  We consider an illiquid financial market with different regimes modeled by a
continuous-time finite-state Markov chain. The investor can trade a stock only
at the discrete arrival times of a Cox process with intensity depending on the
market regime. Moreover, the risky asset price is subject to liquidity shocks,
which change its rate of return and volatility, and induce jumps on its
dynamics. In this setting, we study the problem of an economic agent optimizing
her expected utility from consumption under a non-bankruptcy constraint. By
using the dynamic programming method, we provide the characterization of the
value function of this stochastic control problem in terms of the unique
viscosity solution to a system of integro-partial differential equations. We
next focus on the popular case of CRRA utility functions, for which we can
prove smoothness $C^2$ results for the value function. As an important
byproduct, this allows us to get the existence of optimal
investment/consumption strategies characterized in feedback forms. We analyze a
convergent numerical scheme for the resolution to our stochastic control
problem, and we illustrate finally with some numerical experiments the effects
of liquidity regimes in the investor's optimal decision.
