TIME-DEPENDENT NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS DRIVEN BY A FRACTIONAL BROWNIAN MOTION

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1 Communications on Stochastic Analysis Vol. 1, No. 1 ( Serials Publications TIME-DEPENDENT NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS DRIVEN BY A FRACTIONAL BROWNIAN MOTION B. BOUFOUSSI, S. HAJJI, AND E. LAKHEL Abstract. In this paper we consider a class of time-dependent neutral stochastic functional differential equations with finite delay driven by a fractional Brownian motion with Hurst parameter H ( 1, 1, in a separable real 2 Hilbert space. We prove an existence and uniqueness result of mild solution by means of the Banach fixed point principle. A practical example is provided to illustrate the viability of the abstract result of this work. 1. Introduction The stochastic functional differential equations have attracted much attention because of their practical applications in many areas such as physics, medicine, biology, finance, population dynamics, electrical engineering, telecommunication networks, and other fields. For more details, one can see Da Prato and Zabczyk [7], and Ren and Sun [14] and the references therein. In many areas of science, there has been an increasing interest in the investigation of the systems incorporating memory or aftereffect, i.e., there is the effect of delay on state equations. Therefore, there is a real need to discuss stochastic evolution systems with delay. In many mathematical models the claims often display long-range memories, possibly due to extreme weather, natural disasters, in some cases, many stochastic dynamical systems depend not only on present and past states, but also contain the derivatives with delays. Neutral functional differential equations are often used to describe such systems. Very recently, neutral stochastic functional differential equations driven by fractional Brownian motion have attracted the interest of many researchers. One can see [4, 6, 1, 8, 9] and the references therein. The literature concerning the existence and qualitative properties of solutions of time-dependent functional stochastic differential equations is very restricted and limited to a very few articles. This fact is the main motivation of our work. We mention here the recent paper by Ren et al. [13] concerning the existence of mild solutions for a class of stochastic evolution equations driven by fractional Brownian motion in Hilbert space. Received ; Communicated by T. Duncan. 21 Mathematics Subject Classification. Primary 6H15; Secondary 6J65. Key words and phrases. Neutral stochastic evolution equations, evolution operator, fractional Brownian motion, Wiener integral, Banach fixed point theorem. 1

2 2 B. BOUFOUSSI, S. HAJJI, AND E. LAKHEL Motivated by the above works, this paper is concerned with the existence and uniqueness of mild solutions for a class of time-dependent neutral functional stochastic differential equations described in the form: d[x(t + g(t, x(t r(t] = [A(tx(t + f(t, x(t ρ(t]dt + σ(tdb H (t, t T, x(t = φ(t, τ t, (1.1 in a real Hilbert space X with inner product <.,. > and norm., where A(t, t [, T ] is a family of linear closed operators from a space X into X that generates an evolution system of operators {U(t, s, s t T }. B H is a fractional Brownian motion on a real and separable Hilbert space Y, r, ρ : [, + [, τ] (τ > are continuous and f, g : [, + X X, σ : [, + L 2(Y, X, are appropriate functions. Here L 2(Y, X denotes the space of all Q-Hilbert-Schmidt operators from Y into X (see section 2 below. On the other hand, to the best of our knowledge, there is no paper which investigates the study of time-dependent neutral stochastic functional differential equations with delays driven by fractional Brownian motion. Thus we will make the first attempt to study such problem in this paper. Our results are inspired by the one in [4] where the existence and uniqueness of mild solutions to model (1.1 with A(t = A, t [, T ], is studied, as well as some results on the asymptotic behavior. The substance of the paper is organized as follows. Section 2, recapitulate some notations, basic concepts, and basic results about fractional Brownian motion, Wiener integral over Hilbert spaces and we recall some preliminary results about evolution operator. We need to prove a new technical lemma for the L 2 estimate of stochastic convolution integral which is different from that used by [4]. Section 3, gives sufficient conditions to prove the existence and uniqueness for the problem (1.1. In Section 4 we give an example to illustrate the efficiency of the obtained result. 2. Preliminaries 2.1. Evolution families. In this subsection we introduce the notion of evolution family. Definition 2.1. A set {U(t, s : s t T } of bounded linear operators on a Hilbert space X is called an evolution family if (a U(t, su(s, r = U(t, r, U(s, s = I if r s t, (b (t, s U(t, sx is strongly continuous for t > s. Let {A(t, t [, T ]} be a family of closed densely defined linear unbounded operators on the Hilbert space X and with domain D(A(t independent of t, satisfying the following conditions introduced by [1]. There exist constants λ, θ ( π 2, π, L, K, and µ, ν (, 1] with µ + ν > 1 such that Σ θ {} ρ(a(t λ, R(λ, A(t λ K 1 + λ (2.1

3 TIME-DEPENDENT NEUTRAL SFDE DRIVEN BY FBM 3 and (A(t λ R(λ, A(t λ [ R(λ, A(t R(λ, A(s ] L t s µ λ ν, (2.2 for t, s R, λ Σ θ where Σ θ := { λ C {} : arg λ θ }. It is well known, that this assumption implies that there exists a unique evolution family {U(t, s : s t T } on X such that (t, s U(t, s L(X is continuous for t > s, U(, s C 1 ((s,, L(X, t U(t, s = A(tU(t, s, and A(t k U(t, s C(t s k (2.3 for < t s 1, k =, 1, α < µ, x D((λ A(s α, and a constant C depending only on the constants in (2.1-(2.2. Moreover, + s U(t, sx = U(t, sa(sx for t > s and x D(A(s with A(sx D(A(s. We say that A( generates {U(t, s : s t T }. Note that U(t, s is exponentially bounded by (2.3 with k =. Remark 2.2. If {A(t, t [, T ]} is a second order differential operator A, that is A(t = A for each t [, T ], then A generates a C semigroup {e At, t [, T ]}. For additional details on evolution system and their properties, we refer the reader to [12] Fractional Brownian Motion. Let (Ω, F, P be a complete probability space. A standard fractional Brownian motion (fbm {β H (t, t R} with Hurst parameter H (, 1 is a zero mean Gaussian process with continuous sample paths such that R H (t, s = E[β H (tβ H (s] = 1 2( t 2H + s 2H t s 2H, s, t R. (2.4 Remark 2.3. In the case H > 1 2, it follows from [11]that the second partial derivative of the covariance function R H t s = α H t s 2H 2, where α H = H(2H 2, is integrable, and we can write R H (t, s = α H s u v 2H 2 dudv. (2.5 Let X and Y be two real, separable Hilbert spaces and let L(Y, X be the space of bounded linear operator from Y to X. For the sake of convenience, we shall use the same notation to denote the norms in X, Y and L(Y, X. Let Q L(Y, Y be an operator defined by Qe n = λ n e n with finite trace trq = n=1 λ n <. where λ n (n = 1, 2... are non-negative real numbers and {e n } (n = 1, 2... is a complete orthonormal basis in Y. We define the infinite dimensional fbm on Y with covariance Q as B H (t = BQ H (t = λn e n βn H (t, n=1

