Exact simulation of the first passage time through a given level of jump diffusions

Abstract

Continuous-time stochastic processes play an important role in the description of random phenomena. It is therefore important to study particular stopping times dependent on the trajectories of these processes. Two approaches are possible: introducing an explicit expression of their probability distribution, and evaluating values generated by numerical models. Choosing the second alternative, we propose an algorithm to generate the first passage time through a given level. The stochastic process under consideration is a one-dimensional jump diffusion that satisfies a stochastic differential equation driven by a Brownian motion. It is subject to random shocks that are characterised by an independent Poisson process. The proposed algorithm belongs to the family of rejection sampling procedures: the outcome of the algorithm and the stopping time under consideration are identically distributed. This algorithm is based on both the exact simulation of the diffusion value at a fixed time and on the exact simulation of the first passage time for continuous diffusion processes (see Herrmann and Zucca (2019)). At a fixed point in time, the main challenge is to generate the position of a continuous diffusion that is conditioned not to reach a given level before that time. We present the construction of the algorithm and numerical examples. We also discuss specific conditions leading to the recurrence of a jump diffusion process.