Optimal Control of Linear Stochastic Systems with Singular Costs, and the Mean-Variance Hedging Problem with Stochastic Market Conditions

The optimal control problem is considered for linear stochastic systems with a singular cost. A new uniformly convex structure is formulated, and its consequences on the existence and uniqueness of optimal controls and on the uniform convexity of the value function are proved. In particular, the singular quadratic cost case with random coefficients is discussed and the existence and uniqueness results on the associated nonlinear singular backward stochastic Riccati differential equations are obtained under our structure conditions, which generalize Bismut-Peng's existence and uniqueness on nonlinear regular backward stochastic Riccati equations to nonlinear singular backward stochastic Riccati equations. Finally, applications are given to the mean-variance hedging problem with random market conditions, and an explicit characterization for the optimal hedging portfolio is given in terms of the adapted solution of the associated backward stochastic Riccati differential equation.