4 4 B. BOUFOUSSI, S. HAJJI, AND E. LAKHEL where β H n are real, independent fbm s. This process is Gaussian, it starts from, has zero mean and covariance: E B H (t, x B H (s, y = R(s, t Q(x, y, x, y Y, t, s [, T ]. In order to define Wiener integrals with respect to the Q-fBm, we introduce the space L 2 := L 2(Y, X of all Q-Hilbert-Schmidt operators ψ : Y X. We recall that ψ L(Y, X is called a Q-Hilbert-Schmidt operator, if ψ 2 L := λ n ψe n 2 <, 2 n=1 and that the space L 2 equipped with the inner product φ, ψ L 2 = n=1 φe n, ψe n is a separable Hilbert space. Let ϕ(s; s [, T ] be a function with values in L 2(Y, X, such that K ϕq 1 2 en 2 L <. 2 n=1 The Wiener integral of ϕ with respect to B H is defined by ϕ(sdb H (s = n=1 λn ϕ(se n dβ H n (s. (2.6 Now, we end this subsection by stating the following result which is fundamental to prove our result. It can be proved by similar arguments as those used to prove Lemma 2 in [6]. Lemma 2.4. If ψ : [, T ] L 2(Y, X satisfies T ψ(s 2 <, then the L2ds above sum in (2.6 is well defined as a X-valued random variable and we have E ψ(sdb H (s 2 2Ht 2H 1 ψ(s 2 L2ds The stochastic convolution integral. In this subsection we present a few properties of the stochastic convolution integral of the form Z(t = U(t, sσ(sdb H (s, t [, T ], where σ(s L 2(Y, X and {U(t, s, s t T } is an evolution system of operators. The properties of the process Z are crucial when regularity of the mild solution to stochastic evolution equation is studied, see [7] for asystematic account of the theory of mild solutions to infinite-dimensional stochastic equations. Unfortunately, the process Z is not a martingale, and standard tools of the martingale theory, yielding e.g. continuity of the trajectories or L 2 estimates are not available. The following result on the stochastic convolution integral Z holds. Lemma 2.5. Suppose that σ : [, T ] L 2(Y, X satisfies sup t [,T ] σ(t 2 L 2 <, and suppose that {U(t, s, s t T } is an evolution system of operators

5 TIME-DEPENDENT NEUTRAL SFDE DRIVEN BY FBM 5 satisfying U(t, s Me β(t s, for some constants β > and M 1 for all t s. Then, we have E U(t, sσ(sdb H (s 2 CM 2 t 2H ( sup σ(t L 2 2. t [,T ] Proof. Let {e n } n N be the complete orthonormal basis of Y and {βn H } n N is a sequence of independent, real-valued standard fractional Brownian motion each with the same Hurst parameter H ( 1 2, 1. Thus using fractional Itô isometry one can write E U(t, sσ(sdb H (s 2 = = E n=1 n=1 U(t, sσ(se n dβ H n (s 2 < U(t, sσ(se n, U(t, rσ(re n > H(2H 1 s r 2H 2 dsdr H(2H 1 { U(t, sσ(s U(t, rσ(r s r 2H 2 dr}ds H(2H 1M 2 {e β(t s σ(s L 2 Since σ is bounded, one can then conclude that E U(t, sσ(sdb H (s 2 H(2H 1M 2 ( sup e β(t r s r 2H 2 σ(r L 2 dr}ds. t [,T ] σ(t L 2 2 e β(t r s r 2H 2 dr}ds. {e β(t s Make the following change of variables, v = t s for the first integral and u = t r for the second. One can write E U(t, sσ(sdb H (s 2 H(2H 1M 2 ( sup t [,T ] e βu u v 2H 2 du}dv H(2H 1M 2 ( sup t [,T ] u v 2H 2 dudv. σ(t L 2 2 σ(t L 2 2 {e βv

6 6 B. BOUFOUSSI, S. HAJJI, AND E. LAKHEL By using (2.5, we get that E U(t, sσ(sdb H (s 2 CM 2 t 2H ( sup σ(t L 2 2. t [,T ] Remark 2.6. Thanks to Lemma 2.5, the stochastic integral Z(t is well-defined. 3. Existence and Uniqueness of Mild Solutions In this section we study the existence and uniqueness of mild solutions of equation (1.1. Henceforth we will assume that the family {A(t, t [, T ]} of linear operators generates an evolution system of operators {U(t, s, s t T }. Before stating and proving the main result, we give the definition of mild solutions for equation (1.1. Definition 3.1. A X-valued process {x(t, t [ τ, T ]}, is called a mild solution of equation (1.1 if i x(. C([ τ, T ], L 2 (Ω, X, ii x(t = φ(t, τ t. iii For arbitrary t [, T ], x(t satisfies the following integral equation: x(t = U(t, (φ( + g(, φ( r( g(t, x(t r(t + U(t, sa(sg(s, x(s r(sds + U(t, sσ(sdb H (s P a.s We introduce the following assumptions: (H.1 U(t, sf(s, x(s ρ(sds i The evolution family is exponentially stable, that is, there exist two constants β > and M 1 such that U(t, s Me β(t s, for all t s, ii There exist a constant M > such that A 1 (t M for all t [, T ]. (H.2 The maps f, g : [, T ] X X are continuous functions and there exist two positive constants C 1 and C 2, such that for all t [, T ] and x, y X: i f(t, x f(t, y g(t, x g(t, y C 1 x y. ii f(t, x 2 A k (tg(t, x 2 C 2 (1 + x 2, k =, 1. (H.3 i There exists a constant < L < 1 M such that A(tg(t, x A(tg(t, y L x y, for all t [, T ] and x, y X. ii The function g is continuous in the quadratic mean sense: for all x(. C([, T ], L 2 (Ω, X, we have lim E g(t, x(t g(s, t s x(s 2 =.

7 (H.4 TIME-DEPENDENT NEUTRAL SFDE DRIVEN BY FBM 7 i The map σ : [, T ] L 2(Y, X is bounded, that is: there exists a positive constant L such that σ(t L 2 (Y,X L uniformly in t [, T ]. ii Moreover, we assume that the initial data φ = {φ(t : τ t } satisfies φ C([ τ, ], L 2 (Ω, X. The main result of this paper is given in the next theorem. Theorem 3.2. Suppose that (H.1-(H.4 hold. Then, for all T >, the equation (1.1 has a unique mild solution on [ τ, T ]. Proof. Fix T > and let B T := C([ τ, T ], L 2 (Ω, X be the Banach space of all continuous functions from [ τ, T ] into L 2 (Ω, X, equipped with the supremum norm Let us consider the set x 2 B T = sup E x(t, ω 2. τ t T S T (φ = {x B T : x(s = φ(s, for s [ τ, ]}. S T (φ is a closed subset of B T provided with the norm. BT. We transform (1.1 into a fixed-point problem. Consider the operator ψ on S T (φ defined by ψ(x(t = φ(t for t [ τ, ] and for t [, T ] ψ(x(t = U(t, (φ( + g(, φ( r( g(t, x(t r(t = + 5 I i (t. i=1 U(t, sa(sg(s, x(s r(sds + U(t, sσ(sdb H (s U(t, sf(s, x(s ρ(sds Clearly, the fixed points of the operator ψ are mild solutions of (1.1. The fact that ψ has a fixed point will be proved in several steps. We will first prove that the function ψ is well defined. Step 1: ψ is well defined. Let x S T (φ and t [, T ], we are going to show that each function t I i (t is continuous on [, T ] in the L 2 (Ω, X-sense. We can easily see that E I i (t + h I i (t 2, i = 1, 2, 3 as h. For the fourth term I 4 (h, we suppose h > (similar calculus for h <. We have I 4 (t + h I 4 (t (U(t + h, s U(t, sf(s, x(s ρ(sds +h + (U(t, sf(s, x(s ρ(sds t I 41 (h + I 42 (h.

