Advances in Applied Probability

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

Laura Sacerdote, Ottavia Telve, and Cristina Zucca

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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.

Article information

Adv. in Appl. Probab., Volume 46, Number 1 (2014), 186-202.

First available in Project Euclid: 1 April 2014

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Primary: 60J60: Diffusion processes [See also 58J65] 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 65R20: Integral equations

First hitting time diffusion process Brownian motion Ornstein-Uhlenbeck process copula


Sacerdote, Laura; Telve, Ottavia; Zucca, Cristina. Joint densities of first hitting times of a diffusion process through two time-dependent boundaries. Adv. in Appl. Probab. 46 (2014), no. 1, 186--202. doi:10.1239/aap/1396360109.

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