Dynamic Systems and Applications 16 (2007) 141-154 SET VALUED STRATONOVICH INTEGRAL AND STRATONOVICH TYPE STOCHASTIC INCLUSION

In the paper we discuss the construction of a set-valued stochastic integral of the Stratonovich type driven by a semimartingale. It allows to consider the stochastic inclusion of a type of Stratonovich. The existence of strong solutions to such inclusion with upper separated set-valued functions is investigated. AMS (MOS) Subject Classification. 47H04, 49J53, 60H05. 1. PRELIMINARIES Investigation of stochastic controlled dynamical systems by methods of multivalued analysis requires an appropriate kind of regularity of their multivalued structure. The properties of Lipschitz continuity, lower or upper semicontinuity and monotonicity are often considered. In the paper, we consider a different class of set-valued functions called “upper separated”. The upper separatity of a set-valued function F is necessary and sufficient for the existence of a convex selection of F . We are able to define a stochastic set-valued integral of a Stratonovich type in a proper way. This enables us to investigate a new class of stochastic inclusions, and therefore to consider control problems driven by stochastic Stratonovich equations. As a consequence, we deduce the existence of solutions to stochastic inclusions with right sides taken from the above class of multifunctions. The paper is organized as follows. In the Section 2 we define the new type of a set-valued stochastic integral. We discuss there the existence and properties of the set-valued Stratonovich integral ∫ t 0 Rs ◦ dzs, where R is a set-valued semimartingale while z denotes a single-valued semimartingale. In the first part of the Section 3 we discuss selection properties of upper separated set-valued functions. This enable to define the Stratonovich type stochastic inclusion in a proper way. Next we discuss the existence of exploding and nonexploding solutions to such inclusion. Some examples of both situations are presented also. Received August 30, 2006 1056-2176 $15.00 c ©Dynamic Publishers, Inc. 142 MARIUSZ MICHTA AND JERZY MOTYL 2. SET-VALUED STRATONOVICH INTEGRAL Let I = [0,∞) and let (Ω,F, {Ft}t∈I , P ) be a complete filtered probability space satisfying the usual hypothesis, i.e., {Ft}t∈I} is an increasing and right continuous family of σ-subalgebras of F and F0 contains all P -null sets. Let xt : (Ω,F, P ) → R, t ∈ I be a random variable. The stochastic process x = (xt)t∈I is said to be adapted if xt is Ft-measurable for each t ∈ I. A stochastic process x is called cádlág if it a.s. has sample paths which are right continuous, with left limits. Similarly, a stochastic process x is said to be cáglád if it a.s. has sample paths which are left continuous, with right limits. The family of all adapted cádlág (cáglád) processes is denoted by D [L]. Let P(Ft) denote the smallest σ-algebra on I ×Ω with respect to which every cáglád adapted process is measurable in (t, ω), i.e., P(Ft) = σ(L). A stochastic process x is said to be predictable if x is P(Ft)-measurable. One has P(Ft) ⊂ β ⊗ F , where β denotes the Borel σ-algebra on [0,∞). By H, p ≥ 1, we denote the normed space of semimartingales with finite Hnorm, i.e. ‖x‖Hp = infx=n+a jp(n, a), where jp(n, a) = ‖[n, n] 1 2 ∞ + ∫ ∞ 0 |das|‖Lp, and |at| = ∫ t 0 |das| represents the total variation on [0, t] of the measure induced by the paths of the FV −process a (see [14] for details). Let R = (Rt)t∈I be a set-valued stochastic process with values in Comp(R ), the space of all compact subsets of R considered with a Hausdorff metric h(·, ·), i.e., a family of F measurable set-valued mappings Rt : Ω → Comp(R ), each t ∈ I. We call R measurable if it is β ⊗ F measurable in the sense of set-valued functions (see e.g.: [1]). Similarly, R is {Ft}t∈I-adapted if Rt is Ft-measurable for each t ∈ I. We call R predictable if R is P(Ft) -measurable. A set-valued stochastic process R = (Rt)t≥0 is z-integrably bounded if there exists a predictable and z-integrable process m such that ‖ supt≥0 |mt|‖Lp(Ω) < ∞ and such that the Hausdorff distance h(|Rt|, {0}) ≤ |mt| a.s. , each t ∈ I. A set-valued process R = (Rt)t∈I is a set-valued martingale if Rs = E(Rt|Fs) for s ≤ t, where E denotes the set-valued conditional expectation of R relative to {Ft}t∈I , [4]. R has a finite variation on compacts (is an FV-process) if