Markov-modulated diffusion risk models

Nicole Bäuerle, Mirko Kötter
2007 Scandinavian Actuarial Journal
In this paper we consider Markov-modulated diffusion risk reserve processes. Using diffusion approximation we show the relation to classical Markov-modulated risk reserve processes. In particular we derive a representation for the adjustment coefficient and prove some comparison results. Among others we show that increasing the volatility of the diffusion increases the probability of ruin. We consider a diffusion risk reserve process where the underlying data changes according to a
more » ... g to a continuous-time Markov chain with finite state space. More precisely, we denote by J = {J t , t ≥ 0} an irreducible continuous-time Markov chain with finite state space E = {1, . . . , d} and intensities q ij . If not stated otherwise, the distribution of J 0 is arbitrary. J t can be interpreted as the general economic conditions which are present at time t. J t influences the premium rate, the arrival intensity of claims, the claim size distribution and the volatility of the diffusion process as follows: the premium income rate at time t is c Jt , i.e. as long as J t = i we have a linear income stream at rate c i . Claim arrivals are according to a Poisson-process with rate λ Jt . Thus, N = {N t , t ≥ 0} is a Markovmodulated Poisson-process. A claim U k which occurs at time t has distribution Q Jt , where Q i is some distribution concentrated on (0, ∞) for i ∈ E. As usual claim sizes U 1 , U 2 , . . . are assumed to be conditionally independent given J and µ i is the finite expectation of Q i , for i ∈ E. The volatility of the diffusion at time t is given by σ Jt . If u ≥ 0 denotes the initial risk reserve, and W = {W t , t ≥ 0} is a standard Brownian motion, the Markovmodulated diffusion risk reserve process {X t , t ≥ 0} is given by