Existence of weak solutions to stochastic evolution inclusions

Abstract

We prove the existence of a weak mild solution to the Cauchy problem for the semilinear stochastic differential inclusion in a Hilbert spacewhere W is a Wiener process, A is a linear operator which generates a C 0 -semigroup, F and G are multifunctions with convex compact values satisfying some growth condition and, with respect to the second variable, a condition weaker than the Lipschitz condition. The weak solution is constructed in the sense of Young measures.

Weak solutions of backward stochastic differential equations with continuous generator

“…Proof The boundedness of (Z (n) ) n≥1 and ( Z (n) ) n≥1 is a direct consequence of Lemma 3.4 and Proposition 3.5. Then (23) follows by Itô's isometry, Doob's inequality, and the fact that…”

“…Corollary 3.6 The sequences (Z (n) ) n≥1 and ( Z (n) ) n≥1 are bounded in L 2 L (Ω× [0, T ]), and we have (23) sup…”

“…This part of the construction of a weak solution follows the same lines as in [23], with some complications due to the processes Z (n) .…”

Existence and Dependence Results for Semilinear Functional Stochastic Differential Equations with Infinite Delay in a Hilbert Space

“…As a first step, using techniques of measures of noncompactness, as in [3,5,14,21], we first define a measure of noncompactness adapted to our formulation of the problem; then, we prove that the sequence of approximating solutions constructed via Tonelli's scheme has a subsequence which converges to the unique desired solution (Theorem 3.2). As a second step, we prove that this solution depends continuously on the initial condition (Theorem 4.1).…”

Remarks on unboundedness of set-valued Itô stochastic integrals

“…They are the main tools used in the theory of stochastic differential inclusions, theory of set-valued stochastic differential equations and fuzzy stochastic differential equations (see e.g. [2][3][4][5]8,12,15,16, and references therein). The first formulation was proposed by G. Bocşan in [8].…”