Optimal investment game for two regulated players with regime switching

This paper investigated a zero-sum stochastic investment game for two investors in a regime-switching market with common random time solvency regulations. We considered two types of intensities for the inter-arrival time of regulations: one was modeled as a function of a time-homogeneous Markov chain, while the other was treated as a deterministic function of time $t$. In the first case, the associated Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation was an elliptic partial differential equation (PDE). By solving an auxiliary problem, we demonstrated the existence and regularity of the value function. In the regime-switching model, players' optimal strategies resembled those in a non-regime-switching model but required dynamic adjustments based on the Markov chain state. In the second case, the associated HJBI equation was a parabolic PDE. We provided a numerical method using a Markov chain approximation scheme and presented several numerical examples to illustrate the impact of regime switching and random time solvency on optimal policies.