Existence and convergence results for infinite dimensional nonlinear stochastic equations with multiplicative noise

Viorel Barbu; Zdzisław Brzeźniak; Erika Hausenblas; Luciano Tubaro

paper

Existence and convergence results for infinite dimensional nonlinear stochastic equations with multiplicative noise

arXiv:1210.4578

Abstract

The solution $X_{n}$ to a nonlinear stochastic differential equation of the form $d X_{n} (t) + A_{n} (t) X_{n} (t) d t - \frac{1}{2} \sum_{j = 1}^{N} (B_{j}^{n} (t))^{2} X_{n} (t) d t = \sum_{j = 1}^{N} B_{j}^{n} (t) X_{n} (t) d β_{j}^{n} (t) + f_{n} (t) d t$ , $X_{n} (0) = x$ , where $β_{j}^{n}$ is a regular approximation of a Brownian motion $β_{j}$ , $B_{j}^{n} (t)$ is a family of linear continuous operators from $V$ to $H$ strongly convergent to $B_{j} (t)$ , $A_{n} (t) \to A (t)$ , ${A_{n} (t)}$ is a family of maximal monotone nonlinear operators of subgradient type from $V$ to $V^{'}$ , is convergent to the solution to the stochastic differential equation $d X (t) + A (t) X (t) d t - \frac{1}{2} \sum_{j = 1}^{N} B_{j}^{2} (t) X (t) d t = \sum_{j = 1}^{N} B_{j} (t) X (t) d β_{j} (t) + f (t) d t$ , $X (0) = x$ . Here $V \subset H ≅ H^{'} \subset V^{'}$ where $V$ is a reflexive Banach space with dual $V^{'}$ and $H$ is a Hilbert space. These results can be reformulated in terms of Stratonovich stochastic equation $d Y (t) + A (t) Y (t) d t = \sum_{j = 1}^{N} B_{j} (t) Y (t) \circ d β_{j} (t) + f (t) d t$ .