Exact simulation of the first-passage time of SDEs to time-dependent thresholds

TL;DR

The paper tackles the problem of exactly simulating the first-passage time (FPT) of a diffusion to a time-dependent threshold by extending Beskos–Roberts–Hermann–Zucca–related exact-sampling techniques to moving boundaries. It constructs an acceptance–rejection framework using Girsanov’s transformation to relate the target diffusion’s FPT to the FPT of Brownian motion, with a computable weight function $\eta(t)$ and a practical scheme based on a Brownian first-passage to $\beta(t)$, a Bessel-bridge, and a Poisson thinning step. When the Brownian FPT density is unavailable, it employs the Hermann–Tanré iterative approximation to $τ_{β}^{W}$ and demonstrates the approach on linear and curved thresholds, including a neuron-spike timing application. The paper further analyzes the algorithm’s time complexity, proposes drift-shifting and space-splitting strategies to reduce iterations, and shows the method’s advantages over discretisation-based methods in accuracy, with concrete results on spike-time prediction and thresholds in neuroscience.

Abstract

The first-passage time (FPT) is a fundamental concept in stochastic processes, representing the time it takes for a process to reach a specified threshold for the first time. Often, considering a time-dependent threshold is essential for accurately modeling stochastic processes, as it provides a more accurate and adaptable framework. In this paper, we extend an existing Exact simulation method developed for constant thresholds to handle time-dependent thresholds. Our proposed approach utilizes the FPT of Brownian motion and accepts it for the FPT of a given process with some probability, which is determined using Girsanov's transformation. This method eliminates the need to simulate entire paths over specific time intervals, avoids time-discretization errors, and directly simulates the first-passage time. We present results demonstrating the method's effectiveness, including the extension to time-dependent thresholds, an analysis of its time complexity, comparisons with existing methods through numerical examples, and its application to predicting spike times in a neuron.

Figures (4)

• Figure 1: Density plot of first-passage time for 10,000 simulations with parameters $x_{0}=0$, $K=1.6$, $a=-1$ and $b=0.5$, using three different methods, Euler-Maruyama, Improved Euler and Exact method. The points on the x-axis are the peaks of the density plots.
• Figure 2: First and second moment bias of Euler-Maruyama and Improved Euler-Maruyama method from 10k simulations with step size going from $2^{-4}$ to $2^{-10}$.
• Figure 3: Density plot of first-passage time from 5000 simulations with parameters $x_{0}=0$, $K=1.6$, $a=b=1$ using Exact method (with different $\epsilon$ parameter for generating $\tau_\beta$) and "High-resolution" Improved Euler method.
• Figure 4: Spike times of a neuron with different input current, $I$, observed until 2 seconds. For each $I$, we did 5 trials with parameters $\tau=-1$, $V_r =1$, $\sigma=1$, $v_0 =1$, $\theta_0 =1$, $\Delta=1$ and $\tau_1 =1$.

Theorems & Definitions (3)

• Remark 2.1
• Remark 2.2: Interpretation of Assumption 2.3
• Remark 3.1