# Optimal investment problem for an insurer and a reinsurer

- 179 Downloads
- 2 Citations

## Abstract

This paper studies the optimal investment problem for an insurer and a reinsurer. The basic claim process is assumed to follow a Brownian motion with drift and the insurer can purchase proportional reinsurance from the reinsurer. The insurer and the reinsurer are allowed to invest in a risk-free asset and a risky asset. Moreover, the authors consider the correlation between the claim process and the price process of the risky asset. The authors first study the optimization problem of maximizing the expected exponential utility of terminal wealth for the insurer. Then with the optimal reinsurance strategy chosen by the insurer, the authors consider two optimization problems for the reinsurer: The problem of maximizing the expected exponential utility of terminal wealth and the problem of minimizing the ruin probability. By solving the corresponding Hamilton-Jacobi-Bellman equations, the authors derive the optimal reinsurance and investment strategies, explicitly. Finally, the authors illustrate the equality of the reinsurer’s optimal investment strategies under the two cases.

## Keywords

Hamilton-Jacobi-Bellman equation optimal reinsurance and investment strategies proportional reinsurance ruin probability utility maximization## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Hipp C and Plum M, Optimal investment for insurers,
*Insurance: Mathematics and Economics*, 2000,**27**(2): 215–4.zbMATHMathSciNetGoogle Scholar - [2]Schmidli H, Optimal proportional reinsurance policies in a dynamic setting,
*Scandinavian Actuarial Journal*, 2001,**2001**(1): 55–4.zbMATHMathSciNetCrossRefGoogle Scholar - [3]Promislow S D and Young V R, Minimizing the probability of ruin when claims follow Brownian motion with drift,
*North American Actuarial Journal*, 2005,**9**(3): 110–4.MathSciNetCrossRefGoogle Scholar - [4]Li J and Wu R, Upper bound for finite-time ruin probability in a Markov-modulated market,
*Journal of Systems Science and Complexity*, 2011,**24**(2): 308–4.zbMATHMathSciNetCrossRefGoogle Scholar - [5]Liang Z and Guo J, Optimal investment and proportional reinsurance in the Sparre Andersen model,
*Journal of Systems Science and Complexity*, 2012,**25**(5): 926–4.zbMATHMathSciNetCrossRefGoogle Scholar - [6]Yang H and Zhang L, Optimal investment for insurer with jump-diffusion risk process,
*Insurance: Mathematics and Economics*, 2005,**37**(3): 615–4.zbMATHMathSciNetGoogle Scholar - [7]Wang N, Optimal investment for an insurer with exponential utility preference,
*Insurance: Mathematics and Economics*, 2007,**40**(1): 77–4.zbMATHMathSciNetGoogle Scholar - [8]Cao Y and Wan N,
*Optimal proportional reinsurance and investment based on Hamilton-Jacobi- Bellman equation*, Insurance: Mathematics Economics, 2009,**45**(2): 157–4.zbMATHMathSciNetGoogle Scholar - [9]Bai L and Guo J, Optimal dynamic excess-of-loss reinsurance and multidimensional portfolio selection,
*Science China Mathematics*, 2010,**53**(7): 1787–4.zbMATHMathSciNetCrossRefGoogle Scholar - [10]Liang Z, Yuen K C, and Guo J, Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process,
*Insurance: Mathematics and Economics*, 2011,**49**(2): 207–4.zbMATHMathSciNetGoogle Scholar - [11]Liang Z and Bayraktar E,
*Optimal reinsurance and investment with unobservable claim size and intensity*, Insurance: Mathematics and Economics, 2014,**55**: 156–3.zbMATHMathSciNetGoogle Scholar - [12]Bai L and Zhang H, Dynamic mean-variance problem with constrained risk control for the insurers,
*Mathematical Methods of Operations Research*, 2008,**68**(1): 181–4.zbMATHMathSciNetCrossRefGoogle Scholar - [13]Bi J, Guo J, and Bai L, Optimal multi-asset investment with no-shorting constraint under meanvariance criterion for an insurer,
*Journal of Systems Science and Complexity*, 2011,**24**(2): 291–4.zbMATHMathSciNetCrossRefGoogle Scholar - [14]Zeng Y and Li Z, Optimal time-consistent investment and reinsurance policies for mean-variance insurers,
*Insurance: Mathematics and Economics*, 2011,**49**(1): 145–4.zbMATHMathSciNetGoogle Scholar - [15]Li Y and Li Z, Optimal time-consistent investment and reinsurance strategies for mean-variance insurers with state dependent risk aversion,
*Insurance: Mathematics and Economics*, 2013,**53**(1): 86–4.zbMATHMathSciNetGoogle Scholar - [16]Gu M, Yang Y, Li S, and Zhang J, Constant elasticity of variance model for proportional reinsurance and investment strategies,
*Insurance: Mathematics and Economics*, 2010,**46**(3): 580–4.zbMATHMathSciNetGoogle Scholar - [17]Lin X and Li Y, Optimal reinsurance and investment for a jump diffusion risk process under the CEV model,
*North American Actuarial Journal*, 2011,**15**(3): 417–4.zbMATHMathSciNetCrossRefGoogle Scholar - [18]Liang Z, Yuen K C, and Cheung K C, Optimal reinsurance-investment problem in a constant elasticity of variance stock market for jump-diffusion risk model,
*Applied Stochastic Models in Business and Industry*, 2012,**28**(6): 585–4.zbMATHMathSciNetCrossRefGoogle Scholar - [19]Gu A, Guo X, Li Z, and Zeng Y, Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model,
*Insurance: Mathematics and Economics*, 2012,**51**(3): 674–4.zbMATHMathSciNetGoogle Scholar - [20]Li Z, Zeng Y, and Lai Y, Optimal time-consistent investment and reinsurance strategies for insurers under Heston’s SV model,
*Insurance: Mathematics and Economics*, 2012,**51**(1): 191–4.zbMATHMathSciNetGoogle Scholar - [21]Zhao H, Rong X, and Zhao Y, Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model,
*Insurance: Mathematics and Economics*, 2013,**53**(3): 504–4.zbMATHMathSciNetGoogle Scholar - [22]Browne S, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin,
*Mathematics of Operations Research*, 1995,**20**(4): 937–4.zbMATHMathSciNetCrossRefGoogle Scholar - [23]Bai L and Guo J, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint,
*Insurance: Mathematics and Economics*, 2008,**42**(3): 968–4.zbMATHMathSciNetGoogle Scholar