Joint Densities of First Hitting Times of a Diffusion Process Through Two Time-Dependent Boundaries

Abstract

Consider a one dimensional diffusion process on the diffusion interval I originated in x0 ∈ I. Let a(t) and b(t) be two continuous functions of t, t > t0 with bounded derivatives and with a(t) < b(t) and a(t), b(t) ∈ I, ∀t > t0. We study the joint distribution of the two random variables Ta and T b , first hitting times of the diffusion process through the two boundaries a(t) and b(t), respectively. We express the joint distribution of Ta, T b in terms of P (Ta < t, Ta < T b ) and P (T b < t, Ta > T b ) and we determine a system of integral equations verified by these last probabilities. We propose a numerical algorithm to solve this system and we prove its convergence properties. Examples and modeling motivation for this study are also discussed.

Exact simulation of first exit times for one-dimensional diffusion processes

“…Of course, such an approach is interesting provided that (T n , X n ) is easy to simulate numerically. For the particular Brownian case, the distribution of the exit time from an interval has a quite complicated expression which is difficult to use for simulation purposes (see, for instance [12]) whereas the exit distribution from particular time-dependent domains, for instance the spheroids also called heat balls, can be precisely determined. These timedependent domains are characterized by their boundaries:…”

Exit problem for Ornstein-Uhlenbeck processes: A random walk approach

“…Of course, such an approach is interesting provided that (T n , X n ) is easy to simulate numerically. For the particular Brownian case, the distribution of the exit time from an interval has a quite complicated expression which is difficult to use for simulation purposes (see, for instance [11]) whereas the exit distribution from particular time-dependent domains, for instance the spheroids also called heat balls, can be precisely determined. These timedependent domains are characterized by their boundaries:…”

Approximation of the first passage time density of a Wiener process to an exponentially decaying boundary by two-piecewise linear threshold. Application to neuronal spiking activity

“…For most of the processes arising from applications and for time-varying thresholds, analytical expressions are not available. Numerical algorithms based on solving integral equations have been proposed in [4,5,27,29,32,34], while approximations based on Monte-Carlo path-simulation methods in [13,14,20].…”