We study a stochastic control problem arising from mathematical finance. The goal is to minimize a cost function that is quadratic in both the wealth process and portfolio strategy in a continuous time complete market with random market parameters and portfolio constraints. Following a convex duality approach, we adopt the formulation of the primal and dual problem as introduced in Heunis and Labbe (2007). We then prove that the necessary and sufficient conditions for both the primal and dual problems can be written in terms of processes satisfying a system of FBSDEs together with other conditions. This allows us to explicitly characterize the primal control as a function of adjoint process coming from the dual FBSDEs in a dynamic fashion and vice versa. Moreover, we also find that the optimal primal wealth process coincides with the optimal adjoint process of the dual problem and vice versa. Finally, we apply the dual approach to solve both the unconstrained and cone-constrained quadratic risk minimization problems. (Joint work with Yusong Li).
Topic: Convex Quality Method for Constrained Quadratic Risk Minimization via FBSDEs
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