Remarks on the existence and approximation for semilinear stochastic differential equations in Hilbert spaces

Abstract

In this paper we first shall establish an existence and uniqueness result for the semilinear stochastic differential equations in Hilbert space dX=(AX+f(X))dt+g(X)dW under weaker conditions than the Lipschitz one by investigating the convergence of the successive approximations. Secondly we show, under the assumption of so-called pathwise uniqueness (PU), the convergence of the Euler and Lie-Trotter schemes in L ^p , p>2 and the continuous dependence of the solutions on the initial data and on the coefficients for such equation. Finally we study the existence of the solutions when the coefficients f and g are only defined on a subset of the state Hilbert space.

Keywords:

• Stochastic evolution equations
• Pathwise uniqueness

Acknowledgments

The authors are sincerely grateful to Professor Y. Ouknine for very helpful discussions and remarks. Also the authors would like to thank Professor M. Dozzi for drawing their attention to papers [22] and [26].