Том 218, № 4 (2016)

We investigate two insurance mathematical models of the following behavior of an insurance company in the insurance market: the company invests a constant part of the capital in a risk asset (shares) and invests the remaining part in a risk-free asset (a bank account). Changing parameters (characteristics of shares), this strategy is reduced to the case where all the capital is invested in a risk asset. The first model is based on the classical Cramér–Lundberg risk process for the exponential distribution of values of insurance demands (claims). The second one is based on a modification of the classical risk process (the so-called stochastic premium risk process) where both demand values and insurance premium values are assumed to be exponentially distributed. For the infinite-time nonruin probability of an insurance company as a function of its initial capital, singular problems for linear second-order integrodifferential equations arise. These equations are defined on a semiinfinite interval and they have nonintegrable singularities at the origin and at infinity. The first model yields a singular initial-value problem for integrodifferential equations with a Volterra integral operator with constraints. The second one yields more complicated problem for integrodifferential equations with a non-Volterra integral operator with constraints and a nonlocal condition at the origin. We reduce the problems for integrodifferential equations to equivalent singular problems for ordinary differential equations, provide existence and uniqueness theorems for the solutions, describe their properties and long-time behavior, and provide asymptotic representation of solutions in neighborhoods of singular points. We propose efficient algorithms to find numerical solutions and provide the computational results and their economics interpretation.

Basing on the notion of compact subdifferentials, we develop a subdifferential calculus of the first and the second orders beyond the Taylor expansion and extremum theory. We introduce and investigate a comprehensive class of subsmooth maps such that the constructed theory is applicable to them. We develop a technique to investigate one-dimensional extremal variational problems with subsmooth Lagrangians (including sufficient conditions). A number of examples are considered.

We introduce anti-compact sets (anti-compacts) in Frechét spaces. We thoroughly investigate the properties of anti-compacts and the scale of Banach spaces generated by anti-compacts. Special attention is paid to systems of anti-compact ellipsoids in Hilbert spaces. The existence of a system of anti-compacts is proved for any separable Frechét space E. Using the constructed theory, we obtain analogs of the Lyapunov theorem on the convexity and compactness of the range of vector measures in the class of separable Frechét spaces: We prove the convexity and compactness of the range of vector measure in a space \( {E}_{\overline{C}} \) generated by an anti-compact \( \overline{C} \). Also, the nondifferentiability problem with respect to the upper limit is investigated for the Pettis integral. We obtain differentiability conditions for the indefinite Pettis integrals in terms of the new weak integral boundedness and the σ-compact measurability. We prove an analog of the Lebesgue theorem on the differentiability of the indefinite Pettis integral for any strongly measurable integrand.

Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 53, Proceedings of the Crimean Autumn Mathematical School-Symposium KROMSH-2013, 2014.