# American Institute of Mathematical Sciences

September  2020, 16(5): 2141-2157. doi: 10.3934/jimo.2019047

## Optimal dividend of compound poisson process under a stochastic interest rate

 1 School of Mathematical Sciences, Nankai University, Tianjin 300071, China 2 School of Economics and Management, Hebei University of Technology, Tianjin 300401, China

* Corresponding author: Xiaoyi Zhang

Received  January 2018 Revised  November 2018 Published  September 2020 Early access  May 2019

Fund Project: Research is supported by Chinese NSF Grants No.11471171 and No.11571189

In this paper we assume the insurance wealth process is driven by the compound Poisson process. The discounting factor is modelled as a geometric Brownian motion at first and then as an exponential function of an integrated Ornstein-Uhlenbeck process. The objective is to maximize the cumulated value of expected discounted dividends up to the time of ruin. We give an explicit expression of the value function and the optimal strategy in the case of interest rate following a geometric Brownian motion. For the case of the Vasicek model, we explore some properties of the value function. Since we can not find an explicit expression for the value function in the second case, we prove that the value function is the viscosity solution of the corresponding HJB equation.

Citation: Linlin Tian, Xiaoyi Zhang, Yizhou Bai. Optimal dividend of compound poisson process under a stochastic interest rate. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2141-2157. doi: 10.3934/jimo.2019047
##### References:
 [1] H. Albrecher and S. Thonhauser, Optimal dividend strategies for a risk process under force of interest, Insurance Math. Econom., 43 (2008), 134-149.  doi: 10.1016/j.insmatheco.2008.03.012. [2] H. Albrecher and S. Thonhauser, Optimality results for dividend problems in insurance, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 103 (2009), 295-320.  doi: 10.1007/BF03191909. [3] S. Asmussen and M. Taksar, Controlled diffusion models for optimal dividend pay-out, Insurance Math. Econom., 20 (1997), 1-15.  doi: 10.1016/S0167-6687(96)00017-0. [4] P. Azcue and N. Muler, Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model, Math. Finance, 15 (2005), 261-308.  doi: 10.1111/j.0960-1627.2005.00220.x. [5] P. Azcue and N. Muler, Optimal dividend policies for compound Poisson processes: The case of bounded dividend rates, Insurance Math. Econom., 51 (2012), 26-42.  doi: 10.1016/j.insmatheco.2012.02.011. [6] L. Bai, J. Ma and X. Xing, Optimal dividend and investment problems under Sparre Andersen model, Ann. Appl. Probab., 27 (2017), 3588-3632.  doi: 10.1214/17-AAP1288. [7] A. N. Borodin and P. Salminen, Handbook of Brownian motion-facts and formulae, Birkhäuser Verlag, Basel, 2002. [8] M. G. Crandall and H. Ishii, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5. [9] M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42.  doi: 10.1090/S0002-9947-1983-0690039-8. [10] F. De. Finetti, Su un'impostazione alternativa della teoria collettiva del rischio, Transactions of the XVth International Congress of Actuaries, $\Pi$ (1957), 33-443. [11] J. Eisenberg, Optimal dividends under a stochastic interest rate, Insurance Math. Econom., 65 (2015), 259-266.  doi: 10.1016/j.insmatheco.2015.10.007. [12] J. Eisenberg, Unrestricted consumption under a deterministic wealth and an Ornstein-Uhlenbeck process as a discount rate, Stoch. Models, 34 (2018), 139-153.  doi: 10.1080/15326349.2017.1392867. [13] W. H. Fleming and H. M. Soner, Controlled Markov processes and Viscosity Solutions, 2$^{nd}$ edition, Springer, New York, 2006. [14] H. U. Gerber and E. S. W. Shiu, On optimal dividend strategies in the compound Poisson model, N. Am. Actuar. J., 10 (2006), 76-93. [15] R. Loeffen, On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes, Ann. Appl. Probab., 18 (2008), 1669-1680.  doi: 10.1214/07-AAP504. [16] P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Ⅰ. The dynamic programming principle and applications, Comm. Partial Diff. Eqs., 8 (1983), 1101-1174.  doi: 10.1080/03605308308820297. [17] P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Ⅱ. Viscosity solutions and uniqueness, Comm. Partial Diff. Eqs., 8 (1983), 1229-1276.  doi: 10.1080/03605308308820301. [18] C. Mou and A. $\acute{S}$wiȩch, Uniqueness of viscosity solutions for a class of integro-differential equations, NoDea-Nonlinear Differ. Equ. Appl., 22 (2015), 1851-1882.  doi: 10.1007/s00030-015-0347-9. [19] J. Smoller, Stochastic Control in Insurance, Springer, New York, 2008. [20] H. M. Soner, Optimal control with state-space constraint. Ⅱ, SIAM J. Control Optim., 24 (1986), 1110-1122.  doi: 10.1137/0324067. [21] O. A. Vasicek, An equilibrium characterization of the term structure, Finance, Economics and Mathematics, 5 (1977), 177-188.  doi: 10.1002/9781119186229.ch6. [22] R. L. Wheeden and A. Zygmund, Measure and Integral, Marcel Dekker, Inc., New York-Basel, 1977. [23] J. Yong and X. Y. Zhou, Tochastic Controls. Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999.

