The Annals of Applied Probability

On a class of optimal stopping problems for diffusions with discontinuous coefficients

Ludger Rüschendorf and Mikhail A. Urusov

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In this paper, we introduce a modification of the free boundary problem related to optimal stopping problems for diffusion processes. This modification allows the application of this PDE method in cases where the usual regularity assumptions on the coefficients and on the gain function are not satisfied. We apply this method to the optimal stopping of integral functionals with exponential discount of the form Ex0τeλsf(Xs) ds, λ≥0 for one-dimensional diffusions X. We prove a general verification theorem which justifies the modified version of the free boundary problem. In the case of no drift and discount, the free boundary problem allows to give a complete and explicit discussion of the stopping problem.

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Ann. Appl. Probab., Volume 18, Number 3 (2008), 847-878.

First available in Project Euclid: 26 May 2008

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Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05]

Optimal stopping free boundary problem one-dimensional SDE Engelbert–Schmidt condition local times occupation times formula Itô–Tanaka formula


Rüschendorf, Ludger; Urusov, Mikhail A. On a class of optimal stopping problems for diffusions with discontinuous coefficients. Ann. Appl. Probab. 18 (2008), no. 3, 847--878. doi:10.1214/07-AAP474.

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