Optimal reinsurance and investment with unobservable claim size and intensity

We consider the optimal reinsurance and investment problem in an unobservable Markov-modulated compound Poisson risk model, where the intensity and jump size distribution are not known but have to be inferred from the observations of claim arrivals. Using a recently developed result from filtering theory, we reduce the partially observable control problem to an equivalent problem with complete observations. Then using stochastic control theory, we get the closed form expressions of the optimal strategies which maximize the expected exponential utility of terminal wealth. In particular, we investigate the effect of the safety loading and the unobservable factors on the optimal reinsurance strategies. With the help of a generalized Hamilton–Jacobi–Bellman equation where the derivative is replaced by Clarke’s generalized gradient as in Bauerle and Rieder (2007), we characterize the value function, which helps us verify that the strategies we constructed are optimal.

[1]  S. Haberman,et al.  An Introduction to Mathematical Risk Theory . By Hans U. Gerber [S. S. Huebner Foundation, R. D. Irwin Inc. Homeward Illinois, 1979] , 1981 .

[2]  Robert J. Elliott,et al.  Filtering a Markov Modulated Random Measure , 2010, IEEE Transactions on Automatic Control.

[3]  Utility maximization with partial information: Hamilton-Jacobi-Bellman equation approach , 2007 .

[4]  H. Vincent Poor,et al.  Optimal time to change premiums , 2008, Math. Methods Oper. Res..

[5]  Hailiang Yang,et al.  Optimal investment for insurer with jump-diffusion risk process , 2005 .

[6]  J. Paulsen,et al.  Optimal control of risk exposure, reinsurance and investments for insurance portfolios , 2004 .

[7]  Ulrich G. Haussmann,et al.  Optimizing the terminal wealth under partial information: The drift process as a continuous time Markov chain , 2004, Finance Stochastics.

[8]  Mengdi Gu,et al.  Constant elasticity of variance model for proportional reinsurance and investment strategies , 2010 .

[9]  Hans U. Gerber,et al.  An introduction to mathematical risk theory , 1982 .

[10]  Tak Kuen Siu,et al.  An HMM approach for optimal investment of an insurer , 2012 .

[11]  Jin Ma,et al.  Optimal reinsurance/investment problems for general insurance models. , 2009, 0908.4538.

[12]  Virginia R. Young,et al.  Minimizing the Probability of Ruin When Claims Follow Brownian Motion with Drift , 2005 .

[13]  T. Siu,et al.  On optimal proportional reinsurance and investment in a Markovian regime-switching economy , 2012 .

[14]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[15]  Erhan Bayraktar,et al.  Sequential tracking of a hidden Markov chain using point process observations , 2007 .

[16]  Optimal Proportional Reinsurance for Controlled Risk Process which is Perturbed by Diffusion , 2007 .

[17]  P. Lakner.  Optimal trading strategy for an investor: the case of partial information , 1998 .

[18]  Junyi Guo,et al.  Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint , 2008 .

[19]  C. Hipp,et al.  Optimal investment for insurers , 2000 .

[20]  Walter Schachermayer,et al.  Asymptotic ruin probabilities and optimal investment , 2003 .

[21]  U. Rieder,et al.  Portfolio optimization with unobservable Markov-modulated drift process , 2005, Journal of Applied Probability.

[22]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[23]  A. Müller,et al.  Comparison Methods for Stochastic Models and Risks , 2002 .

[24]  Shangzhen Luo,et al.  On reinsurance and investment for large insurance portfolios , 2008 .

[25]  Hanspeter Schmidli,et al.  Optimal Proportional Reinsurance Policies in a Dynamic Setting , 2001 .

[26]  Sid Browne,et al.  Optimal Investment Policies for a Firm With a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin , 1995, Math. Oper. Res..

[27]  Junyi Guo,et al.  Optimal investment and proportional reinsurance with constrained control variables , 2011 .

[28]  Christian Hipp,et al.  Asymptotics of ruin probabilities for controlled risk processes in the small claims case , 2004 .

[29]  Junyi Guo,et al.  Upper bound for ruin probabilities under optimal investment and proportional reinsurance , 2008 .

[30]  U. Rieder,et al.  PORTFOLIO OPTIMIZATION WITH JUMPS AND UNOBSERVABLE INTENSITY PROCESS , 2007 .

[31]  H. Schmidli,et al.  On minimizing the ruin probability by investment and reinsurance , 2002 .

[32]  Erhan Bayraktar,et al.  Minimizing the probability of lifetime ruin under borrowing constraints , 2007 .

[33]  Peter Lakner,et al.  Utility maximization with partial information , 1995 .

[34]  Junyi Guo,et al.  Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process , 2011 .

[35]  D. S. Mitrinovic,et al.  Classical and New Inequalities in Analysis , 1992 .