t(X) y β = F (X). Si κ < β entonces X tiene una sucesi´on libre de longitud κ+ . Pero esto contradice 4.2. Por tanto κ ≥ β. Si κ > β, entonces, por el corolario 4.5, X tiene una sucesi´on libre de longitud β + ; lo cual contradice la definici´on de β. Por tanto κ = β. ✷

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ˇ Teorema 4.7 (Sapirovsk˘ ı) Para X ∈ C2 , t(X) = hπχ(X). En particular, todo subespacio de un espacio Hausdorff y compacto con estrechez numerable tiene π−caracter numerable.

Demostraci´ on: Puesto que t(X) ≤ hπχ(X), es suficiente demostrar que no ocurre que t(X) < hπχ(X). Si t(X) < hπχ(X), entonces, por el teorema 4.4, X tiene una sucesi´on libre de longitud t(X)+ , contradiciendo el teorema anterior. ✷ Teorema 4.8 (Arkhangel’ski˘ı) Para X ∈ C2 , t(X) ≤ s(X)+. En particular, todo espacio compacto con amplitud numerable tiene estrechez numerable.

Demostraci´ on: Sea κ = s(X). Si t(X) > κ, entonces por el corolario 4.5, X tiene una sucesi´on libre de longitud κ+ . Puesto que toda sucesi´on libre es un conjunto discreto, entonces X tiene un subconjunto discreto de cardinalidad > s(X); lo cual es una contradicci´on. Por tanto t(X) ≤ s(X). ✷ Suponga que {Xs : s ∈ S} es una colecci´on, no vac´ıa, de espacios topol´ogicos. Sup´ongase adem´as que X es el espacio producto de dichos espacios. Bajo el supuesto de que D(2)|S| $→ X (donde D(2)|S| es el cubo de Cantor de peso |S|, vea [2], pg. 84) o F (2)|S| $→ X (donde F (2)|S| es el cubo de Alexandroff de peso |S|, vea [2], pg. 84), es posible demostrar que t(X) ≥ |S| · tS (X), donde tS (X) = {t(Xs ) : s ∈ S}. En general, la igualdad no siempre se da, aun en el caso finito: Ejemplo 4.9 Considere κ un cardinal arbitrario con la topolog´ıa discreta y el espacio compacto Z = { n1 : n ∈ ω}∪{0}. En el producto κ×Z, identificamos los puntos de la forma (α, 0), para α ∈ κ; y denotamos por 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 igualdad en t(X) ≥ |S| · tS (X). La prueba requiere de algunos resultados que a continuaci´on se dan.

El siguiente resultado es una generalizaci´on del lema de la p´agina 113 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 ).

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Demostraci´ on: Sea κ = t(X) · t(Y ) y A un subconjunto κ−cerrado en X × 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 es suficiente con probar que y ∈ prY (B).

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

Supongamos, pues, que B no es vac´ıo. Entonces B es κ−cerrado en {x}×Y . Por supuesto, si C ⊆ B tal que |C| ≤ κ, entonces C ⊆ {x}×Y , C ⊆ A y |B| ≤ κ. Claramente, cl{x}×Y C ⊆ ({x} × Y ); por otro lado, puesto que C ⊆ A y |C| ≤ κ, se tiene que clX×Y (C) ⊆ A; de donde cl{x}×Y C = clX×Y (C) ∩ ({x} × Y ) ⊆ A ∩ ({x} × Y ) = B. As´ı, B es κ−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, que y ∈ / prY (B). Como B es cerrado en {x} × Y y prY : {x} × Y → Y es 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 en X × Y , entonces (X × V ) ∩ A es κ−cerrado en X × Y . En particular, (X ×V )∩A es κ−cerrado en X ×V . Como V es compacto en Y , entonces prX : X × V → X, es cerrada, y por lo tanto prX ((X × V ) ∩ A) es κ cerrado en X. As´ı, por la continuidad de prX : X × V → X, obtenemos que 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; lo que significa que r ∈ V ∩ prY B; lo cual es una contradicci´on. Por tanto y ∈ prY B. La prueba est´a completa. ✷ La siguiente proposici´on es consecuencia inmediata del lema anterior.

Proposici´ on 4.11 Si para cada i ∈ {1, ..., n}, Xi ∈ C2 , entonces !n t( i=1 Xi ) ≤ max{t(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 vac´ıa de espacios topol´ ogicos (no vac´ıos). Si cada Xs ∈ C2 , entonces t(X) = |S| · tS (X).

