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Optimal control of linear backward stochastic differential equations with a quadratic cost criterion. (English) Zbl 1047.93049

Pasik-Duncan, Bozenna (ed.), Stochastic theory and control. Proceedings of the workshop, Lawrence, KS, USA, October 18–20, 2001. Berlin: Springer (ISBN 3-540-43777-0/pbk). Lect. Notes Control Inf. Sci. 280, 301-317 (2002).
The paper deals with backward stochastic differential equations (BSDEs). In general, a BSDE is an ItĂ´-type equation with a terminal condition which is random.
This area, BSDEs, became one of the most important areas of contemporary applied stochastics. Extensive studies led to great theoretical developments with significant implications for applications in other areas, in particular in mathematical finance. It turns out that little has been known for controlled BSDEs, which are the object of study in this paper.
The authors describe in detail the model: \[ dx(t)= \{A(t)x(t)+ B(t)u(t)+ C(t)z(t)\}\,dt+ z(t)dW(t),\;t\in [0,T],\;x(T)= \xi. \] Here \(x(t)\) is the system process, \(z(t)\) a risk-adjustment factor, \(W(t)\) a standard Wiener process and \(u(t)\) a control. All processes are adapted to a given filtration, and the coefficients \(A\), \(B\), \(C\) satisfy appropriate conditions. The goal is to minimize a functional \(J(\xi, u(\cdot))\) which is a quadratic function of \(x\), \(z\) and the control \(u\), where \(u\) is taken from a class of admissible controls.
The problem does not come to be easy, and the authors show that knowing the solution to the deterministic analogue of the above optimization problem, is not enough. It is shown that the solution to the stochastic optimal control problem relies on two Riccati-type equations and also uses the corresponding uncontrolled BSDE.
The authors present also an alternative derivation of the solution based on the idea of the completion of squares technique and a limiting procedure, which is typical for forward linear quadratic problems.
As a whole, the paper is well written, with clear explanation of the main points, including the arising difficulties. The references list seems quite complete. Thus, the results obtained in this paper can be considered as a good achievement in this modern area.
For the entire collection see [Zbl 1005.00048].

MSC:

93E20 Optimal stochastic control
60H30 Applications of stochastic analysis (to PDEs, etc.)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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