8 8 B. BOUFOUSSI, S. HAJJI, AND E. LAKHEL and By Hölder s inequality, we have E I 41 (h te U(t + h, s U(t, sf(s, x(s ρ(s 2 ds. Again exploiting properties of Definition 2.1, we obtain lim (U(t + h, s U(t, sf(s, x(s ρ(s =, h U(t + h, s U(t, sf(s, x(s ρ(s Me β(t s (e βh + 1 f(s, x(s ρ(s L 2 (Ω. Then we conclude by the Lebesgue dominated theorem that lim h E I 41(h 2 =. On the other hand, by (H.1, (H.2, and the Hölder s inequality, we have E I 42 (h M 2 C 2 (1 e 2βh 2β +h Thus lim I 42(h =. h Now, for the term I 5 (h, we have Since and I 5 (h + By Lemma 2.4, we get that +h t (1 + E x(s ρ(s 2 ds. (U(t + h, s U(t, sσ(sdb H (s t I 51 (h + I 52 (h. U(t + h, sσ(sdb H (s E I 51 (h 2 2Ht 2H 1 [U(t + h, s U(t, s]σ(s 2 L2ds. lim [U(t + h, s U(t, h s]σ(s 2 L = 2 (U(t + h, s U(t, sσ(s L 2 MLe β(t s e βh+1 L 1 ([, T ], ds, we conclude, by the dominated convergence theorem that, lim E I 51(h 2 =. h Again by Lemma 2.4, we get that Thus E I 52 (h 2 2Ht2H 1 LM 2 (1 e 2βh. 2β lim E I 52(h 2 =. h

9 TIME-DEPENDENT NEUTRAL SFDE DRIVEN BY FBM 9 The above arguments show that lim h E ψ(x(t + h ψ(x(t 2 =. Hence, we conclude that the function t ψ(x(t is continuous on [, T ] in the L 2 -sense. Step 2: Now, we are going to show that ψ is a contraction mapping in S T1 (φ with some T 1 T to be specified later. Let x, y S T (φ, by using the inequality (a + b + c 2 1 ν a ν b ν c2, where ν := L M < 1, we obtain for any fixed t [, T ] ψ(x(t ψ(y(t 2 1 g(t, x(t r(t g(t, y(t r(t 2 ν ν ν 3 = J k (t. k=1 U(t, sa(s(g(s, x(s r(s g(s, y(s r(sds 2 U(t, s(f(s, x(s ρ(s f(s, y(s ρ(sds 2 By using the fact that the operator (A 1 (t is bounded, combined with the condition (H.3, we obtain that E J 1 (t 1 ν A 1 (t 2 E A(tg(t, x(t r(t A(tg(t, y(t r(t 2 L2 M 2 ν E x(t r(t y(t r(t 2 ν sup E x(s y(s 2. s [ τ,t] By hypothesis (H.3 combined with Hölder s inequality, we get that E J 2 (t E 2 1 ν U(t, s [A(tg(t, x(t r(t A(tg(t, y(t r(t] ds 2M 2 L 2 1 ν M 2 e 2β(t s ds 1 e 2βt t 2β sup s [ τ,t] E x(s r y(s r 2 ds E x(s y(s 2. Moreover, by hypothesis (H.2 combined with the Hölder s inequality, we can conclude that E J 3 (t E 2C2 1 1 ν U(t, s [f(s, x(s ρ(s f(s, y(s ρ(s] ds 2 2M 2 C ν M 2 e 2β(t s ds 1 e 2βt t 2β sup s [ τ,t] E x(s r y(s r 2 ds E x(s y(s 2.

10 1 B. BOUFOUSSI, S. HAJJI, AND E. LAKHEL Hence where sup s [ τ,t] E ψ(x(s ψ(y(s 2 γ(t sup s [ τ,t] E x(s y(s 2, γ(t = ν + [L 2 + C1] 2 2M 2 1 e 2βt t 1 ν 2β By condition (H.3, we have γ( = ν = L M < 1. Then there exists < T 1 T such that < γ(t 1 < 1 and ψ is a contraction mapping on S T1 (φ and therefore has a unique fixed point, which is a mild solution of equation (1.1 on [ τ, T 1 ]. This procedure can be repeated in order to extend the solution to the entire interval [ τ, T ] in finitely many steps. This completes the proof. 4. An Example In recent years, the interest in neutral systems has been growing rapidly due to their successful applications in practical fields such as physics, chemical technology, bioengineering, and electrical networks. We consider the following stochastic partial neutral functional differential equation with finite delays τ 1 and τ 2 ( τ i τ <, i = 1, 2, driven by a cylindrical fractional Brownian motion: [ d [u(t, ζ + G(t, u(t τ 1, ζ] = u(t, ζ + b(t, ζu(t, ζ 2 2 ζ ] +F (t, u(t τ 2, ζ dt + σ(tdb H (t, t T, ζ π, u(t, = u(t, π =, t T, u(t, ζ = φ(t, ζ, t [ τ, ], ζ π, (4.1 where B H is a fractional Brownian motion, b(t, ζ is a continuous function and is uniformly Hölder continuous in t, F, G : R + R R are continuous functions. To study this system, we consider the space X = L 2 ([, π] and the operator A : D(A X X given by Ay = y with D(A = {y X : y X, y( = y(π = }. It is well known that A is the infinitesimal generator of an analytic semigroup (T (t t on X. Furthermore, A has discrete spectrum with eigenvalues n 2, 2 n N and the corresponding normalized eigenfunctions given by e n := π sin nx, n = 1, 2,..., in addition (e n n N is a complete orthonormal basis in X and T (tx = e n2t < x, e n > e n, n=1 for x X and t. Now, we define an operator A(t : D(A X X by A(tx(ζ = Ax(ζ + b(t, ζx(ζ.

