Abstract
In this paper, we study the dividend payments prior to absolute ruin in a Markov-dependent risk process in which the claim occurrence and the claim amount are regulated by an external discrete time Markov chain. A system of integro-differential equations with boundary conditions satisfied by the moment-generating function, the nth moment of the discounted dividend payments prior to absolute ruin and the discounted penalty function, given the initial environment state, are derived. In the two-state risk model, explicit solutions to the integro-differential equations satisfied by the nth moment of the discounted dividend payments prior to absolute ruin are obtained when the claim size distribution is exponentially distributed. Finally, the matrix form of systems of integro-differential equations satisfied by the discounted penalty function are presented.
Highlights
Ruin theory under regime-switching model is becoming a popular topic
We study the dividend payments prior to absolute ruin in a Markov-dependent risk process in which the claim occurrence and the claim amount are regulated by an external discrete time Markov chain
A system of integrodifferential equations with boundary conditions satisfied by the moment-generating function, the nth moment of the discounted dividend payments prior to absolute ruin and the discounted penalty function, given the initial environment state, are derived
Summary
Ruin theory under regime-switching model is becoming a popular topic. This model is proposed in Reinhard [1] and Asmussen [2]. Asmussen [2] calls it a Markov-modulated risk model in which both the frequency of the claim arrivals and the distribution of the claim amounts are influenced by an external environment process. Albrecher and Boxma [9] study the expected discounted penalty function in a semi-Markovian dependent risk model in which at each instant of a claim, the underlying Markov chain jumps to a new state and the distribution of claim depends on this state. Dividend barrier and dividends payments under the threshold strategy in a Markov-dependent risk model, respectively They consider the structure of a semiMarkovian dependence type as follows.
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