Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T12:46:02.149Z Has data issue: false hasContentIssue false
  • English
  • Français

An approximation for the inverse first passage time problem

Part of: Partial differential equations, boundary value problems Markov processes Partial differential equations on manifolds; differential operators

Published online by Cambridge University Press:  01 July 2016

Jing-Sheng Song  and
Paul Zipkin
Jing-Sheng Song*
Affiliation:
Duke University
Paul Zipkin*
Affiliation:
Duke University
*
∗ Postal address: Fuqua School of Business, Duke University, Durham, NC, USA.
∗ Postal address: Fuqua School of Business, Duke University, Durham, NC, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We propose an approximation for the inverse first passage time problem. It is similar in spirit and method to the tangent approximation for the original first passage time problem. We provide evidence that the technique is quite accurate in many cases. We also identify some cases where the approximation performs poorly.

Keywords

Inverse first passage timeWiener processapproximation

MSC classification

Primary: 60J65: Brownian motion
Secondary: 65N21: Inverse problems 58J35: Heat and other parabolic equation methods
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 1 , March 2011 , pp. 264 - 275
Copyright
Copyright © Applied Probability Trust 2011 

References

Abundo, M. (2006). Limit at zero of the first-passage time density and the inverse problem for one-dimensional diffusions. Stoch. Anal. Appl. 24, 1119–1145.CrossRefGoogle Scholar
Cheng, L., Chen, X., Chadam, J. and Saunders, D. (2007). Analysis of an inverse first passage problem from risk management. SIAM J. Math. Anal. 38, 845–873.CrossRefGoogle Scholar
Daniels, H. E. (1974). The maximum size of a closed epidemic. Adv. Appl. Prob. 6, 607–621.CrossRefGoogle Scholar
Daniels, H. E. (1982). Sequential tests constructed from images. Ann. Statist. 10, 394–400.CrossRefGoogle Scholar
Daniels, H. E. (1996). Approximating the first crossing-time density for a curved boundary. Bernoulli 2, 133–143.CrossRefGoogle Scholar
Durbin, R. (1992). The first-passage density of the Brownian motion process to a curved boundary. J. Appl. Prob. 29, 291–304.CrossRefGoogle Scholar
Jaimungal, S., Kreinin, A. and Valov, A. (2009). Integral equations and the first passage time of Brownian motion. Working paper, University of Toronto.Google Scholar
Lerche, H. R. (1986). Boundary Crossing of Brownian Motion. Springer, Berlin.CrossRefGoogle Scholar
Lo, V. S. F., Roberts, G. O. and Daniels, H. E. (2002). Inverse method of images. Bernoulli 8, 53–80.Google Scholar
Peskir, G. (2002). On integral equations arising in the first-passage problem for Brownian motion. J. Integral Equations Appl. 14, 397–423.CrossRefGoogle Scholar
Roberts, G. O. and Shortland, C. F. (1995). The hazard rate tangent approximation for boundary hitting times. Ann. Appl. Prob. 5, 446–460.CrossRefGoogle Scholar
Song, J. and Zipkin, P. (2006). Supply streams. Working paper, Duke University.Google Scholar
Strassen, V. (1967). Almost sure behavior of sums of independent random variables and martingales. In Proc. 5th Berkeley Symp. Math. Statist. Prob., Vol. II, University of California Press, Berkeley, CA, pp. 315–343.Google Scholar
Zucca, C. and Sacerdote, L. (2009). On the inverse first-passage-time problem for a Wiener process. Ann. Appl. Prob. 19, 1319–1346.CrossRefGoogle Scholar