Dynamic programming for stochastic target problems and geometric flows

H. Mete Soner, Nizar Touzi

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)201-236
Number of pages36
JournalJournal of the European Mathematical Society
Issue number3
StatePublished - Sep 2002

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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