The Expected Discounted Penalty Function of M-dimensional Risk Model with Markov-modulated Premium Rate

In this paper, we consider the expected discounted penalty function of m-dimensional risk model with markov-modulated premium rate. Using backward differential argument, we derive the integral equations satisfied by the conditional expected value and the expected value which is in the stationary case respectively. A system of Laplace transforms of the discounted penalty function, given the initial environment state, is established from a system of integro-differential equations. Also, an example where the intensity process is a two-state Markov process and the claims are exponentially distributed is given.