Dynamic convex duality in constrained utility maximization

ABSTRACT

In this paper, we study a constrained utility maximization problem following the convex duality approach. After formulating the primal and dual problems, we construct the necessary and sufficient conditions for both the primal and dual problems in terms of forward and backward stochastic differential equations (FBSDEs) plus some additional conditions. Such formulation then allows us to explicitly characterize the primal optimal control as a function of the adjoint process coming from the dual FBSDEs in a dynamic fashion and vice versa. We also find that the optimal wealth process coincides with the adjoint process of the dual problem and vice versa. Finally we solve three constrained utility maximization problems, which contrasts the simplicity of the duality approach we propose and the technical complexity of solving the primal problem directly.

KEYWORDS:

• Convex duality
• primal and dual FBSDEs
• utility maximization
• convex portfolio constraints

Acknowledgements

The authors are very grateful to the reviewer and the associate editor for their constructive comments and suggestions which have helped to improve the paper of the previous two versions.

Disclosure statement

No potential conflict of interest was reported by the authors.