Vol 23, No 3 (2023)
Articles
Risky investments and survival probability in the insurance model with two-sided jumps: Problems for integrodifferential equations and ordinary differential equation and their equivalence
Abstract
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.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2023;23(3):278-285
278-285
On the iterative method for solution of direct and inverse problems for parabolic equations
Abstract
The paper is devoted to approximate methods for solution of direct and inverse problems for parabolic equations. An approximate method for the solution of the initial problem for multidimensional nonlinear parabolic equation is proposed. The method is based on the reduction of the initial problem to a nonlinear multidimensional intergral Fredholm equation of the second kind which is approximated by a system of nonlinear algebraic equations with the help of the method of mechanical quadratures. For constructing the computational scheme we use the nodes of the local splines which realize order-optimal approximation of the functional class that contains solutions of parabolic equations. For implementation of the computational scheme we use the generalization of the continuous method for solution of nonlinear operator equations that is described in the paper. We also analyse the inverse problem for parabolic equation with fractional order derivative with respect to the time variable. The approximate methods for defining the fractional order of the time derivative and the coeffcient at spatial derivative are proposed.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2023;23(3):286-310
286-310
Classic and generalized solutions of the mixed problem for wave equation with a summable potential. Part I. Classic solution of the mixed problem
Abstract
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 $\psi(x) \in L[0,1]$.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2023;23(3):311-319
311-319
Unitary extension principle on zero-dimensional locally compact groups
Abstract
In this article, we obtain methods for constructing step tight frames on an arbitrary locally compact zero-dimensional group. To do this, we use the principle of unitary extension. First, we indicate a method for constructing a step scaling function on an arbitrary zero-dimensional group. To construct the scaling function, we use an oriented tree and specify the conditions on the tree under which the tree generates the mask $m_0$ of a scaling function. Then we find conditions on the masks $m_0, m_1,\ldots , m_q$ under which the corresponding wavelet functions $\psi_1, \psi_2,\ldots ,\psi_q$ generate a tight frame. Using these conditions, we indicate a way of constructing such masks. In conclusion, we give examples of the construction of tight frames.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2023;23(3):320-338
320-338
On functions of van der Waerden type
Abstract
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 $x\to x_0$.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2023;23(3):339-347
339-347
On the question of the residual of strong exponents of oscillation on the set of solutions of third-order differential equations
Abstract
In this paper, we study various types of exponents of oscillation (upper or lower, strong or weak) of non-strict signs, zeros, and roots of non-zero solutions of linear homogeneous differential equations of the third order with continuous and bounded coefficients on the positive semi-axis. A nonzero solution of a linear homogeneous equation cannot be zeroed due to the existence and uniqueness theorem. Therefore, the spectra of all the listed exponents of oscillation (i.e. their sets of values on nonzero solutions) consist of one zero value. In addition, it is known that the spectra of the oscillation exponents of linear homogeneous equations of the second order also consist of a single value. Consequently, on the set of solutions of equations up to the second order there is a residual of all exponents of oscillation. On the set of solutions of third-order equations, strong exponents vibrations of hyper roots are not residual, i.e. are not invariant with respect to the change in the solution at any finite section of the half-axis of time. In this article, it is proved that on the set of solutions of third-order equations, strong oscillation indices of non-strict signs, zeros, and roots are not residual. In parallel, the existence of a function from the specified set with the following properties is proved: all listed exponents of oscillation are accurate, but not absolute. At the same time, all strong exponents like all weak ones, are equal to each other.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2023;23(3):348-356
348-356
Stability of three-layer differential-difference schemes with weights in the space of summable functions with supports in a network-like domain
Abstract
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.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2023;23(3):357-369
357-369
Simulation models and research algorithms of thin shell structures deformation Part I. Shell deformation models
Abstract
In the article the development of thin shell construction theory is considered according to the contribution of researchers, chronology, including the most accurate and simplified solutions. The review part of the article consists only of those publications which are related to the development of shell theory. The statement is based on the works of famous Russian researchers (V. V. Novozhilov, A. I. Lurie, A. L. Goldenweiser, H. M. Mushtari, V. Z. Vlasov), who developed the specified theory the most. The paper also mentions the researchers who improved the theory, calculation methods in aspects of strength, sustainability and vibrations of thin elastic shell constructions. Separately the application of the models for ribbed shells constructions is shown. It is reporting the basic principles of nonlinear thin shell construction theory development, including the nonlinear relations for deformations. In the article it is shown that if median surface of the shell is referred to the orthogonal coordinate system, then the expressions for deformations, obtained by different authors, practically correspond. The case in which the median surface of the shell is referred to an oblique-angled coordinate system was developed by A. L. Goldenweiser. For static problem, the functional of the total potential energy of deformation, representing the difference between the potential energy and the work of external forces, is used. The equilibrium equations and natural boundary conditions are derived from the minimum condition of this functional. In case of dynamic problem, the functional of the total deformation energy of the shell is described in which it is necessary to consider the kinetic energy of shell deformation. It is necessary to underline that the condition for minimum of the specified functional lets to derive the movement equations and natural boundary and initial conditions. Also, in the article the results of contemporary research of thin shell theory are presented.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2023;23(3):370-410
370-410
Cloud service for interactive simulation of production location
Abstract
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.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2023;23(3):411-417
411-417