Optimal investment-consumption under recursive utility in regime-switching markets with transaction cost

We study an optimal investment-consumption problem with proportional transaction costs formulated as follows: the investor allocates total wealth between a non-risky asset and a risky asset, with the possibility of consuming the non-risky asset and transferring funds between the two at any time. The central question is how the investor should optimally consume and rebalance the portfolio over time so as to maximize recursive utility, and how the corresponding value function can be characterized. Another modelling feature is the empirically realistic assumption that the risky asset is traded in a financial market with regime switches. In a continuous-time Epstein–Zin recursive utility, we formulate this optimal problem as a system of Hamilton–Jacobi–Bellman (HJB) variational inequalities. We then prove the existence and uniqueness of the value function as the viscosity solution, and our results provide further characterizations of the viscosity solution to the HJB system. We develop essential modifications of the common viscosity solution arguments and propose new analytical methods to deal with technical challenges brought by the strong non-linearity, lack of convexity and degenerate diffusion through modelling improvements.