Jump-diffusion processes in random environments

TL;DR

This paper studies jump-diffusion processes in random environments, establishing conditions for existence, uniqueness, and the Markov property, with applications to generalized exponential Levy models.

Contribution

It provides new conditions for existence and uniqueness of solutions to SDEs with jump-diffusions in random environments and explores their Markov properties.

Findings

Established existence and uniqueness conditions for jump-diffusion SDEs in random environments.

Proved the Markov property for these processes.

Applied results to generalized exponential Levy models.

Abstract

In this paper we investigate jump-diffusion processes in random environments which are given as the weak solutions to SDE's. We formulate conditions ensuring existence and uniqueness in law of solutions. We investigate Markov property. To prove uniqueness we solve a general martingale problem for \cadlag processes. This result is of independent interest. In the last section we present application of our results considering generalized exponential Levy model.