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Dynamical insurance models with investment: constrained singular problems for integrodifferential equations. (English. Russian original) Zbl 1349.91129
Comput. Math. Math. Phys. 56, No. 1, 43-92 (2016); translation from Zh. Vychisl. Mat. Mat. Fiz. 56, No. 1, 47-98 (2016).
Summary: Previous and new results are used to compare two mathematical insurance models with identical insurance company strategies in a financial market, namely, when the entire current surplus or its constant fraction is invested in risky assets (stocks), while the rest of the surplus is invested in a risk-free asset (bank account). Model I is the classical Cramér-Lundberg risk model with an exponential claim size distribution. Model II is a modification of the classical risk model (risk process with stochastic premiums) with exponential distributions of claim and premium sizes. For the survival probability of an insurance company over infinite time (as a function of its initial surplus), there arise singular problems for second-order linear integrodifferential equations (IDEs) defined on a semiinfinite interval and having nonintegrable singularities at zero: model I leads to a singular constrained initial value problem for an IDE with a Volterra integral operator, while II model leads to a more complicated nonlocal constrained problem for an IDE with a non-Volterra integral operator. A brief overview of previous results for these two problems depending on several positive parameters is given, and new results are presented. Additional results are concerned with the formulation, analysis, and numerical study of “degenerate” problems for both models, i.e., problems in which some of the IDE parameters vanish; moreover, passages to the limit with respect to the parameters through which we proceed from the original problems to the degenerate ones are singular for small and/or large argument values. Such problems are of mathematical and practical interest in themselves. Along with insurance models without investment, they describe the case of surplus completely invested in risk-free assets, as well as some noninsurance models of surplus dynamics, for example, charity-type models.

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 45J05 Integro-ordinary differential equations 34B16 Singular nonlinear boundary value problems for ordinary differential equations 60H30 Applications of stochastic analysis (to PDEs, etc.)
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##### References:
  J. Grandell, Aspects of Risk Theory (Springer-Verlag, Berlin, 1991). · Zbl 0717.62100  V. Yu. Korolev, V. E. Bening, and S. Ya. Shorgin, Mathematical Foundations of Risk Theory (Fizmatlit, Moscow, 2007) [in Russian]. · Zbl 1234.60004  N. L. Bowers, H. U. Gerber, J. C. Hickman, D. A. Jones, and C. J. Nesbitt, Actuarial Mathematics (Soc. of Actuaries, Itasca, IL, 1986; Yanus-K, Moscow, 2001). · Zbl 0634.62107  S. Asmussen and H. Albrecher, Ruin Probabilities (World Scientific, Singapore, 2010). · Zbl 1247.91080  Belkina, T. A.; Konyukhova, N. B.; Kurkina, A. O., Optimal investment problem in dynamic insurance models: II. cramér-lundberg model with exponential claim size distribution, Obozr. Prikl. Promyshl. Mat. (Sekts. Finans. Strakh. Mat.), 17, 3-24, (2010)  Belkina, T. A.; Konyukhova, N. B.; Kurochkin, S. V., Singular initial value problem for linear integrodifferential equation arising in insurance models, Int. Sci. J. Spectral Evolution Probl., 21, 40-54, (2011)  Belkina, T. A.; Konyukhova, N. B.; Kurochkin, S. V., Singular boundary value problem for the integrodifferential equation in an insurance model with stochastic premiums: analysis and numerical solution, Comput. Math. Math. Phys., 52, 1384-1416, (2012) · Zbl 1274.65334  Belkina, T.; Konyukhova, N.; Kurochkin, S., Singular problems for integro-differential equations in dynamic insurance models, 27-44, (2013) · Zbl 1314.45007  Belkina, T. A.; Konyukhova, N. B.; Kurochkin, S. V., Singular initial and boundary value problems for integro- differential equations in dynamic insurance models with investment, Sovrem. Mat. Fundam. Napravl., 53, 5-29, (2014)  Belkina, T. A.; Belen’kii, V. Z. (ed.), Sufficiency theorems for survival probability in dynamic insurance models with investment, (2011), Moscow  Belkina, T. A., Risky investment for insurers and sufficiency theorems for the survival probability, Markov Processes Related Fields, 20, 505-525, (2014) · Zbl 1310.91074  Paulsen, J.; Gjessing, H. K., Ruin theory with stochastic return on investments, Adv. Appl. Probab., 29, 965-985, (1997) · Zbl 0892.90046  Frolova, A.; Pergamenshchikov, S., In the insurance business risky investments are dangerous, Finance Stoch., 6, 227-235, (2002) · Zbl 1002.91037  Pergamenshchikov, S.; Zeitouny, O., Ruin probability in the presence of risky investments, Stochastic Process. Appl., 116, 267-278, (2006) · Zbl 1088.60076  A. V. Boikov, Candidate’s Dissertation in Mathematics and Physics (Steklov Mathematical Inst., Russ. Acad. Sci., Moscow, 2003).  