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A deterministic approach to stochastic optimal control with application to anticipative control. (English) Zbl 0774.93085
The stochastic optimal control problems are considered as a family of deterministic control problems parametrized by the paths of the driving Wilner process even for anticipative controls. The main tools are stochastic flows of diffeomorphisms for representing solutions and stochastic partial differential equations under the commuting conditions of the basic vector fields. The linear-quadratique and a scalar nonlinear anticipative control problem are considered as examples. The paper has about fifty pages and one may get a lot of useful information when studying stochastic differential equations with anticipative controls.

93E20 Optimal stochastic control
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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