Article
HJB equations with gradient constraint associated with controlled jumpdiusion processes
In this paper, we guarantee the existence and uniqueness (in the almost everywhere
sense) of the solution to a HamiltonJacobiBellman (HJB) equation with gradient
constraint and a partial integrodi erential operator whose Levy measure has bounded
variation. This type of equation arises in a singular control problem, where the state
process is a multidimensional jumpdi usion with jumps of finite variation and infinite
activity. We verify, by means of "penalized controls, that the value function associated
with this problem satis es the aforementioned HJB equation.
This article is about numerical control of HIV propagation. The contribution of the article is threefold: first, a novel model of HIV propagation is proposed; second, the methods from numerical optimal control are successfully applied to the developed model to compute optimal control profiles; finally, the computed results are applied to the real problem yielding important and practically relevant results.
This article gives a brief summary on the main theoretical and practical results for the Scale functions. The article is organized in the following way: the first part describes the main theoretical concepts of Lévy processes, gives the formal definition and analytical properties of the Scale function. The second part describes the most significant practical cases where the Scale functions are applied. Thereafter, the closedform expressions of Scale functions for several classes of spectrally negative Lévy processes are considered. Finally, we concentrate on the very important application of Scale functions, namely, derivation of the distribution of dividends, paid by the insurance company to shareholders. The main contribution of this article is a lemma, describing the asymptotic behaviour of dividends, paid by n identical companies.
In this paper, we introduce a principally new method for modelling the dependence structure between two L{\'e}vy processes. The proposed method is based on some special properties of the timechanged Levy processes and can be viewed as an reasonable alternative to the copula approach.
This collection of articles contain materials of the talks presented at the International Conference "Systems Analysis: Modeling and Control" in memory of Academician A.V. Kryazhimskiy, Moscow, May 31  June 1, 2018

We consider an optimal investment and consumption problem for a BlackScholes financial market with stochastic volatility and unknown stock price appreciation rate. The volatility parameter is driven by an external economic factor modeled as a diffusion process of Ornstein Uhlenbeck type with unknown drift. We use the dynamical programming approach and find an optimal financial strategy which depends on the drift parameter. To estimate the drift coefficient we observe the economic factor Y in an interval [0, T0] for fixed T0 > 0, and use sequential estimation. We show that the consumption and investment strategy calculated through this sequential procedure is δoptimal.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnitedimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasisolutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasisolutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasisolutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasisolutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.
This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents stateofthe art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.