Optimal investment-consumption problem with constraint

In this paper, we consider an optimal investment-consumption problem subject to a closed convex constraint. In the problem, a constraint is imposed on both the investment and the consumption strategy, rather than just on the investment. The existence of solution is established by using the Mar-tingale technique and convex duality. In addition to investment, our technique embeds also the consumption into a family of ctitious markets. However, with the addition of consumption, it leads to nonreflexive dual spaces. This diculty is overcome by employing the so-called technique of "relaxation-projection" to establish the existence of solution to the problem. Furthermore, if the solution to the dual problem is obtained, then the solution to the primal problem can be found by using the characterization of the solution. An illustrative example is given with a dynamic risk constraint to demonstrate the method.