Demostraci´ on: 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,

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para cualquier S0 ∈ [S]<ω , prS0 es una funci´on cerrada; luego, por la proposici´on 3.2, para cualquier S0 ∈ [S]<ω , prS0 (A) es κ− cerrado en XS0 . As´ı, dado que tS0 (X) ≤ κ, se tiene que prS0 (A) es cerrado en XS0 . Consecuentemente, para cada S0 ∈ [S]<ω prS0 (x) ∈ prS0 (A), de donde se deduce que existe un punto a ∈ A tal que x |S0 = a. Sea B el conjunto formado por estos puntos, entonces B ∈ [A]≤κ , puesto que |[S]<ω | = |S| ≤ κ. Luego, claramente se tiene que x ∈ clX B ⊆ A. Por la definici´on de B y el hecho de que A es κ−cerrado. ✷ Agradecimientos A mi novia Homaira Athenea Ram´ırez Guti´errez, a quien adem´as, dedico este trabajo.

Quiero agradecer a Jes´ us F. Tenorio Arvide su revisi´on y sugerencias a este trabajo.

Alejandro Ram´ırez P´ aramo Facultad de Ciencias F´ısico Matem´ aticas, Benem´erita Universidad Aut´ onoma de Puebla, Av. San Claudio y Rio Verde s/n, Puebla Pue., MEXICO, aparamo@fismat1.fcfm.buap.mx.

Referencias [1] Arkhangel’ski˘ı, A. V., Structure and Classification of Topological Spaces 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 linebreak Set-Theoretic Topology, K. Kunen and J. E. Vaughan, eds., linebreak Amsterdam, 1984.

[4] Juh´asz, I., Cardinal Functions in Topology (ten years later), Heldermann Verlag Berlin, 1989.

Morfismos, Vol. 5, No. 2, 2001, pp. 63–74 Morfismos, Vol. 5, No. 2, 2001, pp. 63–74

Medida de colisi´on de un (α, d, β)-superproceso con su medida inicial ∗ Jos´e Villa Morales

1

Resumen Sea X = {Xt : t ≥ 0} un (α, d, β)-Superproceso. Demostraremos que la medida de colisi´on, M (X), de X con su valor inicial X0 , definida heur´ısticamente por Mt (X)(ϕ) := ⟨ϕ((y + x)/2)δ(y − x), X0 (dy)Xt (dx)⟩ existe. M´as precisamente si Jε , con ε > 0, es un molificador, 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 colisi´ on.

1

Introducci´ on

En este art´ıculo demostraremos la existencia de la medida de colisi´on de un (α, d, β)-superproceso con su valor inicial. Intuitivamente esto significa que al tiempo t el soporte de Xt no est´a muy alejado del soporte de X0 . Nuestro problema es m´as sencillo en relaci´on con lo hecho por Mytnik en [9] y se utilizan las mismas ideas empleadas por este autor (el cual a su vez se basa en los trabajos [3] y [4]). Mytnik demuestra que si X (i) , con parametros (αi , d, βi ), i = 1, 2, son dos (α, d, β)-superprocesos independientes, entonces la medida de colisi´on y el tiempo local de colisi´on existen si d < α1 /β1 + α2 /β2 y d < α1 /β1 + α2 /β2 + max{α1 , α2 }, respectivamente. El caso en que α = 2 y β = 1 es estudiado por Evans 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 trabajo representa parte de la tesis de doctorado que el autor se encuentra desarrollando dentro del programa de doctorado del CIMAT. 1 Estudiante inscrito en el programa de doctorado del CIMAT.

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independencia la existencia del l´ımite anterior depende de α, d y β, sin embargo veremos que en nuestro caso no es as´ı.

1.1

Notaci´ on

• Sea E un espacio m´etrico 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 l´ımite por la izquierda (funciones c`adl`ag), con la topolog´ıa de Skorohod.

• λ denota la media de Lebesgue en Rd .

• MF (Rd ) denota el espacio de las medidas no negativas finitas en Rd , con la topolog´ıa d´ebil.

• Sea B(Rd ) (respectivamente Cb (Rd )) el conjunto de las funciones Borel medibles acotadas en Rd (respectivamente continuas y acotadas). B(Rd )+ ⊂ B(Rd ) (resp. Cb (Rd )+ ⊂ Cb (Rd )) denota al conjunto de las funciones no negativas en B(Rd ) (resp. Cb (Rd )).

• ⟨f, µ⟩ =

1.2

!

