Before revision:  To solve this system, we first fix the optimal exercise boundary for each regime resulting in multi-variable fixed domains. We further eliminate the first-order derivatives associated with the regime-switching model by taking derivatives to obtain a system of coupled partial differential equations which we called the asset-delta-gamma-speed option equations.
After revision:  To solve this system, we first fix the optimal exercise boundary for each regime to multi-fixed intervals and then eliminate the first-order derivatives associated with the regime-switching model by taking derivatives to obtain a system of coupled partial differential equations which we called the asset-delta-gamma-speed option equations.