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, a(t) < b(t), and a(t), b(t) ∈ I, for all t > t₀. We study the joint distribution of the two random variables T_a and T_b, the first hitting times of the diffusion process through the two boundaries a(t) and b(t), respectively. We express the joint distribution of T_a and T_b in terms of P(T_a < t, T_a < T_b) and P(T_b < t, T_a > 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.