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

(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 homog´eneo en el tiempo con trayectorias en DMF (Rd ) [0, ∞), tal que para cada f ∈ B(Rd )+ # " (1)

Eµ e−⟨f,Xt ⟩ = e−⟨Vt (f ),µ⟩ ,

donde µ ∈ MF (Rd ) y Vt (f ) es la soluci´on de la ecuaci´on (2)

Vt (f ) = St (f ) −

$

t 0

St−s (Vs (f )1+β )ds, t ≥ 0.

Aqu´ı {St : t ≥ 0} denota el semigrupo de un proceso de Markov en estable y sim´etrico con exponente α. pt (x, y) = pt (y − x) es la densidad de transici´on de probabilidad correspondiente a St , es decir, para cada f ∈ L1+ (Rd ),

Rd ,

St f (x) =

$

Rd

f (y)pt (y − x)dy, t > 0.

Medida de Colisi´ on

65

{St : t ≥ 0} es un semigrupo de contracci´on, tal que St f ≥ Vt f (ver [6], p. 93).

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

∂Vt ∂t V0

= − (−∆)α/2 − Vt1+β , = f,

"

∆ = di=1 ∂ 2 /∂x2i , de la ecuaci´on (2) tiene una u ´nica soluci´on. Por lo tanto (2) tambi´en tiene una u ´nica soluci´on.

Una manera de probar la existencia de un (α, d, β)-superproceso es como l´ımite d´ebil de un sistema de part´ıculas con ramificaci´on. En ´este contexto {St : t ≥ 0} es el semigrupo del proceso de Markov que determina el movimiento de las part´ıculas. Por otra parte, la ramificaci´on de las part´ıculas, es decir, el n´ umero de hijos que tiene cada part´ıcula, esta dado por una distribuci´on cr´ıtica que pertenece al dominio de atracci´on de una ley estable de exponente 1 + β (ver [8]).

1.3

Medida de colisi´ on de Xt con X0 y resultado principal

Formalmente la medida de colisi´on de Xt con X0 es (3)

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

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

Para hacer preciso (3) introduzcamos la siguiente notaci´on. Sea J una funci´on continua, no negativa, sim´etrica, con #soporte contenido en la bola unitaria {x ∈ Rd : ∥x∥ < 1} y tal que Rd J(x)dx = 1. La funci´on Jε (x) := ε−d J(ε−1 x), ε > 0, se llama molificador.

Definici´ on 1.3.1 Sea X un proceso c`adl`ag MF (Rd )-valuado. Para ε > 0 definimos M ε : DMF (Rd ) [0, ∞) → DMF (Rd ) [0, ∞) por Mtε (X)(ϕ) = ⟨ϕ ((y + x)/2) Jε (y − x) , X0 (dy)Xt (dx)⟩ =

$

Rd

$

Rd

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

66

Jos´e Villa Morales

para cada ϕ ∈ B(Rd )+ . La medida de colisi´on de X con su valor inicial X0 es un proceso M (X) progresivamente medible MF (Rd )-valuado tal que Mtε (X)(ϕ) converge a Mt (X)(ϕ) en probabilidad, cuando ε → 0, para cada t > 0 y cada ϕ ∈ Cb (Rd )+ . El l´ımite no depende de la elecci´on del molificador Jε .

El resultado principal es el: Teorema 1.3.2 Sea X un (α, d, β)-superproceso tal que la medida inicial µ es absolutamente continua con respecto a λ y con densidad acotada, g := dµ/dλ ∈ B(Rd )+ . Entonces la medida de colisi´ on M (X) existe y ! −M (X)(ϕ) " t = e−⟨Vt (ϕg),µ⟩ , E e

para cada t > 0 y cada ϕ ∈ Cb (Rd )+ .

2

Demostraci´ on del Teorema

Comenzaremos por demostrar una serie de lemas antes de dar la prueba del resultado principal. Un resultado que ser´a esencial es el siguiente.

Lema 2.0.3 Sea ϕ ∈ B(Rd ), entonces lim || ⟨ϕ ((y + ·)/2) Jε (y − ·) , µ(dy)⟩ − ϕ(·)g(·)||1 = 0

ε→0

Demostraci´ on:

Por el teorema de cambio de variable tenemos que

⟨ϕ ((y + x)/2) Jε (y − x) , µ(dy)⟩ =

#

=

#

1 J ((y − x)/ε) ϕ ((y + x)/2) εd g(y)dy Rd

Rd

ϕ(x + εy/2)g(x + εy)J(y)dy.

