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

Part of: Integral equations, integral transforms Markov processes Stochastic 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 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 Ta and Tb, the first hitting times of the diffusion process through the two boundaries a(t) and b(t), respectively. We express the joint distribution of Ta and 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.

Keywords

First hitting timediffusion processBrownian motionOrnstein-Uhlenbeck processcopula

MSC classification

Primary: 60J60: Diffusion processes 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 65R20: Integral equations
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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