First-passage time for PDifMPs: an Exact simulation approach for time-varying thresholds

TL;DR

This paper introduces an exact simulation method for calculating the first-passage time of Piecewise Diffusion Markov Processes with time-varying thresholds, enabling accurate modeling of systems with dynamic boundaries.

Contribution

It develops a hybrid exact simulation scheme for PDifMPs with time-dependent thresholds, combining diffusion-based methods with new approaches for jump times and convergence proof.

Findings

The method accurately computes first-passage times in complex PDifMP models.

Numerical examples demonstrate the efficiency and accuracy of the proposed simulation scheme.

The approach extends existing diffusion methods to processes with jumps and dynamic thresholds.

Abstract

Piecewise Diffusion Markov Processes (PDifMPs) are valuable for modelling systems where continuous dynamics are interrupted by sudden shifts and/or changes in drift and diffusion. The first-passage time (FPT) in such models plays a central role in understanding when a process first reaches a critical boundary. In many systems, time-dependent thresholds provide a flexible framework for reflecting evolving conditions, making them essential for realistic modelling. We propose a hybrid exact simulation scheme for computing the FPT of PDifMPs to time-dependent thresholds. Exact methods traditionally exist for pure diffusions, using Brownian motion as an auxiliary process and accepting sampled paths with a probability weight. Between jumps, the PDifMP evolves as a diffusion, allowing us to apply the exact method within each inter-jump interval. The main challenge arises when no threshold crossing is detected in an interval: We then need the value of the process at the jump time, and for that, we introduce an approach to simulate a conditionally constrained auxiliary process and derive the corresponding acceptance probability. Furthermore, we prove the convergence of the method and illustrate it using numerical examples.