Ya que µ ∈ MF (Rd ), entonces g ∈ L1 (Rd ), as´ı por el Teorema de Fubini resulta ∥⟨ϕ ((y + ·)/2) Jε (y − ·) , µ(dy)⟩ − ϕ(·)g(·)∥1

= ≤ =

#

d

|

#

#R # #

Rd Rd

Rd

Rd

ϕ(x + εy/2)g(x + εy)J(y)dy −

#

Rd

ϕ (x) g(x)J(y)dy|dx

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

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

Medida de Colisi´ on

Para mostrar la convergencia usaremos el teorema de la convergencia dominada de Lebesgue. Para esto comencemos notando que g ∈ L1 y ϕ ∈ B(Rd ) implican que ϕg ∈ L1 , pues ∥ϕg∥1 ≤ ∥ϕ∥∞ ∥g∥1 . Entonces limz→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]). As´ı de la estimaci´on ||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 adem´as ||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 afirmaci´on. ✷ Lema 2.0.4 Sea 1 ≤ k ≤ N y τ = T /N . Sean ϕi ∈ B(Rd )+ , i = 1, 2, entonces !

≤ 2

kτ (k−1)τ ! kτ

||Vt (ϕ1 )1+β − Vt (ϕ2 )1+β ||1 dt

(k−1)τ ! kτ

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

+2

(k−1)τ

||St (ϕ1 + ϕ2 ) ||β∞ dt

!

kτ 0

||Vs (ϕ1 )1+β − Vs (ϕ2 )1+β ||1 ds.

Demostraci´ on: De la desigualdad |x1+β − y 1+β | ≤ 2|x − y|(x + y)β , x, y ≥ 0, obtenemos !

≤

kτ

(k−1)τ ! kτ (k−1)τ

||Vt (ϕ1 )1+β − Vt (ϕ2 )1+β ||1 dt !

Rd

2| (Vt (ϕ1 ) − Vt (ϕ2 )) (x)| (Vt (ϕ1 ) + Vt (ϕ2 ))β (x)dxdt

67

68

Jos´e Villa Morales

(Usando el Lema 2.1 de [6] y el hecho de que St es un semigrupo lineal, resulta) ≤ 2 ≤ 2

!

!

kτ (k−1)τ kτ (k−1)τ

!

Rd

| (Vt (ϕ1 ) − Vt (ϕ2 )) (x)|St (ϕ1 + ϕ2 )β (x)dxdt

||St (ϕ1 + ϕ2 )||β∞

!

Rd

| (Vt (ϕ1 ) − Vt (ϕ2 )) (x)|dxdt.

Por otra parte, de la relaci´on (2) vemos que |Vt (ϕ1 ) − Vt (ϕ2 )| ≤ |St (ϕ1 − ϕ2 )| +

!

t 0

|St−s (Vs (ϕ1 )1+β − Vs (ϕ2 )1+β )|ds.

As´ı !

≤ 2

kτ (k−1)τ ! kτ

+

||Vt (ϕ1 )1+β − Vt (ϕ2 )1+β ||1 dt

(k−1)τ ! t 0

||St (ϕ1 +

ϕ2 )||β∞

!

Rd

(|St (ϕ1 − ϕ2 )(x)|

|St−s (Vs (ϕ1 )1+β − Vs (ϕ2 )1+β )(x)|ds)dxdt.

Por ser St una contracci´on y t ≤ kτ , implica que !

≤ 2

kτ

(k−1)τ ! kτ

+

||Vt (ϕ1 )1+β − Vt (ϕ2 )1+β ||1 dt

(k−1)τ ! kτ 0

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

||Vs (ϕ1 )1+β − Vs (ϕ2 )1+β ||1 ds)dt.

Con lo cual la afirmaci´on queda demostrada. ✷ Una consecuencia de los lemas anteriores es el siguiente resultado.

Lema 2.0.5 Para toda T > 0 lim

!

ε→0 0

T

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

69

Medida de Colisi´ on !

"

Demostraci´ on: Sea N = 2β+1 ∥ϕ∥β∞ ∥g∥β∞ T + 1 ([·] funci´on mayor entero) y sea τ = T /N . Por el Lema 2.0.4 obtenemos (4) =

#

T

∥Vt (⟨ϕ ((y 0 N # kτ $

k=1 (k−1)τ

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

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

−Vt (ϕ(·)g(·))1+β ∥1 dt ≤

# N $

kτ

2

(k−1)τ

k=1

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

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

k=1 kτ

·

kτ

2

+ #

0

(k−1)τ

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

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

−Vt (ϕ(·)g(·))1+β ∥1 dt.

