Stochastic Differential Equations with Non-Lipschitz Coefficients in Hilbert Spaces

Abstract

In this article, we discuss the successive approximations problem for the solutions of the semilinear stochastic differential equations in Hilbert spaces with cylindrical Wiener processes under some conditions which are weaker than the Lipschitz one. We establish the existence and the uniqueness of the solution and additionally, in our framework we consider a limiting problem for the mild solution. It is shown that the mild solution tends to the solution of the stochastic differential equation of Itô type in finite dimensional space.

Keywords:

• Cylindrical Wiener process
• Limiting problem
• Mild solution
• Space-time white noise
• Stochastic differential equation

Subject Classification (2000):

• Primary 60H20
• Secondary 60H15, 34F05

This work is supported by the scholarship of the Japanese Government (Monbukagakusho).