11 TIME-DEPENDENT NEUTRAL SFDE DRIVEN BY FBM 11 By assuming that b(.,. is continuous and that b(t, ζ γ (γ > for every t R, ζ [, π], it follows that the system { u (t = A(tu(t, t s, u(s = x X, has an associated evolution family given by [ U(t, sx(ζ = T (t s exp( s ] b(τ, ζdτx (ζ. From this expression, it follows that U(t, s is a compact linear operator and that for every s, t [, T ] with t > s U(t, s e (γ+1(t s. In addition, A(t satisfies the assumption H 1 (see [2]. To rewrite the initial-boundary value problem (4.1 in the abstract form we assume the following: i The substitution operator f : [, T ] X X defined by f(t, u(. = F (t, u(. is continuous and we impose suitable conditions on F to verify assumption H 2. ii The substitution operator g : [, T ] X X defined by g(t, u(. = G(t, u(. is continuous and we impose suitable conditions on G to verify assumptions H 2 and H 3. iii The function σ : [, T ] L 2(L 2 ([, π], L 2 ([, π] is bounded, that is, there exists a positive constant L such that σ(t L 2 L <, where L := sup t [,T ] e t. uniformly in t [, T ]. If we put u(t(ζ = u(t, ζ, t [, T ], ζ [, π], u(t, ζ = φ(t, ζ, t [ τ, ], ζ [, π], then the problem (4.1 can be written in the abstract form d[x(t + g(t, x(t r(t] = [A(tx(t + f(t, x(t ρ(t]dt + σ(tdb H (t, x(t = φ(t, τ t. (4.2 Furthermore, if we assume that the initial data φ = {φ(t : τ t } satisfies φ C([ τ, ], L 2 (Ω, X, thus all the assumptions of Theorem 3.2 are fulfilled. Therefore, we conclude that the system (4.1 has a unique mild solution on [ τ, T ]. Acknowledgements: The authors would like to thank the referee for valuable comments and suggestions on this work. References 1. Acquistapace, P. and Terreni, B.: A unified approach to abstract linear parabolic equations, Tend. Sem. Mat. Univ. Padova. 78 (1987,

12 12 B. BOUFOUSSI, S. HAJJI, AND E. LAKHEL 2. Aoued, D. and Baghli, S.: Mild solutions for Perturbed evolution equations with infinite state-dependent delay. Electronic Journal of Qualitative Theory of Differential Equations. (213, No. 59, Boufoussi, B. and Hajji, S.: Functional differential equations driven by a fractional Brownian motion. Computers and Mathematics with Applications. 62 (211, Boufoussi, B. and Hajji, S.: Neutral stochastic functional differential equation driven by a fractional Brownian motion in a Hilbert space. Statist. Probab. Lett. 82 (212, Boufoussi, B., Hajji, S., and Lakhel, E.: Functional differential equations in Hilbert spaces driven by a fractional Brownian motion. Afrika Matematika. Volume 23, Issue 2 (211, Caraballo, T., Garrido-Atienza, M. J., and Taniguchi, T.: The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion. Nonlinear Analysis. 74 ( 211, Da Prato, G. and Zabczyk, J.: Stochastic Equations in Infinite Dimension. Cambridge University Press, Cambridge Lakhel, E.: On the Controllability for Neutral Stochastic Functional Differential Equations Driven by a Fractional Brownian Motion in a Hilbert Space. Chapter 7 In book :Brownian Motion: Elements, Dynamics, and Applications Lakhel, E.: Controllability of Time-dependent Neutral stochastic functional differential equation driven by a fractional Brownian motion in a Hilbert space. Journal of non linear Sciences and Applications. (To appear. 1. Hajji, S. and Lakhel, E.: Existence and uniqueness of mild solutions to neutral Sfdes driven by a fractional Brownian motion with non-lipschitz coefficients. Journal of Numerical Mathematics and Stochastics. 7 (215, (1, Nualart, D.: The Malliavin Calculus and Related Topics, second edition. Springer-Verlag, Berlin Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York Ren, Y., Cheng, X., and Sakthivel, R.: On time-dependent stochastic evolution equations driven by fractional Brownian motion in Hilbert space with finite delay. Mathematical methods in the Applied Sciences Ren, Y. and Sun, D.: Second-order neutral impulsive stochastic differential equations with delay. J. Math. Phys. 5 (29, B. Boufoussi: Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, Marrakesh, 239, Morocco address: boufoussi@uca.ma S. Hajji: Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, Marrakesh, 239, Morocco address: hajjisalahe@gmail.com E. Lakhel: National School of Applied Sciences, Cadi Ayyad University, Safi, 46, Morocco address: e.lakhel@uca.ma

13 Communications on Stochastic Analysis Vol. 1, No. 1 ( Serials Publications STOCHASTIC INTEGRAL REPRESENTATIONS OF F-SELFDECOMPOSABLE AND F-SEMI-SELFDECOMPOSABLE DISTRIBUTIONS NADJIB BOUZAR Abstract. We study two specific mappings defined on the class of infinitely divisible distributions on Z + with finite log-moments. As a consequence, we re-derive some known stochastic integral representations for F- selfdecomposable distributions and obtain some new ones for F-semi-selfdecomposable distributions. Stochastic integral representations for discrete distributions in nested subclasses of F-selfdecomposable and F-semi-selfdecomposable distributions are established via the iterates of the aforementioned mappings. 1. Introduction Let I(R d denote the class of infinitely divisible distributions on R d. Stochastic integral representations for distributions in sub-classes of I(R d have been the object of numerous articles over the last three decades. The subclasses SD(R d of selfdecomposable distributions (see Sato [14], for e.g. and SSD α (R d, α (, 1, of semi-selfdecomposable distributions (Maejima and Naito [11] have been paid particular attention. We cite two important characterization results. For that we will need some notation first. We denote by L(Y the probability law of a random variable Y and by t the greatest integer function at t R. For a probability law µ I(R d, {X (µ t } will designate a Lévy process such that L(X (µ 1 = µ. We define the subclass I log (R d by { } I log (R d = µ I(R d : ln + x µ(dx <, R d where ln + x = max(ln x,. A random variable X on R d has a selfdecomposable distribution if and only if there exits a Lévy process {X (µ t } such that X d = in which case µ = L(X (µ 1 I log (R d. e t dx (µ t, (1.1 Received ; Communicated by the editors. 21 Mathematics Subject Classification. Primary 6E7; Secondary 6G51. Key words and phrases. Lévy processes, stochastic integral, composition semigroups, F- stability, infinite divisibility, probability generating functions, weak convergence. 13

14 14 NADJIB BOUZAR A random variable X on R d has a semi-selfdecomposable distribution with order (span α (, 1 if and only if there exits a Lévy process {X (µ } such that X d = t α t dx (µ t, (1.2 in which case µ = L(X (µ 1 I log (R d. The representation (1.1 was obtained by Wolfe [19] in the case d = 1 and generalized to the case d > 1 by Sato and Yamazato [15], and to Banach spacevalued random variables by Jurek and Vervaat [9]. The result was extended to operator selfdecomposable distributions by Sato and Yamazato [16]. Versions of (1.1 for distributions in nested subclasses of SD(R d were given in [9], [15], and Barndorff-Nielsen et al. [2]. Maejima and Ueda [12] established the representation (1.2 for distributions in SSD α (R d and extended it to distributions in nested subclasses of SSD α (R d. Maejima and Miura [1] obtained a stochastic integral representation for distributions in SSD α (R d similar to (1.1 but where the integration was with respect to a semi-lévy process. For more on the stochastic integrals with respect to Lévy processes, additive processes, and semi-lévy processes, we refer to Rocha-Arteaga and Sato [13] and to references therein. van Harn et al. [7] proposed discrete analogues of self-decomposability and stability for distributions on Z + := {, 1, 2, }. We recall below some important facts on the topic and refer without further mention to [7] and to Steutel and van Harn [18], Chapter 5, Section 8, for more details. The Z + -valued multiple α F X for a Z + -valued random variable X and < α < 1 is defined as follows: X α F X = Y k (t := Z X (t (t = ln α, (1.3 k=1 where Y 1 (, Y 2 (, are independent copies of a continuous-time Markov branching process, independent of X, such that for every k 1, P (Y k ( = 1 = 1. The processes (Y k (, k 1 are driven by a composition semigroup of probability generating functions (pgf s F := (F t, t : F s F t (z = F s+t (z ( z 1; s, t. (1.4 The process Z X ( of (1.3 is itself a Markov branching process driven by F and starting with X individuals (Z X ( = X. Let P (z be the pgf of X. Then the pgf P α F X(z of α F X is given by P α F X(z = P (F t (z ( t = ln α; z 1. (1.5 As an analogue of scalar multiplication, the operation F must satisfy some minimal conditions. In particular, the following regularity conditions are imposed on the composition semigroup F: lim F t (z = F (z = z, t lim F t (z = 1. (1.6 t

