Nonlinear Trend Exclusion Procedure for Models defined by Stochastic Differential and Difference Equations
We consider the diffusion process and its approximation by Markov chain with nonlinear unbounded trends. The usual parametrix method is not applicable because these models have unbounded trends. We describe a procedure that allows to exclude nonlinear unbounded trend and move to stochastic differential equation with bounded drift and diffusion coefficients. A similar procedure is considered for a Markov chain.
This book provides a rigorous yet accessible introduction to the theory of stochastic processes. A significant part of the book is devoted to the classic theory of stochastic processes. In turn, it also presents proofs of well-known results, sometimes together with new approaches. Moreover, the book explores topics not previously covered elsewhere, such as distributions of functionals of diffusions stopped at different random times, the Brownian local time, diffusions with jumps, and an invariance principle for random walks and local times. Supported by carefully selected material, the book showcases a wealth of examples that demonstrate how to solve concrete problems by applying theoretical results. It addresses a broad range of applications, focusing on concrete computational techniques rather than on abstract theory. The content presented here is largely self-contained, making it suitable for researchers and graduate students alike.
This paper studies program implementation problem of pseudo-random number generators in OpenModelica. We give an overview of generators of pseudo-random uniform distributed numbers. They are used as a basis for construction of generators of normal and Poisson distributions. The last step is the creation of Wiener and Poisson stochastic processes generators. We also describe the algorithm to call external C-functions from programs written in Modelica. This allows us to use random number generators implemented in the C language.
This article concerns the problem of predicting the size of company's customer base in case of solving the task of managing its clients. The author purposes a new approach to segment-oriented predicting the size of clients based on adopting the Staroverov's employees moving model. Besides the article includes the limitations of using this model and its modification for each type of relations of the client and the company.
In this paper we present a novel approach towards variance reduction for discretised diffusion processes. The proposed approach involves specially constructed control variates and allows for a significant reduction in the variance for the terminal functionals. In this way the complexity order of the standard Monte Carlo algorithm (ε−3) can be reduced down to ε−2 log(ε−1) in case of the Euler scheme with ε being the precision to be achieved. These theoretical results are illustrated by several numerical examples.
We use a Markov chains models for the analysis of Russian stock market. First problem studied in the paper is the multiperiod portfolio optimization. We show that known approaches applied for the Russian stock market produce the phenomena of non stability and propose a new methods in order to smooth it. The second problem addressed in the paper is a structural changes on the Russian stock market after the financial crisis of 2008.We propose a hidden Markov chains model to analyse a structural changes and apply it for the Russian stock market.
The textbook has passed practical tests and written on the basis of the readable authors for many years. Presented in textbook materials give students orientation in the solution of many practical problems in a number of areas, constitute the initial level to obtain a broader and deeper education in the field of probability theory. The book provides an overview of the theory of stochastic processes, detailed material on the theory of Markov processes with discrete time (Markov chains) and continuous-time. In addition to the solved problems for each Chapter of the textbook suggested problems to solve and theoretical questions to test the quality of the learning material.
This is an advanced text on ordinary differential equations (ODES) in Banach and more general locally convex spaces, most notably the ODEs on measures and various function spaces. It yields the concise exposition of the fundamentals with the fast, but rigorous and systematic transition to the up-fronts of modern research in linear and nonlinear partial and pseudo-differential equations, general kinetic equations and fractional evolutions. The level of generality is chosen to be suitable for the study of the most important nonlinear equations of mathematical physics, such as Boltzmann, Smoluchovskii, Vlasov, Landau-Fokker-Planck, Cahn-Hilliard, Hamilton-Jacobi-Bellman, nonlinear Schroedinger, McKean-Vlasov diffusions and their nonlocal extensions, mass-action-law kinetics from chemistry. It also covers nonlinear evolutions arising in evolutionary biology and mean-field games, optimization theory, epidemics and system biology, in general models of interacting particles or agents describing splitting and merging, collisions and breakage, mutations and the preferential-attachment growth on networks. The book is meant for final year undergraduate and postgraduate students and researchers in differential equations and their applications. A significant amount of attention is paid to the interconnections between various topics revealing where and how a particular result is used in other chapters or may be used in other contexts, as well as to the clarification of the links between the languages of pseudo-differential operators, generalized functions, operator theory, abstract linear spaces, fractional calculus and path integrals.