Some Refinements of Existence Results for SPDEs Driven by Wiener Processes and Poisson Random Measures

Stefan Tappe

doi:10.1155/2012/236327

Abstract

We provide existence and uniqueness of global (and local) mild solutions for a general class of semilinear stochastic partial differential equations driven by Wiener processes and Poisson random measures under local Lipschitz and linear growth (or local boundedness, resp.) conditions. The so-called “method of the moving frame” allows us to reduce the SPDE problems to SDE problems.

Highlights

Semilinear stochastic partial differential equations SPDEs on Hilbert spaces, being of the typeZt AZt α t, Zt dt σ t, Zt dWt, 1.1Z0 z0, have widely been studied in the literature, see, for example, 1–4
We provide existence and uniqueness of global and local mild solutions for a general class of semilinear stochastic partial differential equations driven by Wiener processes and Poisson random measures under local Lipschitz and linear growth or local boundedness, resp. conditions
In 1.1, A denotes the generator of a strongly continuous semigroup, and W is a trace class Wiener process. This framework has been extended by adding jumps to the SPDE 1.1

Summary

Introduction

Semilinear stochastic partial differential equations SPDEs on Hilbert spaces, being of the type. C t, Yt−, x μ dt, dx , By using the technique of interlacing solutions at jump times which, in particular cases has been applied, e.g., in 15, Section 6.2 and 10, Section 9.7 , we can reduce the SDE 1.4 to SDEs of the form dYt a t, Yt dt b t, Yt dWt c t, Yt−, x μ dt, dx − F dx dt , Y0 y0, without large jumps, and for those SDEs, suitable techniques and results are available in the literature This allows us to derive existence and uniqueness results for the SDE 1.4 , which are subject to the regularity conditions described above. We point out that 14 studies Hilbert space-valued SDEs of the type 1.4 and provides an existence and uniqueness result considerably going beyond the classical results which impose global Lipschitz conditions.

Preliminaries and Notation

Existence and Uniqueness of Strong Solutions to Hilbert Space-Valued SDEs

Existence of Strong Solutions to Hilbert Space-Valued SDEs

Comparison with the Method of Successive Approximations

Existence and Uniqueness of Mild Solutions to Hilbert Space-Valued SPDEs