The Annals of Applied Probability

The inverse first-passage problem and optimal stopping

Erik Ekström and Svante Janson

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Given a survival distribution on the positive half-axis and a Brownian motion, a solution of the inverse first-passage problem consists of a boundary so that the first passage time over the boundary has the given distribution. We show that the solution of the inverse first-passage problem coincides with the solution of a related optimal stopping problem. Consequently, methods from optimal stopping theory may be applied in the study of the inverse first-passage problem. We illustrate this with a study of the associated integral equation for the boundary.

Article information

Ann. Appl. Probab., Volume 26, Number 5 (2016), 3154-3177.

Received: August 2015
First available in Project Euclid: 19 October 2016

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Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Inverse first-passage problem optimal stopping nonlinear integral equation


Ekström, Erik; Janson, Svante. The inverse first-passage problem and optimal stopping. Ann. Appl. Probab. 26 (2016), no. 5, 3154--3177. doi:10.1214/16-AAP1172.

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  • [1] Anulova, S. V. (1980). Markov times with given distribution for a Wiener process. Theory Probab. Appl. 25 362–366.
  • [2] Avellaneda, M. and Zhu, J. (2001). Modeling the distance-to-default process of a firm. RISK 14 125–129.
  • [3] Chen, X., Cheng, L., Chadam, J. and Saunders, D. (2011). Existence and uniqueness of solutions to the inverse boundary crossing problem for diffusions. Ann. Appl. Probab. 21 1663–1693.
  • [4] Cheng, L., Chen, X., Chadam, J. and Saunders, D. (2006). Analysis of an inverse first passage problem from risk management. SIAM J. Math. Anal. 38 845–873.
  • [5] Jacka, S. (1991). Optimal stopping and the American put. Math. Finance 1 1–14.
  • [6] Peskir, G. (2005). On the American option problem. Math. Finance 15 169–181.
  • [7] Peskir, G. and Shiryaev, A. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel.
  • [8] Zucca, C. and Sacerdote, L. (2009). On the inverse first-passage-time problem for a Wiener process. Ann. Appl. Probab. 19 1319–1346.