Author:
Sacerdote Laura,Telve Ottavia,Zucca Cristina
Abstract
Consider a one-dimensional diffusion process on the diffusion interval I originated in x
0 ∈ I. Let a(t) and b(t) be two continuous functions of t, t > t
0, with bounded derivatives, a(t) < b(t), and a(t), b(t) ∈ I, for all t > t
0. 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.
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,Statistics and Probability