Consider a one dimensional diffusion process on the diffusion interval $I$ originated in $x_0\in I$. Let $a(t)$ and $b(t)$ be two continuous functions of $t$, $t>t_0$ with bounded derivatives and with $a(t)<b(t)$ and $a(t),b(t)\in I$, $\forall t>t_0$. We study the joint distribution of the two random variables $T_a$ and $T_b$, first hitting times of the diffusion process through the two boundaries $a(t)$ and $b(t)$, respectively. We express the joint distribution of $T_a, T_b$ 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.

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

SACERDOTE, Laura Lea;TELVE, OTTAVIA;ZUCCA, CRISTINA
2014

Abstract

Consider a one dimensional diffusion process on the diffusion interval $I$ originated in $x_0\in I$. Let $a(t)$ and $b(t)$ be two continuous functions of $t$, $t>t_0$ with bounded derivatives and with $a(t)t_0$. We study the joint distribution of the two random variables $T_a$ and $T_b$, first hitting times of the diffusion process through the two boundaries $a(t)$ and $b(t)$, respectively. We express the joint distribution of $T_a, T_b$ in terms of $P(T_aT_b)$ 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.
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http://arxiv.org/abs/1403.1756
http://projecteuclid.org/euclid.aap/1396360109
First-hitting time; Diffusion Process; Brownian motion; Ornstein Uhlenbeck process; copula
L. SACERDOTE; O. TELVE; C. ZUCCA
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2318/148646
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