A numerical method for the expected penalty-reward function in a Markov-modulated jump-diffusion process. (English) Zbl 1218.91075

Summary: 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.


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.)
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[1] Akyuz-Dascioglu, A., A Chebyshev polynomial approach for linear Fredholm-Volterra integro-differential equations in the most general form, Applied Mathematics and Computation, 181, 1, 103-112 (2007) · Zbl 1148.65318
[2] Akyuz-Dascioglu, A.; Sezer, M., Chebyshev polynomial solutions of systems of higher-order linear Fredholm-Volterra integro-differential equations, Journal of the Franklin Institute, 342, 6, 688-701 (2005) · Zbl 1086.65121
[3] Asmussen, S.; Albrecher, H., Ruin Probabilities (2010), World Scientific: World Scientific Singapore · Zbl 1247.91080
[4] Avram, F.; Usabel, M., The Gerber-Shiu expected discounted penalty-reward function under an affine jump-diffusion model, Astin Bulletin, 38, 2, 461-481 (2008) · Zbl 1256.91025
[5] Berk, J.; Green, R., Mutual fund flows and performance in rational markets, Journal of Political Economy, 112, 6, 1269-1295 (2004)
[6] Boyd, J., Chebyshev and Fourier Spectral Methods (2001), Dover: Dover New York · Zbl 0994.65128
[7] Cai, J.; Yang, H., Ruin in the perturbed compound Poisson risk process under interest force, Advances in Applied Probability, 37, 3, 819-835 (2005) · Zbl 1074.60090
[8] Clenshaw, C.; Curtis, A., A method for numerical integration on an automatic computer, Numerische Mathematik, 2, 1, 197-205 (1960) · Zbl 0093.14006
[9] Dufresne, F.; Gerber, H., Risk theory for the compound Poisson process that is perturbed by diffusion, Insurance Mathematics and Economics, 10, 1, 51-59 (1991) · Zbl 0723.62065
[10] Gaier, J.; Grandits, P., Ruin probabilities and investment under interest force in the presence of regularly varying tails, Scandinavian Actuarial Journal, 2004, 4, 256-278 (2004) · Zbl 1091.62102
[11] Gerber, H., An extension of the renewal equation and its application in the collective theory of risk, Skandinavisk Aktuarietidskrift, 53, 205-210 (1970) · Zbl 0229.60062
[12] Gerber, H.; Landry, B., On the discounted penalty at ruin in a jump-diffusion and the perpetual put option, Insurance Mathematics and Economics, 22, 3, 263-276 (1998) · Zbl 0924.60075
[13] Gerber, H.; Shiu, E., The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin, Insurance: Mathematics and Economics, 21, 2, 129-137 (1997) · Zbl 0894.90047
[14] Gerber, H.; Shiu, E., On the time value of ruin, North American Actuarial Journal, 2, 48-71 (1998)
[15] Grandits, P., Minimal ruin probabilities and investment under interest force for a class of subexponential distributions, Scandinavian Actuarial Journal, 2005, 6, 401-416 (2005) · Zbl 1142.91042
[16] Kushner, H. J.J.; Dupuis, P., Numerical Methods for Stochastic Control Problems in Continuous Time (1992), Springer-Verlag: Springer-Verlag New York · Zbl 0754.65068
[17] Le, U.; Pascali, E., Existence theorems for systems of nonlinear integro-differential equations, Ricerche di matematica, 58, 1, 91-101 (2009) · Zbl 1185.45010
[18] Li, S.; Garrido, J., Ruin probabilities for two classes of risk processes, ASTIN Bulletin, 35, 1, 61-77 (2005) · Zbl 1098.62139
[19] Ma, J.; Sun, X., Ruin probabilities for insurance models involving investments, Scandinavian actuarial journal, 2003, 3, 217-237 (2003) · Zbl 1039.91045
[20] Morales, M., On the expected discounted penalty function for a perturbed risk process driven by a subordinator, Insurance Mathematics and Economics, 40, 2, 293-301 (2007) · Zbl 1130.91032
[21] Paulsen, J., Risk theory in a stochastic economic environment, Stochastic Processes and their Applications, 46, 2, 327-361 (1993) · Zbl 0777.62098
[22] Paulsen, J.; Gjessing, H., Optimal choice of dividend barriers for a risk process with stochastic return on investments, Insurance Mathematics and Economics, 20, 3, 215-223 (1997) · Zbl 0894.90048
[23] Ren, J., The expected value of the time of ruin and the moments of the discounted deficit at ruin in the perturbed classical risk process, Insurance Mathematics and Economics, 37, 3, 505-521 (2005) · Zbl 1129.91027
[24] Risken, H., The Fokker-Planck Equation: Methods of Solution and Applications (1996), Springer Verlag: Springer Verlag Berlin · Zbl 0866.60071
[25] Sarkar, J.; Sen, A., Weak convergence approach to compound Poisson risk processes perturbed by diffusion, Insurance Mathematics and Economics, 36, 3, 421-432 (2005) · Zbl 1242.91097
[26] Sezer, M.; Kaynak, M., Chebyshev polynomial solutions of linear differential equations, International Journal of Mathematical Education in Science and Technology, 27, 4, 607-618 (1996) · Zbl 0887.34012
[27] Usabel, M., Calculating multivariate ruin probabilities via Gaver-Stehfest inversion technique, Insurance: Mathematics and Economics, 25, 2, 133-142 (1999) · Zbl 1028.91561
[28] Wang, G., A decomposition of the ruin probability for the risk process perturbed by diffusion, Insurance Mathematics and Economics, 28, 1, 49-59 (2001) · Zbl 0993.60087
[29] Wang, G.; Wu, R., The expected discounted penalty function for the perturbed compound Poisson risk process with constant interest, Insurance: Mathematics and Economics, 42, 1, 59-64 (2008) · Zbl 1141.91551
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