EM algorithm for stochastic hybrid systems

A stochastic hybrid system, also known as a switching diffusion, is a continuous-time Markov process with state space consisting of discrete and continuous parts. We consider parametric estimation of theQmatrix for the discrete state transitions and of the drift coefficient for the diffusion part. First, we derive the likelihood function under the complete observation of a sample path in continuous-time. Then, extending a finite-dimensional filter for hidden Markov models developed by Elliott et al. (Hidden Markov Models, Springer, 1995) to stochastic hybrid systems, we derive the likelihood function and the EM algorithm under a partial observation where the continuous state is monitored continuously in time, while the discrete state is unobserved.