Dynamic programming for stochastic target problems and geometric flows |
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Authors: | H. Mete Soner Nizar Touzi |
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Affiliation: | 1.Ko? University, Department of Mathematics, Rumelifener Yolu, Sariyer, 80910 Istanbul, Turkey, e-mail: msoner@ku.edu.tr,TR;2.CREST, 15 Bd Gabriel Peri, 92245 Malakoff, Paris, France, e-mail: touzi@ensae.fr,FR |
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Abstract: | Given a controlled stochastic process, the reachability set is the collection of all initial data from which the state process can be driven into a target set at a specified time. Differential properties of these sets are studied by the dynamic programming principle which is proved by the Jankov-von Neumann measurable selection theorem. This principle implies that the reachability sets satisfy a geometric partial differential equation, which is the analogue of the Hamilton-Jacobi-Bellman equation for this problem. By appropriately choosing the controlled process, this connection provides a stochastic representation for mean curvature type geometric flows. Another application is the super-replication problem in financial mathematics. Several applications in this direction are also discussed. Received October 24, 2000 / final version received July 24, 2001?Published online November 27, 2001 |
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Keywords: | Mathematics Subject Classification (1991): 49J20, 60J60 49L20, 35K55 |
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