The hybrid exact scheme for the simulation of first-passage times of jump-diffusions with time-dependent thresholds

Abstract: The first-passage time is a key concept in stochastic modeling, representing the time at which a process first reaches a specified threshold. In this work, we consider a jump-diffusion (JD) model with a time-dependent threshold, providing a more flexible framework for describing stochastic dynamics. We are interested in the Exact simulation method developed for JD processes with constant thresholds, where the Exact method for pure diffusion is applied between jump intervals. An adaptation of this method to time-dependent thresholds has recently been proposed for a more general stochastic setting. We show that this adaptation can be applied to JD models by establishing a formal correspondence between the two frameworks. A comparative analysis is then performed between the proposed approach and the constant-threshold version in terms of algorithmic structure and computational efficiency. Finally, we show the applicability of the method by predicting neuronal spike times in a JD model driven by two independent Poisson jump mechanisms.