A numerical method for the expected penalty-reward function in a Markov-modulated jump-diffusion process.

*(English)*Zbl 1218.91075Summary: A generalization of the Cramér-Lundberg risk model perturbed by a diffusion is proposed. Aggregate claims of an insurer follow a compound Poisson process and premiums are collected at a constant rate with additional random fluctuation. The insurer is allowed to invest the surplus into a risky asset with volatility dependent on the level of the investment, which permits the incorporation of rational investment strategies as proposed by Berk and Green (2004). The return on investment is modulated by a Markov process which generalizes previously studied settings for the evolution of the interest rate in time. The Gerber-Shiu expected penalty-reward function is studied in this context, including ruin probabilities (a first-passage problem) as a special case. The second order integro-differential system of equations that characterizes the function of interest is obtained. As a closed-form solution does not exist, a numerical procedure based on the Chebyshev polynomial approximation through a collocation method is proposed. Finally, some examples illustrating the procedure are presented.

##### MSC:

91B30 | Risk theory, insurance (MSC2010) |

60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |

60K10 | Applications of renewal theory (reliability, demand theory, etc.) |

##### Keywords:

expected penalty-reward function; Markov-modulated process; jump-diffusion process; Volterra integro-differential system of equations
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\textit{P. Diko} and \textit{M. Usábel}, Insur. Math. Econ. 49, No. 1, 126--131 (2011; Zbl 1218.91075)

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