A. Ramos, PhD Thesis (Univ. Carlos III de Madrid, Madrid, 2009) (http://e-archivouc3mes/haudle/ 10016/5631).  Bachelier, L., Theorie de la speculation, Ann. Sci. Ecole Norm. Super., 17, 21-86, (1900) · JFM 31.0241.02  Belkina, T. A.; Konyukhova, N. B.; Kurkina, A. O., Optimal investment problem in dynamic insurance models: I. investment strategies and ruin probability, Obozr. Prikl. Promyshl. Mat. (Sekts. Finans. Strakh. Mat.), 16, 961-981, (2009)  R. Bellman, Stability Theory of Differential Equations (McGraw-Hill, New York, 1953; Inostrannaya Literatura, Moscow, 1954). · Zbl 0056.36501  M. V. Fedoryuk, Asymptotic Analysis: Linear Ordinary Differential Equations (Nauka, Moscow, 1983; Springer, Berlin, 1993). · Zbl 0782.34001  E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955; Inostrannaya Literatura, Moscow, 1958). · Zbl 0064.33002  W. R. Wasow, Asymptotic Expansions for Ordinary Differential Equations (Wiley, New York, 1965; Mir, Moscow, 1968). · Zbl 0169.10903  E. Kamke, Differentialgleichungen: Lösungmethoden und Lösungen: I. Gewöhnlishe Differentialgleishungen (Akademie-Verlag, Leipzig, 1959; Nauka, Moscow, 1971).  Birger, E. S.; Lyalikova Konyukhova, N. B., Discovery of the solutions of certain systems of differential equations with a given condition at infinity I, USSR Comput. Math. Math. Phys., 5, 1-17, (1965) · Zbl 0167.08102  Konyukhova, N. B., Singular Cauchy problems for systems of ordinary differential equations, USSR Comput. Math. Math. Phys., 23, 72-82, (1983) · Zbl 0555.34002  Belkina, T. A.; Hipp, C.; Luo, S.; Taksar, M., Optimal constrained investment in the cramér-lundberg model, Scand. Actuarial J., No., 5, 383-404, (2014) · Zbl 1401.91099  Konyukhova, N. B., Singular Cauchy problems for singularly perturbed systems of nonlinear ordinary differential equations, I: Differ. Equations, 32, 54-63, (1996) · Zbl 0874.34007  Higher Transcendental Functions (Bateman Manuscript Project), Ed. by A. Erdelyi (McGraw-Hill, New York, 1953; Nauka, Moscow, 1965). · Zbl 0488.34002  Gingold, H.; Rosenblat, S., Differential equations with moving singularities, SIAM J. Math. Anal., 7, 942-957, (1976) · Zbl 0344.34048  Boikov, A. V., The cramér-lundberg model with stochastic premium process, Theory Probab. Appl., 47, 489-493, (2003) · Zbl 1033.60093  Zinchenko, N.; Andrusiv, A., Risk processes with stochastic premiums, Theory Stoch. Processes, 14, 189-208, (2008) · Zbl 1224.91102  Temnov, G., Risk models with stochastic premium and ruin probability estimation, J. Math. Sci., 196, 84-96, (2014) · Zbl 1307.91101  N. V. Azbelev, V. P. Maksimov, and L. F. Rakhmatulina, Introduction to the Theory of Functional Differential Equations (Nauka, Moscow, 1991) [in Russian]. · Zbl 0725.34071  Konyukhova, N. B., Singular Cauchy problems for some systems of nonlinear functional-differential equations, Differ. Equations, 31, 1286-1293, (1995) · Zbl 0863.34074  Konyukhova, N. B., Singular problems for systems of nonlinear functional-differential equations, Int. Sci. J. Spectral Evolution Probl., 20, 199-214, (2010)  Abramov, A. A., On the transfer of the condition of boundedness for some systems of ordinary linear differential equations, USSR Comput. Math. Math. Phys., 1, 875-881, (1962) · Zbl 0129.09702  Abramov, A. A.; Balla, K.; Konyukhova, N. B., Transfer of boundary conditions from singular points for systems of ordinary differential equations, (1981) · Zbl 0488.34002  Abramov, A. A.; Konyukhova, N. B.; Balla, K., Stable initial manifolds and singular boundary value problems for systems of ordinary differential equations, Comput. Math. Banach Center Publ., 13, 319-351, (1984) · Zbl 0568.34033  Abramov, A. A.; Konyukhova, N. B., Transfer of admissible boundary conditions from a singular point for systems of linear ordinary differential equations, (1985), Moscow · Zbl 0825.34012  Abramov, A. A.; Konyukhova, N. B., Transfer of admissible boundary conditions from a singular point for systems of linear ordinary differential equations, Sov. J. Numer. Anal. Math. Model., 1, 245-265, (1986) · Zbl 0825.34012  Abramov, A. A.; Ditkin, V. V.; Konyukhova, N. B.; Pariiskii, B. S.; Ul’yanova, V. I., Evaluation of the eigenvalues and eigenfunctions of ordinary differential equations with singularities, USSR Comput. Math. Math. Phys., 20, 63-81, (1980) · Zbl 0472.65073  Abramov, A. A., On the transfer of boundary conditions for systems of ordinary linear differential equations (a variant of the dispersive method), USSR Comput. Math. Math. Phys., 1, 617-622, (1961) · Zbl 0129.09701  N. S. Bakhvalov, Numerical Methods: Analysis, Algebra, Ordinary Differential Equations (Nauka, Moscow, 1973; Mir, Moscow, 1977).  Kalashnikov, V.; Norberg, R., Power tailed ruin probabilities in the presence of risky investments, Stoch. Proc. Appl., 98, 211-228, (2002) · Zbl 1058.60095  Laubis, B.; Lin, J.-E., Optimal investment allocation in a jump diffusion risk model with investment: A numerical analysis of several examples, (2008)
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