# Shuanming Li

### On the discounted aggregate claims occurred in certain states in a MAP risk process

This paper studies the moments and the distribution of the discounted aggregate claims occurred in certain states (a subset of the state space) by time t in a Markovian Arrival Process (MAP) risk model under stochastic interests. The paper first shows that the joint Laplace transform of the discounted aggregate claims occurred in each state by time t satisfies a first order partial differential equation through which a recursive formula is de rived for the moments of the discounted aggregate claims occurred in certain states. We also study two types of covariances of the discounted aggregate claims: [1] the covariance of the discounted aggregate claims occurred in any two subsets of the state space; [2] the covariance of discounted aggregate claims occurred in a subset of the state space at two different times.

The distribution of the discounted aggregate claims occurred in certain states by time t is also investigated. For a two-state Markov-modulated model, numerical results are presented for the distribution, the first two moments, and covariance of the discounted aggregate claims occurred in the two states.