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Backward stochastic differential equations and applications to optimal control. (English) Zbl 0769.60054

Summary: We study the existence and uniqueness of the following kind of backward stochastic differential equation,

x(t)+ t T f(x(s),y(s),s)ds+ t T y(s)dW(s)=X,

under local Lipschitz condition, where (Ω,,P,W(·), t ) is a standard Wiener process, for any given (x,y), f(x,y,·) is an t -adapted process, and X is T -measurable. The problem is to look for an adapted pair (x(·), y(·)) that solves the above equation. A generalized matrix Riccati equation of that type is also investigated. A new form of stochastic maximum principle is obtained.

MSC:
60H10Stochastic ordinary differential equations
93E20Optimal stochastic control (systems)
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