Ahora n´otese que (5)

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

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

= sup x

#

Rd

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

≤ ∥ϕ∥∞ ∥g∥∞ sup x

= 2∥ϕ∥∞ ∥g∥∞ .

#

Rd

pt (z − x)(

#

Rd

Jε (y − z)dy + 1)dz

Usando (4) y el hecho que St es una contracci´on nos queda (6)

#

kτ

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

lim sup ε→0

−ϕ(·)g(·))∥1 dt

≤ lim sup 2β ∥ϕ∥β∞ ∥g∥β∞ ε→0

−ϕ(·)g(·)∥1 dt

#

kτ

(k−1)τ

∥ ⟨ϕ ((y + ·)/2) Jε (y − ·), µ(dy)⟩

70

Jos´e Villa Morales

= 2β ∥ϕ∥β∞ ∥g∥β∞ τ lim sup ∥ ⟨ϕ ((y + ·)/2) Jε (y − ·), µ(dy)⟩ ε→0

−ϕ(·)g(·)∥1

= 0.

La u ´ltima 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 segunda serie. Para k ≥ 1 definamos (7)

bk = lim sup ε→0

!

kτ 0

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

−Vt (ϕ(·)g(·))1+β " ds, 1

y b0 = 0. Supongamos que bk = 0 y veamos que bk+1 = 0. Del Lema 2.0.4 y (6) resulta que !

bk+1 = bk + lim sup

(k+1)τ

kτ

ε→0

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

−Vs (ϕ(·)g(·))1+β ∥1 ds !

(k+1)τ

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

≤ lim sup 2 ε→0

−ϕ(·)g(·))∥1 dt + lim sup 2 ε→0

!

(k+1)τ

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

kτ

+ϕ(·)g(·))∥β∞

!

(k+1)τ

0 1+β

−Vs (ϕ(·)g(·))

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

∥1 ds

≤ 0 + 2β+1 ∥ϕ∥β∞ ∥g∥β∞ τ lim sup ε→0

!

(k+1)τ 0

∥Vs (⟨ϕ ((y + ·)/2)

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

En la u ´ltima desigualdad hemos usado (5). As´ı 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 sumatoria de (4) y usando (5), como antes, y (7) obtenemos la prueba del resultado. ✷

71

Medida de Colisi´ on

Lema 2.0.6 Para toda t > 0 lim ∥Vt (⟨ϕ ((y + ·)/2) Jε (y − ·), µ(dy)⟩) − Vt (ϕ(·)g(·))∥1 = 0.

ε→0

Demostraci´ on:

Usando (2) y que St es una contracci´on nos da

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

ε→0

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

+ lim

!

t

ε→0 0

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

−Vs (ϕ(·)g(·))1+β )∥1 ds

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

+ lim

!

ε→0 0

t

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

−Vs (ϕ(·)g(·))1+β ∥1 ds.

Por los Lemas 2.0.3 y 2.0.5 tenemos que los dos sumandos de la derecha de la desigualdad son cero. En consecuencia el l´ımite es cero. ✷ Finalmente veamos el siguiente resultado.

Lema 2.0.7 Sea {Z ε : 0 ≤ ε ≤ 1} una familia de procesos MF (Rd )-valuados progresivamente medibles tal que lim E

ε,ε′ →0

"#

e

−Ztε (ϕ)

−e

′

−Ztε (ϕ)

$2 %

= 0,

para toda t > 0 y cada ϕ ∈ Cb (Rd )+ . Entonces existe un proceso Z progresivamente medible MF (Rd )-valuado tal que Ztε converge en probabilidad a Zt , cuando ε → 0, para toda t > 0.

Demostraci´ on:

Ya que

&

ε

e−Zt (ϕ)

'

es una sucesi´on de Cauchy en L2 , ε

entonces existe una variable aleatoria Yt (ϕ) tal que e−Zt (ϕ) −→Yt (ϕ) ε en L2 , cuando ε → 0. Por lo tanto e−Zt (ϕ) −→Yt (ϕ) en probabilidad, cuando ε → 0. As´ı Ztε (ϕ) → Zt (ϕ) := − log Yt (ϕ), ε → 0, pues − log(·) es una funci´on continua. Es decir, para cada, t > 0 Zt es una variable aleatoria MF (Rd )-valuada tal que lim P {ω : d(Ztε (ω), Zt (ω)) > δ} = 0,

ε→0

72

Jos´e Villa Morales

para cada δ > 0, donde d es una m´etrica en MF (Rd ), la cual corresponde a la topolog´ıa d´ebil (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 teorema de la convergencia dominada de Lebesgue tenemos que lim (λ × P ){d(Ztε , Zt ) > δ} = lim

ε→0

!

