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

Part of:Integral equations, integral transformsMarkov processesStochastic processes

Published online by Cambridge University Press:  22 February 2016

Laura Sacerdote,
Ottavia Telve and
Cristina Zucca
Laura Sacerdote*
Affiliation:
University of Torino
Ottavia Telve*
Affiliation:
University of Torino
Cristina Zucca*
Affiliation:
University of Torino
*
∗ Postal address: Department of Mathematics ‘G. Peano’, University of Torino, Via Carlo Alberto 10, 10123 Torino, Italy.
∗ Postal address: Department of Mathematics ‘G. Peano’, University of Torino, Via Carlo Alberto 10, 10123 Torino, Italy.
∗ Postal address: Department of Mathematics ‘G. Peano’, University of Torino, Via Carlo Alberto 10, 10123 Torino, Italy.
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Abstract

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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.

Keywords

First hitting timediffusion processBrownian motionOrnstein-Uhlenbeck processcopula

MSC classification

Primary:60J60: Diffusion processes60G40: Stopping times; optimal stopping problems; gambling theory
Secondary:60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)65R20: Integral equations

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 46 , Issue 1 , March 2014, pp. 186 - 202
Copyright
© Applied Probability Trust 

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