# zbMATH — the first resource for mathematics

Optimal investment and reinsurance for an insurer under Markov-modulated financial market. (English) Zbl 1394.91238
Summary: This study examines optimal investment and reinsurance policies for an insurer with the classical surplus process. It assumes that the financial market is driven by a drifted Brownian motion with coefficients modulated by an external Markov process specified by the solution to a stochastic differential equation. The goal of the insurer is to maximize the expected terminal utility. This paper derives the Hamilton-Jacobi-Bellman (HJB) equation associated with the control problem using a dynamic programming method. When the insurer admits an exponential utility function, we prove that there exists a unique and smooth solution to the HJB equation. We derive the explicit optimal investment policy by solving the HJB equation. We can also find that the optimal reinsurance policy optimizes a deterministic function. We also obtain the upper bound for ruin probability in finite time for the insurer when the insurer adopts optimal policies.

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 91G10 Portfolio theory 93E20 Optimal stochastic control
Full Text:
##### References:
 [1] Asmussen, S.; Albrecher, H., Ruin probabilities. vol. 14, (2010), World Scientific [2] Azcue, P.; Muler, N., Optimal investment strategy to minimize the ruin probability of an insurance company under borrowing constraints, Insurance Math. Econom., 44, 1, 26-34, (2009) · Zbl 1156.91391 [3] Badaoui, M.; Fernández, B., An optimal investment strategy with maximal risk aversion and its ruin probability in the presence of stochastic volatility on investments, Insurance Math. Econom., 53, 1, 1-13, (2013) · Zbl 1284.91509 [4] Bai, L.; Guo, J., Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance Math. Econom., 42, 3, 968-975, (2008) · Zbl 1147.93046 [5] Berge, C., Topological spaces: including a treatment of multi-valued functions, vector spaces, and convexity, (1963), Courier Corporation · Zbl 0114.38602 [6] Browne, S., Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin, Math. Oper. Res., 20, 4, 937-958, (1995) · Zbl 0846.90012 [7] Castañeda-Leyva, N.; Hernández-Hernández, D., Optimal consumption-investment problems in incomplete markets with stochastic coefficients, SIAM J. Control Optim., 44, 4, 1322-1344, (2005) · Zbl 1140.91381 [8] Fernández, B.; Hernández-Hernández, D.; Meda, A.; Saavedra, P., An optimal investment strategy with maximal risk aversion and its ruin probability, Math. Methods Oper. Res., 68, 1, 159-179, (2008) · Zbl 1175.60069 [9] Fleming, W. H.; Hernández-Hernández, D., The tradeoff between consumption and investment in incomplete financial markets, Appl. Math. Optim., 52, 2, 219-235, (2005) · Zbl 1128.91024 [10] Fleming, W. H.; Pang, T., A stochastic control model of investment, production and consumption, Quart. Appl. Math., 63, 1, 71-87, (2005) · Zbl 1080.91031 [11] Fleming, W. H.; Sheu, S., Risk-sensitive control and an optimal investment model, Math. Finance, 10, 2, 197-213, (2000) · Zbl 1039.93069 [12] French, K. R.; Schwert, G. W.; Stambaugh, R. F., Expected stock returns and volatility, J. Financ. Econ., 19, 1, 3-29, (1987) [13] Frolova, A.; Kabanov, Y.; Pergamenshchikov, S., In the insurance business risky investments are dangerous, Finance Stoch., 6, 2, 227-235, (2002) · Zbl 1002.91037 [14] Gaier, J.; Grandits, P.; Schachermayer, W., Asymptotic ruin probabilities and