Stability of Stochastic Differential Equations with Jumps by the Coupling Method

Abstract

The topic of this thesis is the study of R^d-valued stochastic processes defined as solutions to stochastic differential equations (SDEs) driven by a noise with a jump component. Our main focus are SDEs driven by pure jump Levy processes and, more generally, by Poisson random measures, but our framework includes also cases in which the noise has a diffusion component. We present proofs of results guaranteeing existence of solutions and invariant measures for a broad class of such SDEs. Next we introduce a probabilistic technique known as the coupling method. We present an original construction of a coupling of solutions to SDEs with jumps, which we subsequently apply to study various stability properties of these solutions. We investigate the rates of their convergence to invariant measures, bounds on their Malliavin derivatives (both in the jump and the diffusion case) and transportation inequalities, which characterize concentration of their distributions. In all these cases the use of the coupling method allows us to significantly strengthen results that have been available in the literature so far. We conclude by discussing potential extensions of our techniques to deal with SDEs with jump noise which is inhomogeneous in time and space.