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##### References:
 [1] H. Albrecher and S. Thonhauser, Optimal dividend strategies for a risk process under force of interest, Insurance Math. Econom., 43 (2008), 134-149.  doi: 10.1016/j.insmatheco.2008.03.012. [2] H. Albrecher and S. Thonhauser, Optimality results for dividend problems in insurance, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 103 (2009), 295-320.  doi: 10.1007/BF03191909. [3] S. Asmussen and M. Taksar, Controlled diffusion models for optimal dividend pay-out, Insurance Math. Econom., 20 (1997), 1-15.  doi: 10.1016/S0167-6687(96)00017-0. [4] P. Azcue and N. Muler, Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model, Math. Finance, 15 (2005), 261-308.  doi: 10.1111/j.0960-1627.2005.00220.x. [5] P. Azcue and N. Muler, Optimal dividend policies for compound Poisson processes: The case of bounded dividend rates, Insurance Math. Econom., 51 (2012), 26-42.  doi: 10.1016/j.insmatheco.2012.02.011. [6] L. Bai, J. Ma and X. Xing, Optimal dividend and investment problems under Sparre Andersen model, Ann. Appl. Probab., 27 (2017), 3588-3632.  doi: 10.1214/17-AAP1288. [7] A. N. Borodin and P. Salminen, Handbook of Brownian motion-facts and formulae, Birkhäuser Verlag, Basel, 2002. [8] M. G. Crandall and H. Ishii, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5. [9] M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42.  doi: 10.1090/S0002-9947-1983-0690039-8. [10] F. De. Finetti, Su un'impostazione alternativa della teoria collettiva del rischio, Transactions of the XVth International Congress of Actuaries, $\Pi$ (1957), 33-443. [11] J. Eisenberg, Optimal dividends under a stochastic interest rate, Insurance Math. Econom., 65 (2015), 259-266.  doi: 10.1016/j.insmatheco.2015.10.007. [12] J. Eisenberg, Unrestricted consumption under a deterministic wealth and an Ornstein-Uhlenbeck process as a discount rate, Stoch. Models, 34 (2018), 139-153.  doi: 10.1080/15326349.2017.1392867. [13] W. H. Fleming and H. M. Soner, Controlled Markov processes and Viscosity Solutions, 2$^{nd}$ edition, Springer, New York, 2006. [14] H. U. Gerber and E. S. W. Shiu, On optimal dividend strategies in the compound Poisson model, N. Am. Actuar. J., 10 (2006), 76-93. [15] R. Loeffen, On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes, Ann. Appl. Probab., 18 (2008), 1669-1680.  doi: 10.1214/07-AAP504. [16] P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Ⅰ. The dynamic programming principle and applications, Comm. Partial Diff. Eqs., 8 (1983), 1101-1174.  doi: 10.1080/03605308308820297. [17] P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Ⅱ. Viscosity solutions and uniqueness, Comm. Partial Diff. Eqs., 8 (1983), 1229-1276.  doi: 10.1080/03605308308820301. [18] C. Mou and A. $\acute{S}$wiȩch, Uniqueness of viscosity solutions for a class of integro-differential equations, NoDea-Nonlinear Differ. Equ. Appl., 22 (2015), 1851-1882.  doi: 10.1007/s00030-015-0347-9. [19] J. Smoller, Stochastic Control in Insurance, Springer, New York, 2008. [20] H. M. Soner, Optimal control with state-space constraint. Ⅱ, SIAM J. Control Optim., 24 (1986), 1110-1122.  doi: 10.1137/0324067. [21] O. A. Vasicek, An equilibrium characterization of the term structure, Finance, Economics and Mathematics, 5 (1977), 177-188.  doi: 10.1002/9781119186229.ch6. [22] R. L. Wheeden and A. Zygmund, Measure and Integral, Marcel Dekker, Inc., New York-Basel, 1977. [23] J. Yong and X. Y. Zhou, Tochastic Controls. Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999.
The shape of the value function
Left picture: The sensitivity of $V$ about parameter $\beta$. Right picture: The sensitivity of $V$ about parameter $\lambda$
the realization of $\exp\{-U_s^r\}$, $r = 1, a = 1, {\hat{\delta }} = 1$ for $\hat{b} = 2$ and $\hat{b} = -2$
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