# Journal of the European Mathematical Society

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**Volume 4, Issue 3, 2002, pp. 201–236**

**DOI: 10.1007/s100970100039**

Dynamic programming for stochastic target problems and geometric flows

H. Mete Soner^{[1]}and Nizar Touzi

^{[2]}(1) Departement Mathematik, ETH Zentrum HG G 54.3, Rämistrasse 101, 8092, ZÜRICH, SWITZERLAND

(2) Centre de Mathématiques Appliquées, Ecole Polytechnique, Plateau de Palaiseau, F-91128, PALAISEAU, FRANCE

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.

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Soner H. Mete, Touzi Nizar: Dynamic programming for stochastic target problems and geometric flows. *J. Eur. Math. Soc.* 4 (2002), 201-236. doi: 10.1007/s100970100039