Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudostable and pseudo-unstable manifolds for a class of random partial differential equations and stochastic partial differential equations is shown. Unlike the invarian...

In this paper, we study forward-backward doubly stochastic differential equations driven by Brownian motions and Poisson process (FBDSDEP in short). Both the probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations (SPDIEs in short) and stochastic Hamiltonian systems arising in stochastic optimal control problems with random jum...

The Milstein scheme is the simplest nontrivial numerical scheme for stochastic differential equations with a strong order of convergence one. The scheme has been extended to the stochastic delay differential equations but the analysis of the convergence is technically complicated due to anticipative integrals in the remainder terms. This paper employs an elementary method to derive the Milstein...

Abstract. Malliavin Calculus is about Sobolev-type regularity of functionals on Wiener space, the main example being the Itô map obtained by solving stochastic differential equations. Rough path analysis is about strong regularity of solution to (possibly stochastic) differential equations. We combine arguments of both theories and discuss existence of a density for solutions to stochastic diff...

So far there are not many results on the stability for stochastic functional differential equations with infinite delay. Themain aim of this paper is to establish some new criteria on the stability with general decay rate for stochastic functional differential equations with infinite delay. To illustrate the applications of our theories clearly, this paper also examines a scalar infinite delay ...

This chapter is an introduction and survey of numerical solution methods for stochastic differential equations. The solutions will be continuous stochastic processes that represent diffusive dynamics, a common modeling assumption for financial systems. We include a review of fundamental concepts, a description of elementary numerical methods and the concepts of convergence and order for stochas...