Dynamic programming for an investment/consumption problem in illiquid markets with regime-switching
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. In this paper we perform the first step needed to treat this model: the proof of the dynamic programming principle (DPP) and the characterization of the value function as the unique viscosity solution of the associated Hamilton–Jacobi–Bellman (HJB) equation. This puts the basis for the analysis of the optimal solution of the model which is done in the companion paper of the authors (SIAM J. Control. Optim. 52 (2014)). The proof of the dynamic programming principle is not standard as in this case we do not know a priori if the value function is continuous up to the boundary.