A numerical method for the expected penalty–reward function in a Markov-modulated jump–diffusion process
AbstractA 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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Bibliographic InfoPaper provided by Universidad Carlos III de Madrid in its series Open Access publications from Universidad Carlos III de Madrid with number info:hdl:10016/12757.
Length: 134 p.
Date of creation: Jul 2011
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Publication status: Published in Insurance, Mathematics & Economics (2011-07) v.v. 49, p.126-132
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Expected penalty–reward function; Markov-modulated process; Jump–diffusion process; Volterra integro-differential system of equations;
Find related papers by JEL classification:
- G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies
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