Geodesic
Singular boundary value problem for the integrodifferential equation in an insurance model with stochastic premiums: Analysis and numerical solution
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 10, pp. 1812-1846

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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 $\mathbb{R}_+$, 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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     author = {T. A. Belkina and N. B. Konyukhova and S. V. Kurochkin},
     title = {Singular boundary value problem for the integrodifferential equation in an insurance model with stochastic premiums: {Analysis} and numerical solution},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
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     publisher = {mathdoc},
     volume = {52},
     number = {10},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_10_a5/}
}
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T. A. Belkina; N. B. Konyukhova; S. V. Kurochkin. Singular boundary value problem for the integrodifferential equation in an insurance model with stochastic premiums: Analysis and numerical solution. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 10, pp. 1812-1846. http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_10_a5/