Optimal consumption and portfolio with both fixed and proportional transaction costs

We consider a market model with one risk-free and one risky asset, in which the dynamics of the risky asset are governed by a geometric Brownian motion. In this market we consider an investor who consumes from the bank account and who has the opportunity at any time to transfer funds between the two assets. We suppose that these transfers involve a fixed transaction cost k > 0, independent of the size of the transaction, plus a cost proportional to the size of the transaction. The objective is to maximize the cumulative expected utility of consumption over a planning horizon. We formulate this problem as a combined stochastic control/impulse control problem, which in turn leads to a (nonlinear) quasi-variational Hamilton Jacobi–Bellman inequality (QVHJBI). We prove that the value function is the unique viscosity solution of this QVHJBI. Finally, numerical results are presented.