15 INTEGRAL REPRESENTATIONS 15 The first part of (1.6 implies the continuity of the semigroup F (by way of (1.4 and the second part is equivalent to assuming that m = F 1(1 1, which implies the (sub-criticality of the continuous-time Markov branching process Y k ( in (1.3. We will restrict ourselves to the subcritical case (m < 1 and we will assume without loss of generality that m = e 1. A Z + -valued random variable X, or its distribution, is said to have an F-selfdecomposable distribution if for every t >, X d = e t F X + X t, (1.7 where X t is Z + -valued and X and X t are independent (see [7]. Equivalently, by (1.5 and (1.7, a distribution on Z + with pgf P (z is F-selfdecomposable if for every t >, there exists a pgf P t (z such that P (z = P (F t (zp t (z ( z 1, (1.8 where P and P t are the pgf s of X and X t, respectively, in (1.7. F-self-decomposable distributions are infinitely divisible. Moreover the distribution of X t is infinitely divisible for every t >. Let < α < 1. A Z + -valued random variable X with a nondegenerate distribution is said to be F-semi-selfdecomposable of order α if X satisfies (1.7 for some infinitely divisible Z + -valued random variable X t independent of X and for t = ln α (Bouzar [4]. F-semi-selfdecomposable distributions are infinitely divisible. F-semistable and F-geometric semistable distributions are F- semi-selfdecomposable (see [4]. Steutel et al. [17] introduced a stochastic integral with respect to a Z + - valued Lévy process and used it to give an integral representation of (continuous time sub-critical branching processes with immigration. They showed that F-selfdecomposable distributions arise as the weak limit for such processes (as t. Although not explicitly noted by the authors, but implicit in the proof of their main result (Theorem 3.2, is an integral representation for F- selfdecomposable distributions. The aim of this paper is to obtain the discrete analogues of the representations (1.1 and (1.2 for F-selfdecomposable and F-semi-selfdecomposable distributions and to extend Theorem 3.2 in [17] to F-semi-selfdecomposable distributions. We parallel the treatment for the continuous case, as covered in [12]. In Section 2, we recall some facts on the stochastic integral with respect to a Z + -valued Lévy process and formally describe the associated improper integral. In Section 3, we define the mapping Ψ α (α (, 1 on the class of infinitely divisible distributions on Z + and identify its domain and range. As a corollary, we re-derive a stochastic integral representation for F-selfdecomposable distributions obtained implicitly in [17]. In addition, we describe a fixed-point property for Ψ α in terms of F-stable distributions. In Section 4, we introduce and study the operator Φ α and study its properties. We obtain a stochastic integral representation for F semi-selfdecomposable distributions with order α (, 1 and we show that these distributions arise as weak limits to a specific class of Z + -valued stochastic processes. Finally, in Section 5, stochastic integral representations for distributions in

16 16 NADJIB BOUZAR nested subclasses of F-decomposable and F-semi-selfdecomposable distributions are derived via the iterates of Ψ α and Φ α, respectively. In the rest of this section we recall some additional definitions and results about the continuous composition semigroup of pgf s F := (F t, t. The following characterization of F-self-decomposable distributions (see [18], Chapter V, Theorem 8.3 plays a key role throughout the paper. Theorem 1.1. A function P (z on [, 1] is the pgf of an F-selfdecomposable on Z + if and only if it has the form ln P (z = 1 z ln Q(x U(x dx = ln Q(F t (z dt ( z 1, (1.9 where U(z is the infinitesimal generator of the semigroup F (see definition below and Q(z is the pgf of an infinitely divisible distribution on Z + such that 1 [ ln Q(x] U(x dx <. (1.1 The infinitesimal generator U of the semigroup F is defined by U(z = lim t (F t (z z/t ( z 1, (1.11 and satisfies U(z > for z < 1. Moreover, The related A-function is defined by { A(z = exp U(x 1 x (x 1. (1.12 z } (U(x 1 dx The functions U(z and A(z satisfy for any t >, and ( z < 1. (1.13 t F t(z = U(F t (z = U(zF t(z ( z 1, (1.14a A(F t (z = e t A(z ( z < 1. (1.14b Following standard terminology, we will refer to any Z + -valued Lévy process as a Z + -valued subordinator which we denote by {X (µ t }, with L(X (µ 1 = µ. We will be referring quite frequently to the following sets: - I(Z + : the set of infinitely divisible distributions on Z +. - I log (Z + : the subset of distributions µ = (q n, n in I(Z + such that n= q n ln(n + 1 <. - F-SD(Z + : the subset of F-self-decomposable distributions in I(Z +. - F-SSD α (Z + : the subset of distributions in I(Z + that are F-semiselfdecomposable of order α (, 1.

17 INTEGRAL REPRESENTATIONS A Stochastic Integral With Respect to a Z + -valued Subordinator Let (X (µ t, t be a Z + -valued subordinator. We will assume without loss of generality that X =. Since Z + -valued subordinators are piecewise constant, it follows from Theorem 21.2, p. 135, in [14], that {X (µ t } is a compound Poisson process, i.e., there exists a Poisson process {N t } and a sequence of Z + -valued random variables {C k } with C =, independent of {N t }, such that or, equivalently, X (µ t = X (µ t = N t k=1 C k, {k: <T k t} C k, (2.1 where (T k, k 1 is the sequence of jump times of {X (µ t }. We recall the definition of the stochastic integral with respect to {X (µ t } introduced in [17]. Let f(s be a (Lebesgue measurable function on an interval [t, t 1 ] in [, and taking values in (, 1]. We define 1 f(s F dx s (µ t d = {k: t <T k t 1 } f(t k F C k. (2.2 As noted in [17], the operation A F X for a random variable A taking values in [, 1] has pgf 1 P A F (z = P a F (z dg A (a, where G A is the distribution function of A. We also recall a useful representation theorem obtained in [17]. Theorem 2.1. Let f(s be a measurable function on [t, t 1 ] with range in (, 1]. The stochastic integral 1 t f(s F dx s (µ has an infinitely divisible distribution with pgf { 1 ( P (z = exp ln Q µ F ln f(s (z } ds, (2.3 t where Q µ (z is the (infinitely divisible pgf of X (µ 1. Next, we describe an improper integral. Let {X (µ t } be a Z + -valued subordinator and f(s a measurable function on [, with values in (, 1]. We define the Z + -valued process {Y t } by Y t d = f(s F dx (µ s d = {k: <T k t} f(t k F C k (t. (2.4 If {Y t } converges in distribution to a Z + -valued random variable Y as t, then we set f(s F dx (µ s d = Y. (2.5

