First passage probabilities of one-dimensional diffusion processes

Abstract

This work is devoted to calculating the first passage probabilities of one-dimensional diffusion processes. For a one-dimensional diffusion process, we construct a sequence of Markov chains so that their absorption probabilities approximate the first passage probability of the given diffusion process. This method is especially useful when dealing with time-dependent boundaries.

Keywords

Boundary crossing probability / first passage probability / Markov chain / Skorokhod approximation

References

[1] Anderson T W. A modification of the sequential probability ratio test to reduce the sample size. Ann Math Statist, 1960, 31: 165-197

[2] Borovkov K, Novikov A. Explicit bounds for approximation rates of boundary crossing probabilities for the Wiener process. J Appl Probab, 2005, 42: 82-92

[3] Cai N, Chen N, Wan X. Pricing double-barrier options under a flexible jump diffusion model. Oper Res Lett, 2009, 37: 163-167

[4] Chi Y, Lin X S. On the threshold dividend strategy for a generalized jump-diffusion risk model. Insurance Math Econom, 2011, 48(3): 326-337

[5] Daniels H E. Approximating the first crossing-time density for a curved boundary. Bernoulli, 1996, 2: 133-143

[6] Downes A N, Borovkov K. First passage densities and boundary crossing probabilities for diffusion processes. Methodol Comput Appl Probab, 2008, 10: 621-644

[7] Durbin J. The first-passage density of the Brownian motion process to a curved boundary. J Appl Probab, 1992, 29: 291-304

[8] Ferebee B. The tangent approximation to one-sided Brownian exit densities. Z Wahrschein-lichkeitstheor Verwandte Geb, 1982, 61: 309-326

[9] Fu J C, Wu T L. Linear and nonlinear boundary crossing probabilities for Brownian motion and related processes. J Appl Probab, 2010, 47: 1058-1071

[10] Giraudo M T. An approximation formula for the first-crossing-time density of a Wiener process perturbed by random jumps. Statist Probab Lett, 2009, 79: 1559-1567

[11] Kou S G, Wang H. First passage times of a jump diffusion process. Adv Appl Probab, 2003, 35: 504-531

[12] Novikov A, Frishling V, Kordzakhia N. Approximations of boundary crossing probabilities for a Brownian motion. J Appl Probab, 1999, 99: 1019-1030

[13] Novikov A, Frishling V, Kordzakhia N. Time-dependent barrier options and boundary crossing probabilities. Georgian Math J, 2003, 10: 325-334

[14] Pötzelberger K, Wang L. Boundary crossing probability for Brownian motion. J Appl Probab, 2001, 38: 152-164

[15] Siegmund D. Boundary crossing probabilities and statistical applications. Ann Statist, 1986, 14: 361-404

[16] Skorokhod A V. Studies in the Theory of Random Processes. Boston: Addison-Wesley Publishing Company, INC, 1965

[17] Wang L, Pötzelberger K. Boundary crossing probability for Brownian motion and general boundaries. J Appl Probab, 1997, 34: 54-65

[18] Wang L, Pötzelberger K. Crossing probabilities for diffusion processes with piecewise continuous boundaries. Methodol Comput Appl Probab, 2007, 9: 21-40

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg