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A BSDE approach to stochastic linear quadratic control problem
Author(s) -
Zhang Wei,
Zhang Liangquan
Publication year - 2021
Publication title -
optimal control applications and methods
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.458
H-Index - 44
eISSN - 1099-1514
pISSN - 0143-2087
DOI - 10.1002/oca.2707
Subject(s) - girsanov theorem , mathematics , stochastic differential equation , stochastic control , optimal control , riccati equation , linear quadratic regulator , quadratic equation , hamiltonian (control theory) , algebraic riccati equation , maximum principle , transformation (genetics) , mathematical optimization , differential equation , mathematical analysis , biochemistry , geometry , chemistry , gene
In this article, we study a kind of linear quadratic optimal control problem driven by forward–backward stochastic differential equations (FBSDEs in short) with deterministic coefficients. The cost functional is defined by the solution of FBSDEs. By means of the Girsanov transformation, the original issue is turned equivalently into the classical LQ problem. By functional analysis approach, some necessary and sufficient conditions for the existence of optimal controls have been obtained. Moreover, we investigate the relationship between two groups of first‐order and second‐order adjoint equations. A new stochastic Riccati equation is derived, which leads to the state feedback form of optimal control. By introducing a new Hamiltonian function with an exponential factor, we establish the stochastic maximum principle to deal with the stochastic linear quadratic problem for forward–backward stochastic system with nonconvex control domain using first‐order adjoint equation. An illustrative example is given as well.