### 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.

Original language | English (US) |
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Pages (from-to) | 201-236 |

Number of pages | 36 |

Journal | Journal of the European Mathematical Society |

Volume | 4 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 2002 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Journal of the European Mathematical Society*,

*4*(3), 201-236. https://doi.org/10.1007/s100970100039