Construction of sampling trajectories of coupled diffusion\\ processes and its simulation algorithm

Abstract

Coupled diffusion process is a system in which a diffusion process is coupled with a Markov jumping process. To simulate such a hybrid process, we consider a product space formed by a Lebesgue space on $[0,1]^{\infty}$ and a Wiener space on $\mathcal{C}(\mathbb R_+,\mathbb R^r)$, and intuitively construct a coupled diffusion process on this probability space. We prove that the constructed process is a Markov process, and present the probability distribution of the residual lifetime as well as the joint probability distribution of the next reaction index and the residual lifetime. We further consider the infinitesimal generator and the martingale problem of the process, and prove the solution to the martingale problem is unique in distribution. Based on the sampling process we constructed, we propose an algorithm to simulate the coupled diffusion process.