# Dynamic programming for stochastic target problems and geometric flows

### H. Mete Soner

ETH Zentrum, Zürich, Switzerland### Nizar Touzi

Ecole Polytechnique, Palaiseau, France

## 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.

## Cite this article

H. Mete Soner, Nizar Touzi, Dynamic programming for stochastic target problems and geometric flows. J. Eur. Math. Soc. 4 (2002), no. 3, pp. 201–236

DOI 10.1007/S100970100039