Department of Mathematics2024-11-0920021435-985510.1007/s1009701000392-s2.0-33845799781http://dx.doi.org/10.1007/s100970100039https://hdl.handle.net/20.500.14288/13030Given 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.MathematicsJournal Article1782482000017932