ε→0 [0,T ]×Ω ! T

P {d(Ztε , Zt ) > δ}dt

= lim

ε→0 0 ! T

=

0

1{d(Ztε ,Zt )>δ} (ω, t)dP dt

lim P {d(Ztε , Zt ) > δ}dt = 0.

ε→0

Z ε →Z

Es decir en λ × P , cuando ε → 0. Ya que Z ε es B([0, T ]) × FT medible, entonces tambi´en lo ser´a Z (ver Teorema 5.2.9 de [7]). As´ı el proceso Z es progresivamente medible. ✷ Demostraci´ on: (del Teorema 1.3.2) Comencemos por observar que Mtε (X)(ϕ) es igual a ⟨⟨ϕ ((y + x)/2) Jε (y − x), X0 (dy)⟩ , Xt (dx)⟩, entonces "

#

ε

#

"

E e−Mt (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 )+ (m´as a´ un ∥f ∥∞ ≤ ∥ϕ∥∞ ∥g∥∞ µ(Rd )) y µ = X0 en (1) nos da (8)

"

#

ε

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

An´alogamente vemos que $

E e

′

−(Mtε (X)(ϕ)+Mtε (X)(ϕ))

%

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

En consecuencia E

&'

e

−Mtε (X)(ϕ)

−e

′

−Mtε (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)⟩ .

73

Medida de Colisi´ on

Usando la desigualdad |e−a − e−b | ≤ |a − b|, a, b ∈ R, tenemos por el Lema 2.0.6 que lim E

ε,ε→0

!"

e

−Mtε (X)(ϕ)

−e

′ −Mtε (X)(ϕ)

#2 $

= 0.

Entonces por el Lema 2.0.7 existe un proceso M (X) progresivamente medible y MF (Rd )-valuado, tal que Mtε (X)(ϕ) converge en probabilidad a Mt (X)(ϕ) cuando ε → 0, para cada t > 0. Y adem´as de (8) y el Lema 2.0.6 resulta que %

E e−Mt (X)(ϕ)

&

%

ε

= lim E e−Mt (X)(ϕ)

&

ε→0 −⟨Vt (ϕ(x)g(x)⟩),µ(dx)⟩

= e

.

As´ı queda demostrado el resultado. ✷ Agradecimientos Le agradezco a Jos´e Alfredo Lopez-Mimbela el haber revisado la presente nota.

Jos´e Villa Morales Departamento de Matem´ aticas y F´ısica Universidad de Aut´ onoma de Aguascalientes CIMAT Apartado Postal 402 36000, Guanajuato, Gto. MEXICO, villa@cimat.mx.

Referencias ´ ´ e de [1] Dawson, D., Measure-valued Markov processes, Ecole d’Et´ Probabilit´es de Saint-Flour XXI, Lecture Notes in Math. 1541, Springer, Berlin, 1991.

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

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

[4] Fleischmann, K and G¨artner, J., Occupation time processes at a critical point, Math. Nachr. 125 (1986), 275-290.

74

Jos´e Villa Morales

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

[6] Iscoe, I., A weighted ocupation time for a class of measure-valued branching processes, Probab. Theory Relat. Fields 71 (1986), 85116.

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

[8] M´el´eard, S. and Roelly, S., Discontinuous measure-valued branching 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 Matem´aticas del CINVESTAV, se termin´ o de imprimir en el mes de enero de 2002 en el taller de reproducci´ on del mismo departamento localizado en Av. IPN 2508, Col. San Pedro Zacatenco, M´exico, D.F. 07300. El tiraje en papel opalina importada de 36 kilogramos de 34 × 25.5 cm consta de 500 ejemplares en pasta tintoreto color verde.

Apoyo t´ecnico: Omar Hern´ andez Orozco.

Contenido

Degree and fixed point index. An account Carlos Prieto . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Existence of Nash equilibria in nonzero-sum ergodic stochastic games in Borel spaces Rafael Ben´ıtez-Medina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Monte Carlo approach to insurance ruin problems using conjugate processes Luis F. Hoyos-Reyes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Sobre la estrechez de un espacio topolo ´gico Alejandro Ram´ırez P´ aramo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Medida de colisin de un (a, d, b)-superproceso con su medida inicial Jos´e Villa Morales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63