18 18 NADJIB BOUZAR Theorem 2.2. Let {X (µ t } be a Z + -valued subordinator and f(t a measurable function on [, with values in (, 1]. Assume that the process {Y t } of (2.4 converges weakly as t. Then (i the distribution of f(s F dx s (µ is infinitely divisible and its pgf admits the representation { P (z = exp ( ln Q µ F ln f(s (z } ds. (2.6 (ii f(s F dx s (µ admits the following infinite series representation f(s F dx (µ s d = k=1 f(t k F C k. (2.7 Proof. By Theorem 2.1, L( f(s F dx s (µ is infinitely divisible as the weak limit of the infinitely divisible distributions {L(Y t }, when t. Since the pgf of Y t is (by (2.3 { ( P t (z = exp ln Q µ F ln f(s (z } ds, (2.8 it ensues that the pgf of f(s F dx s (µ admits the representation (2.6 by letting t in (2.8. The infinite series representation (2.7 follows from the second equation in (2.4. Remark 2.3. One can arrive at an equivalent definition of the stochastic integral (2.2 that is similar to the one constructed in the continuous case (see for e.g. [13], by first defining it for step functions and then extending the definition to measurable functions via the standard limit process. 3. A Stochastic Integral Representation for F-selfdecomposable Distributions Let α (, 1. We define the mapping Ψ α ( on I(Z + by ( Ψ α (µ = L α s F dx s (µ, (3.1 where {X (µ t } is a Z + -valued subordinator with L(X (µ 1 = µ. We identify the domain D(Ψ α and range R(Ψ α of the mapping Ψ α and as a consequence we obtain an integral representation for F-selfdecomposable distributions on Z +. We start out with a useful lemma. Lemma 3.1. Assume µ I(Z + with pgf Q µ (z. The following assertions are equivalent. (i µ I log (Z +. (ii (1.1 holds for Q = Q µ (z. (iii [ ln Q µ(f s (z] ds < for every z [, 1].

19 INTEGRAL REPRESENTATIONS 19 Proof. (i (ii: By Proposition 4.2, Appendix A, in [18], µ I log (Z + if and only if 1 [ ln Q µ (z] 1 x dx <. The conclusion follows by applying (1.12. The statement (ii (iii follows essentially from the proof of Theorem 8.3, Chapter V, in [18]. Next, we give the main result of the section. Parts of the statement of Theorem 3.2 below can be derived from the proof of Theorem 3.2 in [17]. We provide a somewhat abbreviated proof for the sake of completeness. Theorem 3.2. D(Ψ α = I log (Z + and R(Ψ α = F-SD(Z +. Proof. Assume µ D(Ψ α. By applying (2.6 to f(s = α s, one can show that the pgf P Ψα(µ(z of Ψ α (µ admits the representation (1.9 with Q(z = 1/ ln α Qµ (z. Therefore, by Theorem 1.1, Ψ α (µ is F-selfdecomposable and Q µ (z satisfies (1.1. It follows by Lemma 3.1[(ii (i] that µ I log (Z +. We have thus shown that D(Ψ α I log (Z + and R(Ψ α F-SD(Z +. Assume now that µ I log (Z +. By Lemma 3.1[(i (iii] and Theorem 1.1, the function P (z = exp { ln Q µ (F s ln α (z ds } is the pgf of an F-selfdecomposable distribution. Letting P t (z be the pgf of Y t = αs F dx s (µ and applying Theorem 2.1, as well as (2.8, to f(s = α s, we conclude that lim t P t (z = P (z. Therefore, L( α t F dx s (µ exists as the weak limit of {Y t } as t, or equivalently, µ D(Ψ α. We have thus shown I log (Z + D(Ψ α. It remains to prove that F-SD(Z + R(Ψ a. Let P (z be the pgf of an F-selfdecomposable distribution on Z +. Then P (z admits the canonical representation (1.9 for some infinitely divisible pgf Q(z. Let µ I(Z + with pgf Q µ (z = Q ln α (z. It follows that P (z satisfies (2.6 with f(s = α s. Let {X (µ t } be a Z + -valued subordinator with L(X (µ 1 = µ and let {Y t } be as above. It is clear that {Y t } converges weakly and therefore L( α s F dx s (µ is definable and that P (z is its pgf. Corollary 3.3. Let α (, 1. The mapping Ψ α is one-to-one from I log (Z + onto F-SD(Z +. Proof. We only need prove Ψ α is one-to-one. Let µ 1, µ 2 I log (Z + such that Ψ α (µ 1 = Ψ α (µ 2. Denote by P (z the pgf of Ψ α (µ 1 (and thus of Ψ α (µ 2. It follows by (2.6 and (1.4 that ln P (z P (F t ln α (z = ln Q µi (F s ln α (z ds (z [, 1]; i = 1, 2. Therefore, Q µ1 (F t ln α (z = Q µ2 (F t ln α (z for every z [, 1] and t. Letting t and using (1.6 leads to Q µ1 (z = Q µ2 (z, or µ 1 = µ 2. The corollary below follows straightforwardly from Theorem 3.2 and the infinite series representation (2.7. Corollary 3.4. A Z + -valued random variable X has an F-selfdecomposable distribution if and only if for some α (, 1, and therefore for every α (, 1, there

20 2 NADJIB BOUZAR exists a Z + -valued subordinator {X (µ t }, with µ = L(X (µ 1 I log (Z + (depending on α, such that X d = α s F dx (µ s d = α T k F C k. (3.2 Here the sequences {T k } and {C k } respectively represent the jump times and jump sizes of {X (µ t } (see(2.1. For the sake of completeness, and as a consequence of Theorem 3.2, we restate the convergence result obtained in [17] (Theorem 3.2, therein. Corollary 3.5. Let α (, 1. Let {X (µ t } be a Z + -valued subordinator and Y t = αs F dx s (µ. Then Y t converges in distribution to some Z + -valued random variable Y (as t if and only if E(ln(1 + X (µ 1 <, in which case the distribution of Y is F-selfdecomposable. Next, we derive a fixed point property for the mapping Ψ a that is the analogue of a result obtained in [9] for distributions on Banach and Euclidean spaces. First, we briefly recall some results on F-stable distributions (see Chapter V, Section 8, in [18]. A Z + -valued random variable X, or its distribution, is said to be F-stable with exponent γ (, 1] if for every α (, 1, k=1 X d = α F X + (1 α γ 1/γ F X, (3.3 where X d = X and X and X are independent. An F stable distribution µ on Z + is F-selfdecomposable and thus necessarily infinitely divisible. Moreover, its pgf P µ admits the following canonical representation: P µ (z = exp[ λa(z γ ] ( z 1, (3.4 where A(z is the A-function given in (1.13 and λ is a positive constant. Theorem 3.6. Let α (, 1. A distribution µ on Z + is F-stable if and only if µ I log (Z + and there exists c > such that the pgf s P µ and P Ψα(µ of µ and Ψ α (µ, respectively, satisfy the equation ln P Ψα (µ(z = c ln P µ (z, ( z 1, (3.5 in which case Ψ α (µ is also F-stable with the same exponent γ as µ, γ = (, 1]. 1 c ln α Proof. The only if part: assume that µ is F-stable with exponent γ (, 1]. We have by (3.4 and the fact that A (x A(x = 1 U(x, x (, 1 (see (1.13, 1 [ ln P µ (x] 1 A(x γ 1 dx = λ U(x U(x dx = λ A(x γ 1 A (x dx (λ >.

