Optimal investment and proportional reinsurance with constrained control variables. (English) Zbl 1237.91133
Summary: In this paper, under the criterion of maximizing the expected exponential utility from terminal wealth, we study the optimal investment and proportional reinsurance strategy for an insurance company. The closed-form expressions for the optimal strategy and value function are derived not only for the compound Poisson risk model but also for the Brownian motion model. We can see that, with normal constraints on the control variables, the value function is still a classical solution to the corresponding Hamilton-Jacobi-Bellman equation.
Keywords:
Hamilton-Jacobi-Bellman equation; compound Poisson process; Brownian motion; exponential utility; proportional reinsurance; investmentReferences:
| [1] | Browne, Optimal investment policies for a firm with random risk process: exponential utility and minimizing the probability of ruin, Mathematics of Operations Research 20 pp 937– (1995) · Zbl 0846.90012 · doi:10.1287/moor.20.4.937 |
| [2] | Hipp, Optimal investment for insurers, Insurance: Mathematics and Economics 27 pp 215– (2000) · Zbl 1007.91025 · doi:10.1016/S0167-6687(00)00049-4 |
| [3] | Hipp, Optimal investment for investors with state dependent income, and for insurers, Finance and Stochastics 7 pp 299– (2003) · Zbl 1069.91051 · doi:10.1007/s007800200095 |
| [4] | Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal 1 pp 55– (2001) · Zbl 0971.91039 · doi:10.1080/034612301750077338 |
| [5] | Schmidli, On minimizing the ruin probability by investment and reinsurance, Annals of Applied Probability 12 pp 890– (2002) · Zbl 1021.60061 · doi:10.1214/aoap/1031863173 |
| [6] | Liu, Optimal investment for a insurer to minimize its probability of ruin, North American Acruarial Journal 8 (2) pp 11– (2004) · Zbl 1085.60511 · doi:10.1080/10920277.2004.10596134 |
| [7] | Yang, Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics 37 pp 615– (2005) · Zbl 1129.91020 · doi:10.1016/j.insmatheco.2005.06.009 |
| [8] | Liang, Optimal proportional reinsurance for controlled risk process which is perturbed by diffusion, Acta Mathematicae Applicatae Sinica, English Series 23 (3) pp 477– (2007) · Zbl 1127.62100 · doi:10.1007/s10255-007-0387-y |
| [9] | Irgens, Optimal control of risk exposure, reinsurance and investments for insurance portfolios, Insurance: Mathematics and Economics 35 (1) pp 21– (2004) · Zbl 1052.62107 · doi:10.1016/j.insmatheco.2004.04.004 |
| [10] | Bai, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance: Mathematics and Economics 42 (3) pp 968– (2008) · Zbl 1147.93046 · doi:10.1016/j.insmatheco.2007.11.002 |
| [11] | Cao, Optimal proportional reinsurance and investment based on Hamilton-Jacobi-Bellman equation, Insurance: Mathematics and Economics 45 pp 157– (2009) · Zbl 1231.91150 · doi:10.1016/j.insmatheco.2009.05.006 |
| [12] | Hipp, Asymptotics of ruin probabilities for controlled risk processes in the small claims case, Scandinavian Actuarial Journal 5 pp 321– (2004) · Zbl 1087.62116 · doi:10.1080/03461230410000538 |
| [13] | Gaier, Asymptotic ruin probabilities and optimal investment, Annals of Applied Probability 13 pp 1054– (2003) · Zbl 1046.62113 · doi:10.1214/aoap/1060202834 |
| [14] | Waters, Excess of loss reinsurance limits, Scandinavian Actuarial Journal 1 pp 37– (1979) · Zbl 0399.62106 · doi:10.1080/03461238.1979.10413708 |
| [15] | Centeno M, Excess of loss reinsurance and Gerber’s inequality in the Sparre Anderson model, Insurance: Mathematics and Economics 31 pp 415– (2002) · Zbl 1074.91567 · doi:10.1016/S0167-6687(02)00187-7 |
| [16] | Liang, Optimal proportional reinsurance and ruin probability, Stochastic Models 23 (2) pp 333– (2007) · Zbl 1253.60092 · doi:10.1080/15326340701300894 |
| [17] | Liang, Upper bound for ruin probabilities under optimal investment and proportional reinsurance, Applied Stochastic Models in Business and Industry 24 (2) pp 109– (2008) · Zbl 1199.91088 · doi:10.1002/asmb.694 |
| [18] | Moore, Optimal insurance in a continuous-time model, Insurance: Mathematics and Economics 39 (1) pp 47– (2006) · Zbl 1147.91342 · doi:10.1016/j.insmatheco.2006.01.009 |
| [19] | Zhang, Optimal proportional reinsurance and investment with transaction costs, I: maximizing the terminal wealth, Insurance: Mathematics and Economics 44 pp 473– (2009) · Zbl 1162.91446 · doi:10.1016/j.insmatheco.2009.01.004 |
| [20] | Asmussen, Ruin Probabilities (2000) · doi:10.1142/9789812779311 |
| [21] | Gerber H An introduction to mathematical risk theory 1979 |
| [22] | Grandell, Aspects of Risk Theory (1991) · doi:10.1007/978-1-4613-9058-9 |
| [23] | Merton, Optimum consumption and protfolio rules in a continuous time model, Journal of Economic Theory 3 pp 373– (1971) · Zbl 1011.91502 · doi:10.1016/0022-0531(71)90038-X |
| [24] | Karatzas, Optimization problems in the theory of continuous trading, SIAM Journal on Control and Optimization 7 pp 1221– (1989) · Zbl 0701.90008 · doi:10.1137/0327063 |
| [25] | Fleming, Controlled Markov Processes and Viscosity Solutions (1993) · Zbl 0773.60070 |
| [26] | Cont, Chapman and Hall/CRC Financial Mathematics Series, in: Financial Modeling with Jump Processes (2003) · Zbl 1052.91043 · doi:10.1201/9780203485217 |
| [27] | Rolski, Stochastic Processes for Insurance and Finance (1999) · doi:10.1002/9780470317044 |
| [28] | Hald, On the maximization of the adjustment coefficient under proportioal reinsurance, ASTIN Bulletin 34 pp 75– (2004) · Zbl 1095.91033 · doi:10.2143/AST.34.1.504955 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
