  We develop a probabilistic approach to continuous-time finite state mean
field games. Based on an alternative description of continuous-time Markov
chain by means of semimartingale and the weak formulation of stochastic optimal
control, our approach not only allows us to tackle the mean field of states and
the mean field of control in the same time, but also extend the strategy set of
players from Markov strategies to closed-loop strategies. We show the existence
and uniqueness of Nash equilibrium for the mean field game, as well as how the
equilibrium of mean field game consists of an approximative Nash equilibrium
for the game with finite number of players under different assumptions of
structure and regularity on the cost functions and transition rate between
states.
