Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-14T23:23:24.715Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal insurance control for insurers with jump-diffusion risk processes

Published online by Cambridge University Press:  16 July 2018

Linlin Tian  and
Lihua Bai
Linlin Tian
Affiliation:
School of Mathematical Sciences, Nankai University, Tianjin 300071, P. R. China
Lihua Bai*
Affiliation:
School of Mathematical Sciences, Nankai University, Tianjin 300071, P. R. China
*
*Correspondence to: Lihua Bai, School of Mathematical Sciences, Nankai University, Tianjin 300071, P. R. China. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we model the surplus process as a compound Poisson process perturbed by diffusion and allow the insurer to ask its customers for input to minimize the distance from some prescribed target path and the total discounted cost on a fixed interval. The problem is reduced to a version of a linear quadratic regulator under jump-diffusion processes. It is treated using three methods: dynamic programming, completion of square and the stochastic maximum principle. The analytic solutions to the optimal control and the corresponding optimal value function are obtained.

Keywords

Compound Poisson processDiffusionHamilton-Jacobi-Bellman equationCompletion of squareStochastic maximum principle93E2091B30
Type
Paper
Information
Annals of Actuarial Science , Volume 13 , Issue 1 , March 2019 , pp. 198 - 213
Copyright
© Institute and Faculty of Actuaries 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Asmussen, S. & Taksar, M. (1997). Controlled diffusion models for optimal dividend pay-out. Insurance: Mathematics and Economics, 20, 1–15.Google Scholar
Asmussen, S., Hϕjgaard, B. & Taksar, M. (2000). Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation. Finance and Stochastic, 4, 299–324.Google Scholar
Bai, L.H. & Guo, J.Y. (2008). Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint. Insurance Mathematics and Economics, 42(3), 968–975.Google Scholar
Bai, L.H. & Zhang, H.Y. (2008). Dynamic Mean-Variance Problem with Constrained Risk Control for the Insurers. Mathematical Methods of Operations Research, 68(1), 181–205.Google Scholar
Brémaud, P. (1981). Point Processes and Queues. Springer-Verlag, New York.Google Scholar
Browne, S. (1995). Optimal investment policies for a firm with random risk process: exponential utility and minimizing the probability of ruin. Mathematics of Operations Research, 20(4), 937–958.Google Scholar
Choulli, T., Taksar, M. & Zhou, X.Y. (2003). Optimal dividend distribution and risk control. SIAM Journal on Control and Optimization, 41(6), 1946–1979.Google Scholar
Fleming, W.H. & Soner, H.M. (1993). Controlled Markov Processes and Viscosity Solutions. Springer, Berlin, New York.Google Scholar
Framstad, N.C., Øksendal, B. & Sulem, A. (2004). Sufficient stochastic maximum principle for optimal control of jump diffusions and applications to finance. Journal of Optimization Theory and Applications, 121(1), 77–98.Google Scholar
Frangos, N.E., Vrontos, S. & Yannacopoulos, A.N. (2008). Insurance control for a simple model with liabilities of the fractional Brownian motion type. Preprint.Google Scholar
Hipp, C. & Taksar, M. (2000). Stochastic control for optimal new business. Insurance: Mathematics and Economics, 26, 185–192.Google Scholar
Hipp, C. & Plum, M. (2000). Optimal investment for insurers. Insurance: Mathematics and Economics, 27, 215–228.Google Scholar
Hipp, C. & Vogt, M. (2003). Optimal dynamic XL reinsurance. ASTIN Bulletin, 33(2), 93–207.Google Scholar
Hϕjgaard, B. & Taksar, M. (1999). Controlling risk exposure and dividends pay-out schemes: Insurance company example. Mathematical Finance, 9(2), 153–182.Google Scholar
Hϕjgaard, B. & Taksar, M. (2001). Optimal risk control for a large corporation in the presence of returns on investments. Finance and Stochastics, 5, 527–547.Google Scholar
Hϕjgaard, B. & Taksar, M. (2004). Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy. Quantitative Finance, 4, 315–327.Google Scholar
Paulsen, J. & Gjessing, H.K. (1997). Optimal choice of dividend barriers for a risk process with stochastic return on investments. Insurance: Mathematics and Economics, 20, 215–223.Google Scholar
Paulsen, J. (2003). Optimal dividend payouts for diffusions with solvency constraints. Finance and Stochastics, 4, 457–474.Google Scholar
Promislow, S.D. & Young, V.R. (2005). Minimizing the Probability of ruin when Claims follow Brownian Motion with Drift. North American Actuarial Journal, 9(3), 109–128.Google Scholar
Schmidli, H. (2001). Optimal Proportional Reinsurance Policies in a Dynamic Setting. Scandinavaian Actuarial Journal, 1, 55–68.Google Scholar
Schmidli, H. (2002). On minimizing the ruin probability by investment and reinsurance. The Annals of Applied Probability, 12(3), 890–907.Google Scholar
Taksar, M. & Markussen, C. (2003). Optimal dynamic reinsurance policies for large insurance portfolios. Finance and Stochastic, 7, 97–121.Google Scholar
Tang, S.J. & Li, X.J. (1994). Necessary conditions for optimal control of stochastic systems with random jumps. SIAM Journal on Control and Optimization, 32(5), 1447–1475.Google Scholar
Yang, H. & Zhang, L. (2005). Optimal investment for insurer with jump-diffusion risk process. Insurance: Mathematics and Economics, 37, 615–634.Google Scholar
Yong, J.M. & Zhou, X.Y. (1999). Stochastic Controls. Springer-Verlag, New York.Google Scholar