Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: A viscosity solution approach

We study a problem of optimal consumption and portfolio selection in a market where the logreturns of the uncertain assets are not necessarily normally distributed. The natural models then involve pure-jump L evy processes as driving noise instead of Brownian motion like in the Black and Scholes model. The state constrained optimization problem involves the notion of local substitution and is of singular type. The associated Hamilton-Jacobi-Bellman equation is a nonlinear rst order integro-diierential equation subject to gradient and state constraints. We prove that the value function of the singular stochastic control problem is the unique constrained viscosity solution of the Hamilton-Jacobi-Bellman equation. To this end, we prove a new comparison (uniqueness) result for the state constraint problem for a class of integro-diierential variational inequalities. We generalize our results to the second order case, where we in addition allow for a Brownian motion in the noise term. Here too we are able to prove existence and comparison results for the corresponding second order integro-diierential variational inequality. Finally, we discuss related models and present two speciic examples. In the rst we show that our control problem has an explicit solution when the utility function is of HARA type. In the second example, we consider Merton's problem, which is a special case of our stochastic control problem. We also here provide explicit results for HARA utility.