optimal investment, Ann. Appl. Probab., 13, 3, 1054-1076, (2003) · Zbl 1046.62113 [15] Grandell, J., Aspects of risk theory, (1991), Springer · Zbl 0717.62100 [16] Hipp, C.; Plum, M., Optimal investment for insurers, Insurance Math. Econom., 27, 2, 215-228, (2000) · Zbl 1007.91025 [17] Hipp, C.; Plum, M., Optimal investment for investors with state dependent income, and for insurers, Finance Stoch., 7, 3, 299-321, (2003) · Zbl 1069.91051 [18] Højgaard, B.; Taksar, M., Optimal proportional reinsurance policies for diffusion models with transaction costs, Insurance Math. Econom., 22, 1, 41-51, (1998) · Zbl 1093.91518 [19] Ikeda, N.; Watanabe, S., Stochastic differential equations and diffusion processes. vol. 24, (2014), Elsevier [20] Irgens, C.; Paulsen, J., Optimal control of risk exposure, reinsurance and investments for insurance portfolios, Insurance Math. Econom., 35, 1, 21-51, (2004) · Zbl 1052.62107 [21] Jean-Peirre, F.; Papanicolaou, G.; Sircar, K. R., Derivatives in financial markets with stochastic volatility, (2000), Cambridge University Press · Zbl 0954.91025 [22] Li, Z.; Zeng, Y.; Lai, Y., Optimal time-consistent investment and reinsurance strategies for insurers under hestons sv model, Insurance Math. Econom., 51, 1, 191-203, (2012) · Zbl 1284.91250 [23] Liang, Z., Optimal investment and reinsurance for the jump-diffusion surplus processes, Acta Math. Sinica (Chin. Ser.), 51, 6, 1195-1204, (2008) · Zbl 1174.62555 [24] Luo, S.; Taksar, M.; Tsoi, A., On reinsurance and investment for large insurance portfolios, Insurance Math. Econom., 42, 1, 434-444, (2008) · Zbl 1141.91532 [25] Meng, H.; Yuen, F. L.; Siu, T. K.; Yang, H., Optimal portfolio in a continuous-time self-exciting threshold model, J. Ind. Manag. Optim., 9, 2, 487-504, (2013) · Zbl 1274.91389 [26] Pham, H., Smooth solutions to optimal investment models with stochastic volatilities and portfolio constraints, Appl. Math. Optim., 46, 1, 55-78, (2002) · Zbl 1014.91038 [27] Rudin, W., (Principles of Mathematical Analysis, International Series in Pure & Applied Mathematics, (1976)) [28] Schmidli, H., Optimal proportional reinsurance policies in a dynamic setting, Scand. Actuar. J., 2001, 1, 55-68, (2001) · Zbl 0971.91039 [29] Schmidli, H., On minimizing the ruin probability by investment and reinsurance, Ann. Appl. Probab., 12, 3, 890-907, (2002) · Zbl 1021.60061 [30] Schmidli, H., Asymptotics of ruin probabilities for risk processes under optimal reinsurance and investment policies: the large claim case, Queueing Syst., 46, 1-2, 149-157, (2004) · Zbl 1056.90037 [31] Taksar, M. I.; Markussen, C., Optimal dynamic reinsurance policies for large insurance portfolios, Finance Stoch., 7, 1, 97-121, (2003) · Zbl 1066.91052 [32] Tankov, P., Financial modelling with jump processes. vol. 2, (2003), CRC Press [33] Xu, L.; Yang, H.; Wang, R., Cox risk model with variable premium rate and stochastic return on investment, J. Comput. Appl. Math., 256, 52-64, (2014) · Zbl 1314.91147 [34] Yang, H.; Zhang, L., Optimal investment for insurer with jump-diffusion risk process, Insurance Math. Econom., 37, 3, 615-634, (2005) · Zbl 1129.91020 [35] Zeng, Y.; Li, Z., Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance Math. Econom., 49, 1, 145-154, (2011) · Zbl 1218.91167 [36] Zhang, X.; Siu, T. K., Optimal investment and reinsurance of an insurer with model uncertainty, Insurance Math. Econom., 45, 1, 81-88, (2009) · Zbl 1231.91257
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.