Consumption and Investment Optimization Under Constraints

We analyze a problem of maximization of expected terminal wealth and consumption under constraints in a general financial framework, which includes models with constrained portfolios, labor income and large investor models. By introducing the new finite probability space, as well as a new utility function, the considered problem is converted to the one studied by Pham and Mnif (2002) [48]. By using general optional decomposition under constraints, we can develop a dual formulation under minimal assumption modeled as in Pham and Mnif (2002) [48]. We then are able to prove an existence and uniqueness of an optimal solution to primal problem. Under the assumption that there exists a solution to the corresponding dual problem, an optimal consumption plan can be found by convex duality