Some Results on Set-valued Stochastic Integrals with Respect to Poisson Jump in an M-type 2 Banach Space

Probability theory is an important tool of modeling randomness in a practical problem. But besides randomness, in the real world, there exists other kind of uncertainties such as impreciseness or vagueness. Set-valued functions are employed to model the impreciseness in applied field such as in Economics, control theory (see for example [1]). Integrals of set-valued functions have been received much attention with widespread applications, see for example [2, 7, 9, 10] etc. Recently, stochastic integrals for set-valued stochastic processes with respect to the Brownian motion and martingales have been received much attention, e.g. see [12, 13, 18, 23, 32, 37]. Correspondingly, the set-valued stochastic differential equations are studied, e.g. see [23, 25, 33, 34, 35, 36]. Michta (2011) [22] extended the integrator to a larger class: semimartingales. But the integrably boundedness of the corresponding set-valued stochastic integrals are not obtained since the semimartingales may not be of finite variation. In such cases, the set-valued stochastic integrals may not be well defined as Ogura pointed out [25]. The Poisson stochastic processes are special. They play important roles both in the random mathematics (c.f. [11, 8, 17]) and in applied fields, for example, in the financial mathematics [17]. If the characteristic measure ν of a stationary Poisson process p is finite, then both of the Poisson random measure N(dsdz) (where z ∈ Z, the state space of p) and the compensated Poisson random measure Ñ(dsdz) are of finite variation a.s. We will give some results (without giving proof since the page limitation) on the set-valued stochastic integrals with respect to the Poisson random measure N(dsdz), Ñ(dsdz). For the detail proof, the reader can refer to [31, 38]. For example, the stochastic integrals for set-valued S -predictable (see Definition 3.2) processes with respect to N(dsdz) and Ñ(dsdz) are L2-integrably bounded. For Brownian or Martingale integrator with continuous part, the integrable boundedness are not obtained until now. Furthermore, if the σ-algebra F is separable, then the integral {It(F )} of convex set-valued stochastic process will not become a set-valued martingale, which is very different from single valued case. We would like to pointed out that there is a gap in the proof of Theorem 3.7 in [31] about the set-valued martingale property of set-valued stochastic integral with respect to the compensated Poisson measure, which is corrected and proven in [38].