Optimal Proportional Reinsurance in a Bivariate Risk Model

ABSTRACT

The paper deals with the optimal proportional reinsurance in a collective risk theory model involving two classes of insurance business. These classes are dependent through the number of claims. The objective of the insurer is to choose an optimal reinsurance strategy that maximizes the expected exponential utility of terminal wealth. We are able to derive the evolution of the insurer surplus process under the assumption that the number of claims of the two classes of the insurance business has a Poisson bivariate distribution. We face the problem of finding the optimal strategy using the dynamic programming approach. Therefore, we determine the infinitesimal generator for the surplus process and for the value function, and we give the Hamilton Jacobi Bellmann (HJB) equation. Under particular assumptions, we obtain explicit form of the optimal reinsurance strategy on correspondent value function.

The paper deals with the optimal proportional reinsurance in a collective risk theory model involving two classes of insurance business. These classes are dependent through the number of claims. The objective of the insurer is to choose an optimal reinsurance strategy that maximizes the expected exponential utility of terminal wealth. We are able to derive the evolution of the insurer surplus process under the assumption that the number of claims of the two classes of the insurance business has a Poisson bivariate distribution. We face the problem of finding the optimal strategy using the dynamic programming approach. Therefore, we determine the infinitesimal generator for the surplus process and for the value function, and we give the Hamilton Jacobi Bellmann (HJB) equation. Under particular assumptions, we obtain explicit form of the optimal reinsurance strategy on correspondent value function.

Cite this paper

Gosio, C. , Lari, E. and Ravera, M. (2015) Optimal Proportional Reinsurance in a Bivariate Risk Model.*Modern Economy*, **6**, 664-671. doi: 10.4236/me.2015.66062.

Gosio, C. , Lari, E. and Ravera, M. (2015) Optimal Proportional Reinsurance in a Bivariate Risk Model.

References

[1] Ambagaspitiya, R.S. (1998) Compound Bivariate Lagrange Poisson Distributions. Insurance: Mathematics and Economics, 23, 21-31. http://dx.doi.org/10.1016/S0167-6687(98)00020-1

[2] Ambagaspitiya, R.S. (1998) On the Distribution of a Sum of Correlated Aggregate Claims. Insurance: Mathematics and Economics, 23, 15-19. http://dx.doi.org/10.1016/s0167-6687(98)00018-3

[3] Centeno, M.L. (2005) Dependent Risks and Excess of Loss Reinsurance. Insurance: Mathematics and Economics, 37, 229-238. http://dx.doi.org/10.1016/j.insmatheco.2004.12.001

[4] Gosio, C., Lari, E.C. and Ravera, M. (2013) Optimal Expected Utility of the Wealth for Two Dependent Classes of Insurance Business. Theoretical Economics Letters, 3, 90-95. http://dx.doi.org/10.4236/tel.2013.32015

[5] Yuen, K.C., Guo, J. and Wu, X. (2002) On Correlated Aggregate Claims Model with Poisson and Erlang Risk Processes. Insurance: Mathematics and Economics, 31, 205-214. http://dx.doi.org/10.1016/s0167-6687(02)00150-6

[6] Yuen, K.C., Guo, J. and Wu, X. (2006) On the First Time of Ruine in the Bivariate Compound Poisson Model. Insurance: Mathematics and Economics, 38, 298-308. http://dx.doi.org/10.1016/j.insmatheco.2005.08.011

[7] Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J. (1986) Actuarial Mathematics. Society of Actuaries, Itaxa, Illinois.

[8] Johnson, N.L. and Balakrishnan, N. (1997) Discrete Multivariate Distributions. Wiley & Sons, Hoboken.

[9] Lin, X. and Li, Y. (2011) Optimal Reinsurance and Investment for a Jump Diffusion Risk Process under the CEV Model. North American Actuarial Journal, 15, 417-431. http://dx.doi.org/10.1080/10920277.2011.10597628

[10] Pham, H. (2010) Continuous-Time Stochastic Control and Optimization Problem with Financial Applications. Springer Verlag, Berlin.

[11] Cetin, U. (2014) An Introduction to Markov Processes and Their Applications in Mathematical Economics. Lecture Notes, Department of Statistic, London School of Economics and Political Science, London.

[12] Hipp, C. (2004) Stochastic Control with Applications in Insurance, Stochastic Methods in Finance. Lecture Notes in Mathematics, 1856, 127-164.

[1] Ambagaspitiya, R.S. (1998) Compound Bivariate Lagrange Poisson Distributions. Insurance: Mathematics and Economics, 23, 21-31. http://dx.doi.org/10.1016/S0167-6687(98)00020-1

[2] Ambagaspitiya, R.S. (1998) On the Distribution of a Sum of Correlated Aggregate Claims. Insurance: Mathematics and Economics, 23, 15-19. http://dx.doi.org/10.1016/s0167-6687(98)00018-3

[3] Centeno, M.L. (2005) Dependent Risks and Excess of Loss Reinsurance. Insurance: Mathematics and Economics, 37, 229-238. http://dx.doi.org/10.1016/j.insmatheco.2004.12.001

[4] Gosio, C., Lari, E.C. and Ravera, M. (2013) Optimal Expected Utility of the Wealth for Two Dependent Classes of Insurance Business. Theoretical Economics Letters, 3, 90-95. http://dx.doi.org/10.4236/tel.2013.32015

[5] Yuen, K.C., Guo, J. and Wu, X. (2002) On Correlated Aggregate Claims Model with Poisson and Erlang Risk Processes. Insurance: Mathematics and Economics, 31, 205-214. http://dx.doi.org/10.1016/s0167-6687(02)00150-6

[6] Yuen, K.C., Guo, J. and Wu, X. (2006) On the First Time of Ruine in the Bivariate Compound Poisson Model. Insurance: Mathematics and Economics, 38, 298-308. http://dx.doi.org/10.1016/j.insmatheco.2005.08.011

[7] Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J. (1986) Actuarial Mathematics. Society of Actuaries, Itaxa, Illinois.

[8] Johnson, N.L. and Balakrishnan, N. (1997) Discrete Multivariate Distributions. Wiley & Sons, Hoboken.

[9] Lin, X. and Li, Y. (2011) Optimal Reinsurance and Investment for a Jump Diffusion Risk Process under the CEV Model. North American Actuarial Journal, 15, 417-431. http://dx.doi.org/10.1080/10920277.2011.10597628

[10] Pham, H. (2010) Continuous-Time Stochastic Control and Optimization Problem with Financial Applications. Springer Verlag, Berlin.

[11] Cetin, U. (2014) An Introduction to Markov Processes and Their Applications in Mathematical Economics. Lecture Notes, Department of Statistic, London School of Economics and Political Science, London.

[12] Hipp, C. (2004) Stochastic Control with Applications in Insurance, Stochastic Methods in Finance. Lecture Notes in Mathematics, 1856, 127-164.