On the solution of stochastic differential inclusion
Let \((\Omega, \Lambda,p)\) be a probability space, \(I = [0,T]\), \(\{\Lambda_t \}_{t \in I}\) an increasing family of \(\sigma\)- subalgebras such that \(\bigcap_{\alpha > 0} \Lambda _{t + \alpha} = \Lambda_t\), and \(\beta_t\) a \(\sigma\)-algebra of all Borel subsets of \([0,t]\) for fixed \(t \in I\). Let us denote by \(\beta \Lambda\) a \(\sigma\)-algebra of all sets \(A \subset I \otimes \Omega\) such that \(A \cap ([0,t] \times \Omega) \in \beta_t \otimes \Lambda_t\) for each \(t \in I\). \(\text{Comp} (\mathbb{R}^n)\) denotes the space of all nonempty compact subsets of the \(n\)-dimensional Euclidean space \(\mathbb{R}^n\). Let \({\mathcal L}^2 = L^2 (I \times \Omega, \beta \Lambda, dt \times dp; \mathbb{R}^n)\) denote the space of classes of equivalence of all \(\{\Lambda_t\}_{t \in I}\)-progressively measurable random processes such that \(E (\int^T_0 |X (t; \omega) |^2 dt) < \infty\) with the usual norm \(|X |= E (\int^T_0 |X (t; \omega) |^2 dt)^{1/2}\). Let \(F,G: I \times \Omega \times {\mathcal D} \to \text{Comp} (\mathbb{R}^n)\) (where \({\mathcal D}\) be a nonempty subset in \({\mathcal L}^2)\) and let \(B = \{B_t\}_{t \in I}\) be a one- dimensional \(\{\Lambda _t \}_{t \in I}\)-progressively measurable Wiener process \(B_0 = 0\). Let us consider the stochastic functional- differential inclusion \[ dX (t; \omega) \in F(t; \omega, X) dB (t; \omega) + G(t; \omega, X) dt, \quad X_0 = 0 \;\text{for a.e.} \;\omega \in \Omega. \tag{*} \] The main result of this paper is the following theorem: Let \(F,G: I \times \Omega \times {\mathcal D} \to \text{Comp} (\mathbb{R}^n)\) be such that (i) \(F\) and \(G\) are \(\{\Lambda_t\}_{t \in I}\)-progressively measurable with respect to their first two variables, (ii) \(G\) is uniformly 2-integrably bounded and there exists \(p > 2\) such that \(F\) is uniformly \(p\)-integrably bounded, (iii) \(F\) and \(G\) are \({\mathcal L}^2\)-strongly dissipative and \({\mathcal L}^2\)-maximal, (iv) \(F\) is \({\mathcal L}^2\)-locally relatively compact. Then there exists a solution to problem (*).
stochastic functional-differential inclusion
scientific article; zbMATH DE number 786122