On time hybrid controlled Martingale problem

A time hybrid controlled martingale problem (THCMP) is defined. The THCMP extends the classical concept of the controlled martingale problem. The relevance of THCMP to time hybrid control problems, that incorporate both discrete and continuous time scales, is demonstrated. Bellman-type optimality equations and a verification theorem that extend, and unite, the classical optimality results of continuous time and discrete time control theories are provided. Some illustrative examples are also given.