Calcilating optimal dividend payment, reinsurance, and investment strategies in a~diffusion model

D. B. Rokhlin ; G. V. Mironenko

Sibirskij žurnal industrialʹnoj matematiki, Tome 18 (2015) no. 1, pp. 110-122.

Résumé

We consider the problem of the maximization of the total expected discounted amount of dividends paid by an insurance company up to the bankruptcy. It is assumed that the reinsurance is allowed and the wealth can be invested in a risky asset whose dynamics is described by the Black–Scholes model with random drift obeying the Ornstein–Uhlenbeck process. In accordance with the general scheme of dynamic programming, the problem is reduced to solving a Dirichlet problem for the corresponding Hamilton–Jacobi–Bellman equation in the half-plane. The problem is solved numerically by means of a monotone finite-difference scheme, which is proved to converge to a unique viscosity solution of the mentioned equation. We present and discuss the results of numerical experiments which indicate some nontrivial properties of optimal strategies.

Détail

Keywords: Hamilton–Jacobi–Bellman equation, viscosity solution, monotone scheme, dividend, reinsurance, investment, random factor.

Bibliographie

[1] De Finetti B., “Sur un'impostazione alternativa della teoria collettiva del rischio”, Trans. 15 Internat. Congress of Actuaries, v. 2, 1957, 433–443

[2] Avanzi B., “Strategies for dividend distribution: a review”, North American Actuarial J., 13:2 (2009), 217–251 | DOI | MR

[3] Taksar M. I., “Optimal risk and dividend distribution control models for an insurance company”, Math. Meth. Operations Research, 51:1 (2000), 1–42 | DOI | MR | Zbl

[4] Albrecher H., Thonhauser S., “Optimality results for dividend problems in insurance”, RACSAM – Revista de la Real Academia de Ciencias Exactas. Fisicas y Naturales. Ser. A. Matematicas, 103:2 (2009), 295–320 | DOI | MR | Zbl

[5] Zhanblan-Pike M., Shiryaev A. N., “Optimizatsiya potoka dividendov”, Uspekhi mat. nauk, 50:2 (1995), 25–46 | MR | Zbl

[6] Radner R., Shepp L., “Risk vs. profit potential: a model for corporate strategy”, J. Economic Dynam. Control, 20:8 (1996), 1373–1393 | DOI | Zbl

[7] Asmussen S., Taksar M., “Controlled diffusion models for optimal dividend pay-out”, Insurance: Mathematics and Economics, 20:1 (1997), 1–15 | DOI | MR | Zbl

[8] Shreve S. E., Lehoczky J. P., Gaver D. P., “Optimal consumption for general diffusions with absorbing and reflecting barriers”, SIAM J. Control Optim., 22:1 (1984), 55–75 | DOI | MR | Zbl

[9] Meng H., Siu T. K., “On optimal reinsurance, dividend and reinvestment strategies”, Econom. Modelling, 28:1 (2011), 211–218 | DOI

[10] Pelsser A. A. J., Laeven R. J. A., “Optimal dividends and ALM under unhedgeable risk”, Insurance: Mathematics and Economics, 53:3 (2013), 515–523 | DOI | MR | Zbl

[11] Hojgaard B., Taksar M., “Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy”, Quantitative Finance, 4:3 (2004), 315–327 | DOI | MR

[12] Hojgaard B., Taksar M., “Controlling risk exposure and dividends payout schemes: insurance company example”, Math. Finance, 9:2 (1999), 153–182 | DOI | MR | Zbl

[13] Asmussen S., Hojgaard B., Taksar M., “Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation”, Finance and Stochastics, 4:3 (2000), 299–324 | DOI | MR | Zbl

[14] Cadenillas A., Choulli T., Taksar M., Zhang L., “Classical and impulse stochastic control for the optimization of the dividend and risk policies of an insurance firm”, Math. Finance, 16:1 (2006), 181–202 | DOI | MR | Zbl

[15] Zhu J., “Dividend optimization for a regime-switching diffusion model with restricted dividend rates”, Astin Bull., 44:2 (2014), 459–494 | DOI | Zbl

[16] Zhu J., Chen F., “Dividend optimization for regime–switching general diffusions”, Insurance: Mathematics and Economics, 53:2 (2013), 439–456 | DOI | MR | Zbl

[17] Jiang Z., Pistorius M., “Optimal dividend distribution under Markov regime switching”, Finance and Stochastics, 16:3 (2012), 449–476 | DOI | MR | Zbl

[18] Sotomayor L. R., Cadenillas A., “Classical and singular stochastic control for the optimal dividend policy when there is regime switching”, Insurance: Mathematics and Economics, 48:3 (2011), 344–354 | DOI | MR | Zbl

[19] Liang Z., Yuen K. C., Guo J., “Optimal proportional reinsurance and investment in a stock market with Ornstein–Uhlenbeck process”, Insurance: Mathematics and Economics, 49:2 (2011), 207–215 | DOI | MR | Zbl

[20] Krylov N. V., Upravlyaemye protsessy diffuzionnogo tipa, Nauka, M., 1977 | MR

[21] Pham H., Continuous-time Stochastic Control and Optimization with Financial Applications, Springer, Berlin, 2009 | MR

[22] Touzi N., Optimal stochastic control, stochastic target problems, and backward SDE, Ser. Fields Institute Monographs, 29, Springer, N.Y., 2013 | DOI | MR | Zbl

[23] Crandall M., Ishii H., Lions P.-L., “User's guide to viscosity solutions of second-order partial differential equations”, Bull. Amer. Math. Soc., 27:1 (1992), 1–67 | DOI | MR | Zbl

[24] Jakobsen E. R., “Monotone Schemes”, Encyclopedia of Quantitative Finance, 2010, 1253–1263

[25] Jensen R., “Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential equations”, Indiana Univ. Math. J., 38:3 (1989), 629–667 | DOI | MR | Zbl

[26] Bardi M., Bottacin S., “On the Dirichlet problem for nonlinear degenerate elliptic equations and applications to optimal control”, Rend. Sem. Mat. Univ. Politec. Torino, 56 (1998), 13–39 | MR | Zbl

[27] Ishii H., “On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDE's”, Comm. Pure Appl. Math., 42:1 (1989), 15–45 | DOI | MR | Zbl

[28] Barles G., Rouy E., “A strong comparison result for the Bellman equation arising in stochastic exit time control problems and its applications”, Comm. Partial Differential Equations, 22:11–12 (1998), 1995–2033 | MR

[29] Barles G., Burdeau J., “The Dirichlet problem for semilinear second-order degenerate elliptic equations and applications to stochastic exit time control problems”, Comm. Partial Differential Equations, 20:1–2 (1995), 129–178 | DOI | MR | Zbl

[30] Rokhlin D. B., “Verification by stochastic Perron's method in stochastic exit time control problems”, J. Math. Anal. Appl., 419:1 (2014), 433–446 | DOI | MR | Zbl

[31] Oberman A., “Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton–Jacobi equations and free boundary problems”, SIAM J. Numer. Anal., 44:2 (2006), 879–895 | DOI | MR | Zbl

[32] Barles G., Souganidis P. E., “Convergence of approximation schemes for fully nonlinear second order equations”, Asymptotic Anal., 4:3 (1991), 271–283 | MR | Zbl

[33] Rishel R., “Optimal portfoliomanagement with partial observations and power utility function”, Stochastic analysis, control, optimization and applications, Volume in Honor of W. H. Fleming, Birkhauser, Boston, 1999, 605–619 | DOI | MR | Zbl