Consumption and Investment Optimization under Constraints

We analyze a problem of maximization of expected terminal wealth and consumption under constraints in a general framework including financial models with constrained portfolios, labor income and large investor models. By using general optional decomposition under constraints in a multiplicative form, we first develop a dual formulation under minimal assumption modeled as in Pham and Mnif (2002). We then are able to prove an existence and uniqueness of an optimal solution to primal and to the corresponding dual problem. An optimal investment and consumption plan to the original problem then can be found by convex duality, similarly to the case considered by Kramkov and Schachermayer (1999).