Some Optimal Dividend Problems in aMarkov-modulated Risk Model

ISBN 07340 3564 0In this paper, we derive some results on the dividend payments prior toruin in a Markov-modulated risk model in which the claim inter-arrivals,claim sizes and premiums are influenced by an external Markovian environmentprocess. A system of integro-differential equations with boundaryconditions satisfied by the n-th moment of present value of the total dividendpayments prior to ruin, given the initial environment state, is derivedand solved. We show that both the probabilities that the surplus processattains a given level from the initial surplus without first falling below zeroand the Laplace transforms of the time that the surplus process first hits abarrier without ruin occuring can be expressed in term of the solution of theabove mentioned system of integro-differential equations. In the two-statemodel, explicit solutions to the integro-differential equations are obtainedwhen both claim size distributions are exponentially distributed. Finally, anumerical example and the comparison with the results obtained from theassociated averaged compound Poisson risk model are also given