21 INTEGRAL REPRESENTATIONS 21 A simple change of variable argument shows that the third integral in the above equation is finite, and thus µ I log (Z +. By (2.6 and (3.4, ln P Ψα (µ(z = ln P µ (F s ln α (z ds = λ for some λ > and γ (, 1]. It follows by (1.14b that [A(F s ln α (z] γ ds, ln P Ψα (µ(z = λa(z γ α γs ds = λ γ ln α A(zγ. Therefore (3.5 holds with c = 1/γ ln α and Ψ α (µ is F-stable with exponent γ (by (3.4. We now prove the if part. Assume that µ I log (Z + and that (3.5 holds. We have via (2.6 and (1.14a ln P Ψα (µ(z = 1 ln α ln P µ (F s (z ds = 1 ln α ln P µ (F s (z U(F s (z The change of variable x = F t (z and (1.6 yield ln P Ψα (µ(z = 1 ln α which, combined with (3.5, yields in turn the differential equation c d dx ln P µ(x = 1 ln α ln P µ (x. U(x s F s(z ds. 1 z ln P µ (x U(x Since A (x A(x = 1 U(x, one easily deduces that the solution P µ(z takes the form (3.4 with λ = ln P µ ( and γ = 1/(c ln α. The fact that γ (, 1] follows from a result in van Harn and Steutel [8] (see the proof of Lemma 4.2 therein. dx, 4. A Stochastic Integral Representation for F-semi-selfdecomposable Distributions Let α (, 1. We define the mapping Φ α ( on I(Z + by ( Φ α (µ = L α s F dx s (µ, (4.1 where {X (µ t } is a Z + -valued subordinator with L(X (µ 1 = µ and, recall, s denotes the greatest integer function at s. The first two results identify the domain D(Φ α and range R(Φ α of the mapping Φ α. Theorem 4.1. D(Φ α = I log (Z +. Proof. Let f(s = α s for s [, t], and let {s, s 1,, s t, s t +1 } be the subdivision of [, t] such that s j = j, j =, 1,, t, and s t +1 = t. The function f(s can be rewritten as t f(s = α j I [sj,s j+1 (s, j= (s [, t].

22 22 NADJIB BOUZAR A straightforward pgf argument, by way of Theorem 2.1 and equation (2.3, leads to t α s F dx s (µ = α j F (X s (µ j+1 X s (µ j. j= Moreover, the pgf P t (z of α s F dx s (µ simplifies to P t (z = { t 1 j= } Q µ (F j ln α (z Q t t µ (F t ln α (z. (4.2 Since t t 1 and lim F t ln α (z = 1, we have lim t µ (F t ln α (z n = 1. Therefore, lim P t (z exists if and only if lim t n Q µ (F j ln α (z is finite and lim P t(z = lim t n j= t Q t t n Q µ (F j ln α (z. (4.3 j= Now n j= Q µ(f j ln α (z is the pgf of the size of the n-th generation of a Galton- Watson branching process with stationary immigration described by the pgf Q µ (z and an offspring distribution with pgf F ln α (z (see Athreya and Ney [1], Chapter 6. By Corollary 2 in Foster and Williamson [5], n j= Q µ(f j ln α (z converges to a pgf if and only if n= q n ln n <, where {q n } is the distribution with pgf Q µ, or equivalently (since ln(n+1 ln n as n, if and only if E(ln(1+X (µ 1 <. Therefore, Φ α (µ exists if and only if µ I log (Z +. Theorem 4.2. R(Φ α = F-SSD α (Z +. Proof. Assume that Φ α (µ exists for some µ I(Z +. Let P t (z be the pgf of α s F dx s (µ. Then by (4.3, the pgf P Φα (µ(z of Φ α (µ can be written as n [ n ] P Φα (µ(z = lim Q µ (F j ln α (z = lim Q µ (F j ln α (z Q µ (z, (4.4 or, n j= [ P Φα (µ(z = lim n n j=1 = P (F ln α (zq µ (z. n j=1 ] Q µ (F (j 1 ln α (F ln α (z Q µ (z (4.5 Since Q µ (z is infinitely divisible, it follows by (1.8 (applied at t = ln α that Φ α (µ F-SSD α (Z + and thus R(Φ α F-SSD α (Z +. Assume now that P (z is the pgf of an F-semi-selfdecomposable distribution on Z + with order α. We have by (1.8 that P (z = P (F ln α (zq(z for some infinitely divisible pgf Q(z. It follows by equation (2.2 in [4] that n P (z = lim Q(F j ln α (z. (4.6 n j=

23 INTEGRAL REPRESENTATIONS 23 Let µ I(Z + with pgf Q(z(= Q µ (z and let {X (µ t } be a Z + -valued subordinator with L(X (µ 1 = µ. We conclude by (4.3 and (4.6 that L( α s F dx s (µ is definable and that P (z is its pgf. We have thus shown F-SSD α (Z + R(Φ α. Corollaries 4.3 and 4.4 below are straightforward consequences of Theorems 4.1 and 4.2 and the infinite series representation (2.7. Corollary 4.3. A Z + -valued random variable X with pgf P (z has an F-semiselfdecomposable distribution with order α (, 1 if and only if there exists a Z + -valued subordinator {X (µ t } such that X = d α s F dx s (µ d = α Tk F C k, (4.7 in which case µ = L(X (µ 1 I log (Z +. Here the sequences {T k } and {C k } respectively represent the jump times and jump sizes of {X (µ t } (see(2.1. Corollary 4.4. Let α (, 1. Let {X (µ t } be a Z + -valued subordinator and Y t = α s F dx s (µ. Then Y t converges in distribution to some Z + -valued random variable Y (as t if and only if E(ln(1 + X (µ 1 <, in which case the distribution of Y is F-semi-selfdecomposable with order α. The proof of the next result is similar to that of Corollary 3.3 (by way of (4.5. Corollary 4.5. Let α (, 1. The mapping Φ α is one-to-one from I log (Z + onto F-SSD α (Z +. k=1 5. Integral Representations for Elements in Nested Subclasses of F-SD(Z + and F-SSD α (Z + We define nested subclasses of F-SD(Z + and F-SSD α (Z + and study the actions of the mappings Ψ α and Φ a, respectively, on their elements. First, we formulate an equivalent definition of F-selfdecomposability and F- semi-selfdecomposability (see (1.7 and (1.8. A distribution µ on Z + belongs to F-SD(Z + (resp. F-SSD α (Z + for some α (, 1 if and only if for every t > (resp. for t = ln α, µ = (µ ν t µ t, (5.1 where ν t is the distribution with pgf F t (z and µ t I(Z +. The binary operation is the compounding operation and is the convolution operation. We define F-SD (Z + = I(Z + and for m, F-SD m+1 (Z + as the subset of distributions µ I(Z + such that every t >, there exists µ t F-SD m (Z + such that (5.1 holds. Likewise, we define for α (, 1, F-SSD α, (Z + = I(Z + and for m, F-SSD α,m+1 (Z + as the subset of distributions µ I(Z + that satisfy (5.1 for t = ln α and some µ t F-SSD α,m (Z +.

