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...

The following backward stochastic Riccati differential equation (BSRDE in short) is motivated, and is then studied. Some properties are presented. The existence and uniqueness of a global adapted solution to a BSRDE has been open for the case D i 6= 0 for more than two decades. Our recent...

We obtain the global existence and uniqueness result for a one-dimensional back- ward stochastic Riccati equation, whose generator contains a quadratic term of L (the second unknown component). This solves the one-dimensional case of Bismut- Peng's problem which was initially proposed by Bismut...

We review the relations between adjoints of stochastic control problems with the derivative of the value function, and the latter with the value function of a stopping problem. These results are applied to the pricing of contingent claims.

We consider an optimal control problem for a one-dimensional Itô diffusion and a stochastic game of optimal stopping associated with it. Their value functions satisfy ... and an optimal control defines a saddle point for the game. This extends earlier results to the case of bounded variation...