卷 23, 编号 3 (2023)

We consider a model of an insurance portfolio that includes both non-life and life annuity insurance while assuming  that the surplus (or some of its fraction) is invested in risky assets with the price dynamics given by a geometric Brownian motion. The portfolio  surplus (in the absence of investments)  is described by a stochastic process involving two-sided jumps and a continuous drift. Downward jumps correspond to the claim sizes and upward jumps are interpreted as random gains  that arise at the final moments of the life annuity contracts realizations (i.e. at the moments of the death of policyholders). The drift is determined by the difference between premiums in the non-life insurance contracts and the annuity payments. We study the ruin problem for the model with investment using an approach based on integrodifferential equations (IDE) for the survival probabilities as a function of initial surplus. The main problem in calculating the survival probability as a solution of the IDE is that the initial value of the probability itself or its derivative at a zero initial surplus is priori unknown.  For the case of the exponential distributions of the jumps, we propose a solution to this problem based on the assertion that the problem for an IDE  is equivalent to a problem for an ordinary differential equation (ODE) with some nonlocal condition added. As a result,  a solution to the original problem can be obtained as a solution to the ODE problem with an unknown parameter, which is finally determined using the specified nonlocal condition and a normalization condition.

The resolvent approach and the using of the idea of A. N. Krylov on the acceleration of convergence of Fourier series, the properties of a formal solution of a mixed problem for a homogeneous wave equation with a summable potential and a zero initial function are studied. This method makes it possible to obtain deep results on the convergence of a formal series with arbitrary boundary conditions and without overestimating the requirements for the smoothness of the initial data. The different-order boundary conditions considered in the article are such that the operator corresponding to the spectral problem may have an infinite set of multiple eigenvalues and their associated functions. A classical solution is obtained without overstating the requirements for the initial velocity $u'_t(x,0) = \psi(x)$. It is shown that for $\psi(x) \in L[0,1]$ the formal solution, being the uniform limit of the classical ones, is a generalized solution, and when $\psi(x) \in L_p[0,1], ~ 1 < p\leqslant 2$, the formal solution has much smoother properties than the case $\psi(x) \in L[0,1]$.

Let $\omega(t)$ be an arbitrary modulus of continuity type function, such that $\omega(t)/t\to+\infty$, as $t\to+0$. We construct a continuous nowhere-differentiable function $V_\omega(x)$, $x\in[0;1]$, that satisfies the following conditions: 1)  its modulus of continuity satisfies the estimate $O(\omega(t))$ as $t\to+0$; 2) for some positive $c$ at each point $x_0$ holds $\limsup{|V_\omega(x){-}V_\omega(x_0)|}\big/{\omega(|x{-}x_0|)}>c$ as $x\to x_0$; 3) at each point $x_0$ holds $\liminf{|V_\omega(x){-}V_\omega(x_0)|}\big/{\omega(|x{-}x_0|)}=0$ as $x\to x_0$.

The work is a natural continuation of the authors' earlier studies in the analysis of the conditions for the weak solvability of one-dimensional initial-boundary value problems with a space variable changing on a graph (network) in the direction of increasing the dimension $n$ ($n>1$) of the network-like domain of change of this variable. The first results in this direction (for $n = 3$) were obtained by one of the authors for the linearized Navier–Stokes system, later for a much more complex nonlinear Navier–Stokes system. The analysis was carried out in the classical way, using a priori estimates for the norms of weak solutions in Sobolev spaces of functions. In this study (for arbitrary $n>1$) another approach is proposed to obtain conditions for the weak solvability of linear initial-boundary value problems reduction of the original problem to a differential-difference system, the idea of which goes back to E. Rothe's method of semi-discretization of the initial-boundary value problem by temporary variable. A differential-difference system of equations with weighted parameters and its corresponding three-layer differential-difference scheme (a set of schemes) are considered. The resulting system is an analog of the initial-boundary value problem for a parabolic type equation with a space variable changing in a network-like domain of an n-dimensional Euclidean space. The main aim is to establish a domain of the range of weight parameters that guarantees the stability of the differential-difference scheme (continuity by the initial data of the problem), to obtain estimates for the operator norms of the weak solutions of the scheme, to construct a sequence of solutions for a differential-difference system that is weakly compact in its state space. The latter is an important element when using numerical methods of analysis of a wide class of applied multidimensional problems and constructing computational algorithms for finding approximations to their solutions. The results are applicable in applied optimization problems arising from modeling network processes of continuum transport with the help of the formalisms of differential-difference systems.

The paper deals with a problem of decision-making support for production location problem. The paper describes the mathematical model of production location. Minimization of total cost of delivery of raw materials to the place of production is used as a criterion of potential production location. The problem belongs to the class of binary mathematical programming problems with linear constraints but could be reduced to a set of linear programming problems solved by sequential or parallel computing. Based on the mathematical model а software tool is implemented as a cloud service on heterogeneous computing architecture. The software architecture includes the simulation module and modules for control and visualization. The ontology and declarative model for information exchange between the modules are designed with JSON format. This declarative model includes the objects considered in the mathematical model which are “products”, “areas” and “communications”. The simulation module is implemented on a high-performance server platform. Visualization module allows us to present graphically the original and the resulting matrix data and to modify the input parameters of the model interactively. The control and visualization modules are produced within IACPaaS cloud platform. Communication between the modules is established via asynchronous http-queries. The paper demonstrates the use of the software tool for the simulation of production location for the Russian Far East regions based on input data provided by open statistics sources.