Joint densities of first hitting times of a diffusion process through two time dependent boundaries

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 Tb, first hitting times of the diffusion process through the two boundaries a(t) and b(t), respectively. We express the joint distribution of Ta, Tb in terms of P (Ta < t, Ta < Tb) and P (Tb < t, Ta> Tb) 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