A variational approach to nonlinear stochastic differential equations with linear multiplicative noise

Octav Mayer Institute of Mathematics of Romanian Academy, Iaşi, Romania.

^* Corresponding author: vb41@uaic.ro

Received: 27 February 2018
Accepted: 13 November 2018

Abstract

One introduces a new concept of generalized solution for nonlinear infinite dimensional stochastic differential equations of subgradient type driven by linear multiplicative Wiener processes. This is defined as solution of a stochastic convex optimization problem derived from the Brezis-Ekeland variational principle. Under specific conditions on nonlinearity, one proves the existence and uniqueness of a variational solution which is also a strong solution in some significant situations. Applications to the existence of stochastic total variational flow and to stochastic parabolic equations with mild nonlinearity are given.