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A Markov decision problem in a risk model with interest rate and Markovian environment. (English) Zbl 1343.60108

Summary: We consider the compound binomial model in a Markovian environment presented by H. Cossette et al. [Insur. Math. Econ. 35, No. 2, 425–443 (2004; Zbl 1079.91049)]. We modify the model via assuming that the company receives interest on the surplus and a positive real-valued premium per unit time, and introducing a control strategy of periodic dividend payments. A Markov decision problem arises and the control objective is to maximize the cumulative expected discounted dividends paid to the shareholders until ruin minus a discounted penalty for ruin. We show that under the absence of a ceiling of dividend rates the optimal strategy is a conditional band strategy given the current state of the environment process. Under the presence of a ceiling for dividend rates, the character of the optimal control strategy is given. In addition, we offer an algorithm for the optimal strategy and the optimal value function. Numerical results are provided to illustrate the algorithm and the impact of the penalty.

MSC:

60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
90C40 Markov and semi-Markov decision processes
60G51 Processes with independent increments; Lévy processes
49J55 Existence of optimal solutions to problems involving randomness
93E20 Optimal stochastic control
91B30 Risk theory, insurance (MSC2010)

Citations:

Zbl 1079.91049
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References:

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