Combined Stochastic Control and Optimal Stopping, and Application to Numerical Approximation of Combined Stochastic and Impulse Control

This paper is twofold. The first aim is to study a combined stochastic control and optimal stopping problem: we prove a verification theorem and give a characterization of the value function as a unique viscosity solution to the associated Hamilton–Jacobi–Bellman variational inequality (HJBVI). Although these results have independent interest, they are also motivated by the fact that they are the main ingredients in solving a combined stochastic control and impulse control problem. Indeed, this problem can be reduced to an iterative sequence of combined stochastic control and optimal stopping problems. This method is implemented to solve numerically the quasi-variational inequality (QVI) associated with the problem of portfolio optimization with both fixed and proportional transaction costs. Numerical results are provided.