Shanjian Tang

Global adapted solution of one-dimensional backward stochastic Riccati equations, with application to the mean–variance hedging☆

Backward stochastic Riccati equations are motivated by the solution of general linear quadratic optimal stochastic control problems with random coefficients, and the solution has been open in the general case. One distinguishing difficult feature is that the drift contains a quadratic term of the second unknown variable. In this paper, we obtain the global existence and uniqueness result for a general one-dimensional backward stochastic Riccati equation. This solves the one-dimensional case of Bismut-Pengs problem which was initially proposed by Bismut (Lecture Notes in Math. 649 (1978) 180). We use an approximation technique by constructing a sequence of monotone drifts and then passing to the limit. We make full use of the special structure of the underlying Riccati equation. The singular case is also discussed. Finally, the above results are applied to solve the mean-variance hedging problem with general random market conditions.

Minimization of risk and linear quadratic optimal control theory

This article is concerned with the optimal control problem for the linear stochastic system Xt = x + � t 0 (AsXs + Bsus + fs) ds + � t 0 � d i=1 (Ci(s)Xs + Di(s)us + gi(s)) dwi(s) with the convex risk functional J(u )= EM(XT )+ ET 0 G(t, Xt ,u t) dt. In order to guarantee the existence of an optimal control without any(weak) compactness assumption on the admissible control set, we assume that the risk function M is coercive and thatd=1 D ∗ Di is uniformlypositive, rather than to assume like in the control literature that the running risk function G is coercive with respect to the control variable. In this new setting, the running risk function G maybe independent of the control variable, and therefore the so-called singular linear-quadratic (LQ) stochastic control problem is included. A rigorous theoryis developed for the general stochastic LQ problem with random coefficients, and the bounded mean oscillation-martingale theoryis used to account for the concerned integrability. It plays a crucial role in the following exposition: (a) to connect the stochastic LQ problem to two associated backward stochastic differential equations (BSDEs)—one is an n × n symmetric matrix-valued nonlinear Riccati BSDE and the other is an n-dimensional linear BSDE with unbounded coefficients; (b) to show that the latter BSDE has an adapted solution pair of the suitablynecessaryregularity . This seems to be the first application in a stochastic LQ theory of the BMO-martingale theory, which roots in harmonic analysis. Furthermore, with the help of an a priori estimate on the risk functional, existence and uniqueness of the solutions of backward stochastic Riccati differential equations (BSRDEs) in the singular case is reduced to the regular case via a perturbation method, and then a new existence and uniqueness result on BSRDEs is obtained for the singular case.