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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, \[ x(t)+\int^ T_ tf(x(s),y(s),s)ds+\int^ T_ ty(s)dW(s)=X, \] under local Lipschitz condition, where \((\Omega,{\mathcal F},P,W(\cdot),{\mathcal F}_ t)\) is a standard Wiener process, for any given \((x,y)\), \(f(x,y,\cdot)\) is an \({\mathcal F}_ t\)-adapted process, and \(X\) is \({\mathcal F}_ T\)-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 (aspects of stochastic analysis)
93E20 Optimal stochastic control
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References:

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