Dynamical insurance models with investment: Constrained singular problems for integrodifferential equations

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...

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Vydáno v:	Computational mathematics and mathematical physics Ročník 56; číslo 1; s. 43 - 92
Hlavní autoři:	Belkina, T. A., Konyukhova, N. B., Kurochkin, S. V.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Moscow Pleiades Publishing 01.01.2016 Springer Nature B.V
Témata:	Algorithms Computational mathematics Computational Mathematics and Numerical Analysis Constraints Financing Insurance Insurance companies Integrals Mathematical analysis Mathematical models Mathematics Mathematics and Statistics Operators Ordinary differential equations Risk Securities markets exponential distributions of premium and claim sizes behavior of solutions investments in risky and risk-free assets related singular problems for ordinary differential equations numerical solution algorithms existence singular initial value and nonlocal constrained problems numerical results Cramér–Lundberg-type dynamical insurance models with deterministic and stochastic premiums survival probability of an insurance company as a function of its initial surplus uniqueness degenerate problems second-order linear IDEs on a half-line with Volterra and non-Volterra integral operators comparison of models
ISSN:	0965-5425, 1555-6662
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Popis
Shrnutí:	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.
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 content type line 23
ISSN:	0965-5425 1555-6662
DOI:	10.1134/S0965542516010073