Publication: A numerical method for the expected penalty–reward function in a Markov-modulated jump–diffusion process
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2011-07
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Elsevier
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Abstract
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
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Keywords
Expected penalty–reward function, Markov-modulated process, Jump–diffusion process, Volterra integro-differential system of equations
Bibliographic citation
Insurance, Mathematics & Economics, 2011, v. 49, n. 1, pp. 126-132