Robust classical-impulse stochastic control problems in an infinite horizon

This paper establishes a general analytical framework for classical and impulse stochastic control problems in the presence of model uncertainty. We consider a set of dominated models, which are induced by the measures equivalent to that of a reference model. The state process under the reference model is a multidimensional Markov process with multidimensional Brownian motion, controlled by continuous and impulse control variates. We propose quasi-variational inequalities (QVI) associated with the value function of the control problem and prove a verification theorem for the solution to the QVI. With the relative entropy constraints and piecewise linear intervention penalty, we show that the QVI can be degenerated to the non-robust case and it can be solved via the solution to a free boundary problem. To illustrate the tractability of the proposed framework, we apply it to a linear-quadratic setting, which covers a broad class of problems including robust mean-reverting inventory controls.