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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, $$x(t)+\int\sp T\sb tf(x(s),y(s),s)ds+\int\sp T\sb ty(s)dW(s)=X,$$ under local Lipschitz condition, where $(\Omega,{\cal F},P,W(\cdot),{\cal F}\sb t)$ is a standard Wiener process, for any given $(x,y)$, $f(x,y,\cdot)$ is an ${\cal F}\sb t$-adapted process, and $X$ is ${\cal F}\sb T$-measurable. The problem is to look for an adapted pair $(x(\cdot)$, $y(\cdot))$ 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:
 60H10 Stochastic ordinary differential equations 93E20 Optimal stochastic control (systems)
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##### References:
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