24 24 NADJIB BOUZAR For m 1, we define I log m(z + = {µ = (q n, n I(Z + : q n (ln(n + 1 m < }. The sequences {I log m(z + }, {F-SD m (Z + }, and {F-SSD α,m (Z + } are easily checked to be decreasing in I(Z + by an induction argument. Hansen [6] studied extensively the subclasses {F-SD m (Z + } (see also Berg and Forst [3]. We start out by recalling the discrete versions of two important results obtained in [6]. Theorem 5.1. (Theorem 7.4 in [6] Let µ I(Z + with pgf P (z and m. Then µ F-SD m+1 (Z + if and only if ln P (z = n= ln Q(F s (zs m ds, where Q(z is the pgf of a distribution in I(Z +. Theorem 5.2. (Theorem 7.5 in [6] Let µ I(Z + with pgf Q(z and m. The following assertions are equivalent. (i [ ln Q(F s (z]s m ds < ( z 1 (ii The canonical sequence {r k } of µ defined by Q (z Q(z (ln(k + 1 m+1 r k k + 1 < (r k. k= = k= r kz k satisfies Let {X (µ t } be a Z + -valued subordinator with L(X (µ 1 = µ I(Z +. For α (, 1, we define the sequences of mappings {Ψ m α (µ} and {Φ m α (µ} recursively as follows: Ψ α(µ = Φ α(µ = µ and for every m, Ψ m+1 α (µ = Ψ α (Ψ m α (µ and Φ m+1 α (µ = Φ α (Φ m α (µ where Ψ α and Φ α are defined by (3.1 and (4.1, respectively. The aim of this section is to identify the domains and ranges of Ψ m α and Φ m α. Next, we derive a useful lemma. Lemma 5.3. Let α (, 1 and m a positive integer. Let µ I(Z + with pgf Q(z. Then the following assertions are equivalent. ( s + m (i J 1 (z = [ ln Q(F s ln α (z] ds < ( z 1. m (ii J 2 (z = (iii µ I log m+1(z + [ ln Q(F s (z]s m ds < ( z 1. Proof. First, we note that for every s >, s s s + 1 and for every z [, 1], F s (z is increasing as a function of s (, (see 1.14a. Moreover,

25 ( s +m m INTEGRAL REPRESENTATIONS 25 sm m! as s. Rewriting J 2 (z as J 2 (z = t m+1 [ ln Q(F st (z]s m ds, where t = ln α, and using a standard integration argument, one can easily show that (i (ii. The equivalence (ii (iii follows by Theorem 5.2, combined with the discrete versions of Theorem 25.3 and Proposition 25.4 in [14]. The next result identifies the domain D(Ψ m α and the range R(Ψ m α of the mapping Ψ m α, m 1. Theorem 5.4. For every positive integer m, D(Ψ m α = I log m(z + and R(Ψ m α = F-SD m (Z +. Moreover, if µ I log m(z +, then ln P Ψ m α (µ(z = s m 1 (m 1! ln Q µ(f s ln α (z ds. (5.2 Proof. We prove the theorem by induction. The case m = 1 follows from Theorem 3.2 and equation (2.6 (applied to f(s = α s. Assume that Theorem 5.4 holds for m 1. Suppose that µ D(Ψ m+1 α. Let Q µ (z be the pgf of X (µ 1. Then by (2.6, (5.2, and (1.4, ln P Ψ m+1 α (µ (z = ln P Ψ m α (µ(f t ln α (z dt = ( which implies ( s ln P Ψ m+1 α (µ (z = s m 1 (m 1! ln Q µ(f (s+t ln α (z ds dt, (s t m 1 (m 1! dt ln Q µ (F s ln α (z ds, which in turn implies that (5.2 holds for m + 1. By making the change of variable t = s ln α in (5.2, with m + 1 in place of m, we obtain ln P Ψ m+1 α (µ (z = B m t m ln Q µ (F t (z dt, (5.3 where ( ln α m 1 B m =. (5.4 m! Since the integral in (5.3 converges, we conclude by Lemma 5.3[(ii (iii] that µ I log m+1(z +. Moreover, noting that Q B m µ (z is an infinitely divisible pgf, the representation (5.3 implies that Ψ m+1 α (µ F-SD m+1 (Z + by Theorem 5.1. We have thus shown that D(Ψ m+1 α I log m+1(z + and R(Ψ m+1 α F-SD m+1 (Z +. Assume now that µ I log m+1(z +. By Lemma 5.3[(iii (ii], the integral J 2 (z converges for every z [, 1]. Since I log m+1(z + I log m(z +, Ψ m α (µ exists by the induction hypothesis. Using the same argument as above, we deduce [ ln P Ψ m α (µ(f t ln α (z] dt = B m J 2 (z < (z [, 1],

26 26 NADJIB BOUZAR where B m is given by (5.4. It ensues by Lemma 3.1[(iii (i], along with an obvious change of variable argument, that Ψ m α (µ I log (Z +. Therefore Ψ m+1 α (µ = ψ α (Ψ m α (µ exists by Theorem 3.2 and thus I log m+1(z + D(Ψ m+1 α. What remains to be shown is F-SD m+1 (Z + R(Ψ m+1 α. Let P (z be the pgf of a distribution in F-SD m+1 (Z +. By Theorem 5.1, P (z admits the representation ln P (z = t m ln Q(F t (z dt (5.5 for some infinitely divisible pgf Q(z. Let µ I(Z + with pgf Q µ (z = Q 1/Bm (z, where B m is as in (5.4. We have by (5.5 that the integral J 2 (z of Lemma 5.3 converges for every z 1. By Lemma 5.3[(ii (iii], µ I log m+1(z + and thus, as shown above, Ψ m+1 α ln P (z = (µ exists. It is easily seen that s m m! ln Q µ(f s ln α (z ds, or, P (z = P Ψ m+1 α (µ (z. The following corollary gives an integral representation of the iterate Ψ m α (µ. We will need the following sequence of functions {f m } defined on [, as follows: f m (s = ((m + 1!s 1/(m+1 (m. (5.6 Corollary 5.5. Let {X (µ t } be a Z + -valued subordinator with L(X (µ 1 = µ and m a positive integer. If µ I log m(z +, then ( Ψ m α (µ = L α f m 1(s F dx (µ s. (5.7 Proof. Assume µ I log m(z + (m 1. We define the process {Y t } by Y t = α f m 1(s F dx (µ s. (5.8 We have by Theorem 2.1 and (2.3 that the pgf P t (z of Y t is { ( P t (z = exp ln Q µ F fm 1 (s ln α(z } ds. (5.9 Making the change of variable u = f m 1 (s in (5.9 and letting t leads to { lim P u m 1 } t(z = exp t (m 1! ln Q µ(f u ln α (z du, or, by Theorem 5.3 and (5.2, lim t P t (z = P Ψ m α (µ(z. It follows that the ( process {Y t } converges weakly and therefore, by Theorem 2.2, L α fm 1(s F is definable and has pgf P Ψ m α (µ(z. dx (µ s We now turn to the problem of characterizing the domain and range of the iterates {Φ m α (µ}. We denote by D(Φ m α and R(Φ m α, the domain and range of Φ m α, respectively. First, we give a useful characterization of the distributions in F-SSD α,m (Z +.