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March 2014 Joint densities of first hitting times of a diffusion process through two time-dependent boundaries
Laura Sacerdote, Ottavia Telve, Cristina Zucca
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Adv. in Appl. Probab. 46(1): 186-202 (March 2014). DOI: 10.1239/aap/1396360109

Abstract

Consider a one-dimensional diffusion process on the diffusion interval I originated in x0I. 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.

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Laura Sacerdote. Ottavia Telve. Cristina Zucca. "Joint densities of first hitting times of a diffusion process through two time-dependent boundaries." Adv. in Appl. Probab. 46 (1) 186 - 202, March 2014. https://doi.org/10.1239/aap/1396360109

Information

Published: March 2014
First available in Project Euclid: 1 April 2014

zbMATH: 1304.60086
MathSciNet: MR3189054
Digital Object Identifier: 10.1239/aap/1396360109

Subjects:
Primary: 60G40 , 60J60
Secondary: 60J70 , 65R20

Keywords: Brownian motion , copula , diffusion process , First hitting time , Ornstein-Uhlenbeck process

Rights: Copyright © 2014 Applied Probability Trust

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Vol.46 • No. 1 • March 2014
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