Singular boundary value problem for the integrodifferential equation in an insurance model with stochastic premiums: Analysis and numerical solution

A singular boundary value problem for a second-order linear integrodifferential equation with Volterra and non-Volterra integral operators is formulated and analyzed. The equation is defined on ℝ + , has a weak singularity at zero and a strong singularity at infinity, and depends on several positive...

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Veröffentlicht in:Computational mathematics and mathematical physics Jg. 52; H. 10; S. 1384 - 1416
Hauptverfasser: Belkina, T. A., Konyukhova, N. B., Kurochkin, S. V.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Boston Springer US 01.10.2012
Springer Nature B.V
Schlagworte:
Asymptotic properties
Bank accounts
Boundary value problems
Computational mathematics
Computational Mathematics and Numerical Analysis
Insurance
Insurance companies
Interest rates
Investment policy
Investments
Markets
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Ordinary differential equations
Prices
Random variables
Securities markets
Singularities
Stock exchanges
Uniqueness theorems
Cramer-Lundberg model with stochastic premiums
existence, uniqueness, and behavior of a solution
survival probability of an insurance company as a function of its initial surplus
numerical solution algorithm
dynamic insurance models
related singular boundary value problems for ordinary differential equations
second-order linear integrod-ifferential equation on a half-line
singular boundary value problem with constraints
ISSN:0965-5425, 1555-6662
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Zusammenfassung:A singular boundary value problem for a second-order linear integrodifferential equation with Volterra and non-Volterra integral operators is formulated and analyzed. The equation is defined on ℝ + , has a weak singularity at zero and a strong singularity at infinity, and depends on several positive parameters. Under natural constraints on the coefficients of the equation, existence and uniqueness theorems for this problem with given limit boundary conditions at singular points are proved, asymptotic representations of the solution are given, and an algorithm for its numerical determination is described. Numerical computations are performed and their interpretation is given. The problem arises in the study of the survival probability of an insurance company over infinite time (as a function of its initial surplus) in a dynamic insurance model that is a modification of the classical Cramer-Lundberg model with a stochastic process rate of premium under a certain investment strategy in the financial market. A comparative analysis of the results with those produced by the model with deterministic premiums is given.
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ISSN:0965-5425
1555-6662
DOI:10.1134/S0965542512100077