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Relationship between backward stochastic differential equations and stochastic controls: A linear-quadratic approach. (English) Zbl 0960.60052
Let \(W=(W_t)_{t\in [0,T]}\) be a \(d\)-dimensional Brownian motion, \(A:[0,T]\to \mathbb{R}^{n\times n}\), \(B:[0,T]\to \mathbb{R}^{d\times n\times n}\) bounded deterministic functions, and \(f\) a square integrable process. Given an initial condition \(x\in\mathbb{R}^n\) and a random variable \(\xi\in L^2 (\sigma\{W\}; \mathbb{R}^n)\), the authors consider the controlled system \[ dX_t=(A_tX_t+ u_tB_t+ f_t)dt +u_tdW_t,\;t\in [0,T], \quad X_0=x,\tag{1} \] with cost functional \(J(x,u)= E[|X_T-\xi |^2]\). They show that if the associated deterministic Riccati equation (involving the inverse of its unknown) has a solution, then the above control problem possesses an optimal feedback control \(u^*\) of (1), which can be expressed with the help of the Riccati equation, interpreted as backward stochastic differential equation with terminal condition \(X_T=\xi\) (for this \(X_0\) is viewed as unknown) and the system (1) interpreted as forward equation as above. The techique employed is similar to that recently developed by Chen/Zhou for the problem of stochastic linear quadratic regulators and associated stochastic Riccati equations. After the proof of the main result the solvability of the Riccati equation is studied, and an application to the hedging problem in the Black-Scholes model is given.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
49N10 Linear-quadratic optimal control problems
91G80 Financial applications of other theories
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