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Motivated by applications in option pricing theory (Peskir, 1997b), (Research Report No. 386, Dept. Theoret. Statist. Aarhus, 19 pp.) we formulate and solve the following problem. Given a standard Brownian motion B = (Bt)t¿0 and a centered probability measure on R having the distribution function F with a strictly positive density F satisfying ∞ 0 x log x (dx) ¡ ∞; there exists a cost function x → c(x) in the optimal stopping problem sup E max 06t6 Bt -0 c(Bt) dt such that for the optimal stopping time \* we have B \* ∼ : The cost function is explicitly given by the formula: where one incidentally recognizes x → F (x)=(1 -F(x)) as the Hazard function of . There is also a simple explicit formula for the optimal stopping time \* , but the main emphasis of the result is on the existence of the underlying functional in the optimal stopping problem. The integrability condition on is natural and cannot be improved. The condition on the existence of a strictly positive density is imposed for simplicity, and more general cases could be treated similarly. The method of proof combines ideas and facts on optimal stopping of the maximum process (Peskir, 1997a), (

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Author

Goran Peskir

Publisher

Elsevier Science; Elsevier ; Elsevier BV (ISSN 0304-4149)

Published

1999

Language

EN

Field

Mathematics (Physical Sciences)