Exact solutions to first-passage problems for jump-diffusion processes

• Autorzy: Lefebvre, Mario
• Języki publikacji: EN

Abstrakty

EN

Let T(x) be the first time the time-homogeneous jump-diffusion process X(t), starting from X(0) = x, leaves the interval (a, b). The jump size is assumed to have an asymmetric double exponential distribution. The integro-differential equation satisfied by the moment-generating function of T(x) is transformed into an ordinary differential equation and is solved explicitly in particular cases. Explicit and exact results are also obtained for the mean of T(x) as well as the probability P[X(T(x)) ≤ a].

Słowa kluczowe

EN

first-passage time; Brownian motion; Poisson process; jump size; integro-differential equation

• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2024
• Tom: Vol. 72, no 1
• Strony: 81--95
• Opis fizyczny: Bibliogr. 9 poz., wykr.

Twórcy

autor

Lefebvre, Mario

• Department of Mathematics and Industrial Engineering, Polytechnique Montréal, Montréal, Québec, Canada H3C 3A7,

Bibliografia

• [1] M. Abundo, On first-passage times for one-dimensional jump-diffusion processes, Probab. Math. Statist. 20 (2000), 399–423.
• [2] M. Abundo, On the first hitting time of a one-dimensional diffusion and a compound Poisson process, Methodology Computing Appl. Probab. 12 (2010), 473–490.
• [3] L. Bo and M. Lefebvre, Mean first passage times of two-dimensional processes with jumps, Statist. Probab. Lett. 81 (2011), 1183–1189.
• [4] H. Ghamlouch, A. Grall and M. Fouladirad, On the use of jump-diffusion process for maintenance decision-making: A first step, in: Annual Reliability and Maintainability Symposium (RAMS), IEEE, 2015, 6 pp.
• [5] I. I. Gihman and A. V. Skorohod, Stochastic Differential Equations, Springer, New York, 1972.
• [6] S. G. Kou and H. Wang, First passage times of a jump diffusion process, Adv. Appl. Probab. 35 (2003), 504–531.
• [7] M. Lefebvre, LQG homing for jump-diffusion processes, ROMAI J. 10 (2014), 147–152.
• [8] R. C. Merton, Option pricing when underlying stock returns are discontinuous, J. Financial Economics 3 (1976), 125–144.
• [9] J. Peng and Z. Liu, First passage time moments of jump-diffusions with Markovian switching, Int. J. Stochastic Anal. 2011, art. 501360, 11 pp.

Identyfikatory

DOI

10.4064/ba190812-11-6