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

Sascha Desmettre, Devika Khurana, Amira Meddah

TL;DR

This work addresses the challenge of exactly simulating first-passage times for jump-diffusion processes with time-dependent boundaries. By reformulating the JD model as a PDifMP and applying a Lamperti transform together with Girsanov-based exact sampling, the authors develop the Hybrid Exact Scheme (HEx) to compute FPTs between jumps without discretization error. They analyze algorithmic differences from constant-threshold methods, provide bounds on iteration and rejection-sampling costs, and validate the approach on a JD example with time-dependent thresholds. The methodology is demonstrated in a neuronal context, showing how time-dependent thresholds and jump inputs shape spike-time predictions and suggesting extensions to more complex resetting and threshold scenarios.

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.

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

TL;DR

This work addresses the challenge of exactly simulating first-passage times for jump-diffusion processes with time-dependent boundaries. By reformulating the JD model as a PDifMP and applying a Lamperti transform together with Girsanov-based exact sampling, the authors develop the Hybrid Exact Scheme (HEx) to compute FPTs between jumps without discretization error. They analyze algorithmic differences from constant-threshold methods, provide bounds on iteration and rejection-sampling costs, and validate the approach on a JD example with time-dependent thresholds. The methodology is demonstrated in a neuronal context, showing how time-dependent thresholds and jump inputs shape spike-time predictions and suggesting extensions to more complex resetting and threshold scenarios.

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.

Paper Structure

This paper contains 12 sections, 44 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model structure and reformulation via PDifMPs
  3. The FPT of jump diffusions
  4. Exact simulation approach
  5. Comparison analysis with Herrmann and Massin's approach
  6. Algorithmic differences
  7. Consequences for bias and complexity
  8. Iteration complexity
  9. Extended rejection sampling
  10. Numerical comparison
  11. Neuroscience application: predicting spike times in a neuron using the HEx scheme
  12. Conclusion

Figures (3)

  • Figure 1: Estimated FPT density from $10^3$ samples of FPT. Benchmark results from herrmann2023exact are shown in solid black, while the HEx Scheme with $\epsilon=10^{-3}$ and $s_{\min}=-1$ is shown in dashed blue.
  • Figure 2: Effect of positive jump parameters on the firing rate. The firing rate of a neuron is estimated from $2\cdot 10^3$ spike time samples using HEx Scheme. The parameters for the continuous part of the process were taken as follows: $\tau=2$, $\sigma=1$, $V_\mathrm{rest}=1$, $I=3$, $v_0=1/e$.
  • Figure 3: Effect of negative jump parameters on the firing rate. The firing rate of a neuron is estimated from $2\cdot 10^3$ spike time samples using HEx Scheme. The parameters for the continuous part of the process were taken same as in Figure \ref{['fig:application_plots_1']}.

Theorems & Definitions (4)

  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 5.1