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

• Published in: Computational mathematics and mathematical physics Vol. 52; no. 10; pp. 1384 - 1416
• Main Authors: Belkina, T. A., Konyukhova, N. B., Kurochkin, S. V.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.10.2012; Springer Nature B.V
• Subjects: 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
• DOI: 10.1134/S0965542512100077

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.