Abstract
The survival probability of an insurance company in a collective pension insurance model (so-called dual risk model) is investigated in the case when the whole surplus (or its fixed fraction) is invested in risky assets, which are modeled by a geometric Brownian motion. A typical insurance contract for an insurer in this model is a life annuity in exchange for the transfer of the inheritance right to policyholder’s property to the insurance company. The model is treated as dual with respect to the Cramér–Lundberg classical model. In the structure of an insurance risk process, this is expressed by positive random jumps (compound Poisson process) and a linearly decreasing deterministic component corresponding to pension payments. In the case of exponentially distributed jump sizes, it is shown that the survival probability regarded as a function of initial surplus defined on the nonnegative real half-line is a solution of a singular boundary value problem for an integro-differential equation with a non-Volterra integral operator. The existence and uniqueness of a solution to this problem is proved. Asymptotic representations of the survival probability for small and large values of the initial surplus are obtained. An efficient algorithm for the numerical evaluation of the solution is proposed. Numerical results are presented, and their economic interpretation is given. Namely, it is shown that, in pension insurance, investment in risky assets plays an important role in an increase of the company’s solvency for small values of initial surplus.
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REFERENCES
S. Asmussen and H. Albrecher, Ruin Probabilities, Advanced Series on Statistical Science and Applied Probability (World Scientific, Singapore, 2010), Vol. 14.
Yu. Kabanov and S. Pergamenshchikov, “In the insurance business risky investments are dangerous: The case of negative risk sums,” Finance Stochast. 20 (2), 355–379 (2016).
J. Grandell, Aspects of Risk Theory (Springer-Verlag, New York, 1991).
H. Cramér, Collective Risk Theory: A Survey of the Theory from the Point of View of the Theory of Stochastic Processes (AbNordiska Bokhandeln, Stockholm, 1955), pp. 1–92.
V. Yu. Korolev, V. E. Bening, and S. Ya. Shorgin, Mathematical Foundations of Risk Theory (Fizmatlit, Moscow, 2007) [in Russian].
J. Paulsen and H. K. Gjessing, “Ruin theory with stochastic return on investments,” Adv. Appl. Probab. 29 (4), 965–985 (1997).
A. Frolova, Yu. Kabanov, and S. Pergamenshchikov, “In the insurance business risky investments are dangerous,” Finance Stochast. 6 (2), 227–235 (2002).
S. Pergamenshchikov and O. Zeitouny, “Ruin probability in the presence of risky investments,” Stochastic Process. Appl. 116 (2), 267–278 (2006).
J. Paulsen, “On Cramér-Like asymptotics for risk processes with stochastic return on investments,” Ann. Appl. Probab. 12 (4), 1247–1260 (2002).
C. Hipp and M. Plum, “Optimal investment for insurers,” Insur. Math. Econ. 27 (2), 215–228 (2000).
C. Hipp and M. Plum, “Optimal investment for investors with state dependent income, and for insurers,” Finance Stochast. 7 (3), 299–321 (2003).
P. Azcue and M. Muler, “Optimal investment strategy to minimize the ruin probability of an insurance company under borrowing constraints,” Insur. Math. Econ. 44 (1), 26–34 (2009).
T. Belkina, C. Hipp, S. Luo, and M. Taksar, “Optimal constrained investment in the Cramér–Lundberg model,” Scand. Actuarial J., No. 5, 383–404 (2014).
T. A. Belkina, N. B. Konyukhova, and S. V. Kurochkin, “Singular boundary value problem for the integrodifferential equation in an insurance model with stochastic premiums: Analysis and numerical solution,” Comput. Math. Math. Phys. 52 (10), 1384–1416 (2012).
T. Belkina, N. Konyukhova, and S. Kurochkin, “Singular problems for integro-differential equations in dynamic insurance models,” in Differential and Difference Equations with Applications (Springer, Berlin, 2013), Vol. 47, pp. 27–44. http://arxiv.org/abs/1511.08666
T. A. Belkina, N. B. Konyukhova, and S. V. Kurochkin, “Singular initial and boundary value problems for integro-differential equations in dynamic insurance models with investment,” Sovrem. Mat. Fundam. Napravl. 53, 5–29 (2014).
T. A. Belkina, N. B. Konyukhova, and S. V. Kurochkin, “Singular initial-value and boundary-value problems for integrodifferential equations in dynamical insurance models with investments,” J. Math. Sci. 218 (4), 369–394 (2016). https://doi.org/10.1007/s10958-016-3037-1
T. A. Belkina, N. B. Konyukhova, and S. V. Kurochkin, “Dynamical insurance models with investment: Constrained singular problems for integrodifferential equations,” Comput. Math. Math. Phys. 56 (1), 43–92 (2016).
G. Wang and R. Wu, “Distributions for the risk process with a stochastic return on investments,” Stochastic Process. Appl. 95, 329–341 (2001).
T. Belkina, “Risky investment for insurers and sufficiency theorems for the survival probability,” Markov Processes Relat. Fields 20, 505–525 (2014).
T. A. Belkina, N. B. Konyukhova, and B. V. Slavko, “Risky investments and survival in the dual risk model,” in Proceedings of the 8th Moscow International Conference on Operations Research (ORM-2016), Moscow, October 17–22,2016 (MAKS, Moscow, 2016), Vol. 1, pp. 130–133.
T. A. Belkina, N. B. Konyukhova, and B. V. Slavko, “Analytic-numerical investigations of singular problems for survival probability in the dual risk model with simple investment strategies,” in Analytical and Computational Methods in Probability Theory and Its Applications, Ed. by V. V. Rykov et al., Lect. Notes Comput. Sci. 10684, 236–250 (2017). https://doi.org/10.1007/978-3-319-71504-9
T. A. Belkina and N. B. Konyukhova, “On sufficient conditions for survival probability in the life annuity insurance model with risk-free investment income,” in Proceedings of the 9th Moscow International Conference on Operations Research (ORM-2018), Moscow, October 22–27,2018, Ed. by F. Ereshko (MAKS, Moscow, 2018), Vol. 1, pp. 213–218.
T. A. Belkina and Yu. M. Kabanov, “Viscosity solutions of integro-differential equations for nonruin probabilities,” Theory Probab. Appl. 60 (4), 671–679 (2016).
I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus (Springer-Verlag, New York, 1988).
R. Bellman, Stability Theory of Differential Equations (McGraw-Hill, New York, 1953).
E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955).
W. R. Wasow, Asymptotic Expansions for Ordinary Differential Equations (Wiley, New York, 1965).
N. B. Konyukhova, “Singular Cauchy problems for systems of ordinary differential equations,” USSR Comput. Math. Math. Phys. 23 (3), 72–82 (1983).
E. S. Birger and N. B. Lyalikova (Konyukhova), “Discovery of the solutions of certain systems of differential equations with a given condition at infinity I,” USSR Comput. Math. Math. Phys. 5 (6), 1–17 (1965); “On finding the solutions for a given condition at infinity of certain systems of ordinary differential equations II,” USSR Comput. Math. Math. Phys. 6 (3), 47–57 (1966).
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Belkina, T.A., Konyukhova, N.B. & Slavko, B.V. Solvency of an Insurance Company in a Dual Risk Model with Investment: Analysis and Numerical Study of Singular Boundary Value Problems. Comput. Math. and Math. Phys. 59, 1904–1927 (2019). https://doi.org/10.1134/S0965542519110022
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DOI: https://doi.org/10.1134/S0965542519110022