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Mathematics
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New Trends in Random Evolutions and Their Applications
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New Trends in Random Evolutions and Their Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Financial Mathematics".

Deadline for manuscript submissions: closed (31 December 2020) | Viewed by 13061

Special Issue Editor

Prof. Dr. Anatoliy Swishchuk

E-Mail Website
Guest Editor
Department of Mathematics and Statistics, University of Calgary, Calgary, AB T2N 1N4, Canada
Interests: mathematical finance; energy finance; stochastic modelling; risk theory; random evolutions and their applications; modeling high-frequency and algorithmic trading; deep and machine learning in quantitative finance
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

It is my pleasure to announce a forthcoming Special Issue in Mathematics devoted to the new trends in random evolutions (REs) and their many applications. REs began to be studied in the 1970s, because of their potential applications in finance, insurance, biology, signal processing, quantum physics, traffic, storage, queuing, and risk theories, to name a few.

In mathematical language, a RE is an operator integro-differential equation with generator depending on a parameter, and this parameter is a stochastic process. The stochastic processes define the name for the REs: Markov, semi-Markov, stationary, Levy, etc. Additionally, depending on structure of the operator equation, we have continuous, discontinuous/jump, discrete, homogeneous, inhomogeneous REs, etc. Markov REs in Euclidian spaces are usually called in the literature hidden Markov or regime-switching models. In physical language, a RE is a model for a dynamical system in random environment, in which equation of state is subject to random variation.

In this Special Issue, we introduce new directions in modern theory of REs and their many applications.

The deadline for papers submission is 30 June 2019.

Thank you in advance for your participation in this project.

Prof. Dr. Anatoliy Swishchuk
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Random evolutions (RE)
  • Limit theorems for RE
  • Applications of RE in finance and insurance
  • Applications of RE in biology, medicine, epidemiology, branching theory

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Published Papers (6 papers)

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Research

24 pages, 1534 KiB  
Article
Modelling of Fuel- and Energy-Switching Prices by Mean-Reverting Processes and Their Applications to Alberta Energy Markets
by Weiliang Lu, Alexis Arrigoni, Anatoliy Swishchuk and Stéphane Goutte
Mathematics 2021, 9(7), 709; https://doi.org/10.3390/math9070709 - 25 Mar 2021
Cited by 5 | Viewed by 2143
Abstract
This paper introduces a fuel-switching price to the Alberta market, which is designed for encouraging power plant companies to switch from coal to natural gas when they produce electricity; this has been successfully applied to the European market. Moreover, we consider an energy-switching [...] Read more.
This paper introduces a fuel-switching price to the Alberta market, which is designed for encouraging power plant companies to switch from coal to natural gas when they produce electricity; this has been successfully applied to the European market. Moreover, we consider an energy-switching price which considers power switch from natural gas to wind. We modeled these two prices using five mean reverting processes including a Regime-switching processes, Lévy-driven Ornstein–Uhlenbeck process and Inhomogeneous Geometric Brownian Motion, and estimate them based on multiple procedures such as Maximum likelihood estimation and Expectation-Maximization algorithm. Finally, this paper proves previous results applied to the Albertan Market where the jump modeling technique is needed when modeling fuel-switching data. In addition, it not only gives promising conclusions on the necessity of introducing Regime-switching models to the fuel-switching data, but also shows that the Regime-switching model is better fitted to the data. Full article
(This article belongs to the Special Issue New Trends in Random Evolutions and Their Applications)
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26 pages, 350 KiB  
Article
Controlled Discrete-Time Semi-Markov Random Evolutions and Their Applications
by Anatoliy Swishchuk and Nikolaos Limnios
Mathematics 2021, 9(2), 158; https://doi.org/10.3390/math9020158 - 13 Jan 2021
Viewed by 1636
Abstract
In this paper, we introduced controlled discrete-time semi-Markov random evolutions. These processes are random evolutions of discrete-time semi-Markov processes where we consider a control. applied to the values of random evolution. The main results concern time-rescaled weak convergence limit theorems in a Banach [...] Read more.
In this paper, we introduced controlled discrete-time semi-Markov random evolutions. These processes are random evolutions of discrete-time semi-Markov processes where we consider a control. applied to the values of random evolution. The main results concern time-rescaled weak convergence limit theorems in a Banach space of the above stochastic systems as averaging and diffusion approximation. The applications are given to the controlled additive functionals, controlled geometric Markov renewal processes, and controlled dynamical systems. We provide dynamical principles for discrete-time dynamical systems such as controlled additive functionals and controlled geometric Markov renewal processes. We also produce dynamic programming equations (Hamilton–Jacobi–Bellman equations) for the limiting processes in diffusion approximation such as controlled additive functionals, controlled geometric Markov renewal processes and controlled dynamical systems. As an example, we consider the solution of portfolio optimization problem by Merton for the limiting controlled geometric Markov renewal processes in diffusion approximation scheme. The rates of convergence in the limit theorems are also presented. Full article
(This article belongs to the Special Issue New Trends in Random Evolutions and Their Applications)
16 pages, 1117 KiB  
Article
Insurance Contracts for Hedging Wind Power Uncertainty
by Guglielmo D’Amico, Fulvio Gismondi and Filippo Petroni
Mathematics 2020, 8(8), 1376; https://doi.org/10.3390/math8081376 - 17 Aug 2020
Cited by 1 | Viewed by 2361
Abstract
This paper presents an insurance contract that the supplier of wind power may subscribe to with an insurance company in order to immunize his/her revenue against the volatility of wind power and prices. Based on empirical evidence, we found that wind power and [...] Read more.
This paper presents an insurance contract that the supplier of wind power may subscribe to with an insurance company in order to immunize his/her revenue against the volatility of wind power and prices. Based on empirical evidence, we found that wind power and electricity prices are correlated. Then, we adopted a joint stochastic process to model both time series, which is based on indexed semi-Markov chains for the wind power generation process and on a general Markovian process for the electricity price process. Using a joint stochastic model allows the insurance company to compute the fair premium that the wind power producer has to pay in order to hedge the risk against inadequate revenues. Recursive type equations are obtained for the prospective mathematical reserves of the insurance contract. The model and the validity of the results are illustrated through a real data application. Full article
(This article belongs to the Special Issue New Trends in Random Evolutions and Their Applications)
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16 pages, 287 KiB  
Article
Discrete-Time Semi-Markov Random Evolutions in Asymptotic Reduced Random Media with Applications
by Nikolaos Limnios and Anatoliy Swishchuk
Mathematics 2020, 8(6), 963; https://doi.org/10.3390/math8060963 - 12 Jun 2020
Cited by 2 | Viewed by 1825
Abstract
This paper deals with discrete-time semi-Markov random evolutions (DTSMRE) in reduced random media. The reduction can be done for ergodic and non ergodic media. Asymptotic approximations of random evolutions living in reducible random media (random environment) are obtained. Namely, averaging, diffusion approximation and [...] Read more.
This paper deals with discrete-time semi-Markov random evolutions (DTSMRE) in reduced random media. The reduction can be done for ergodic and non ergodic media. Asymptotic approximations of random evolutions living in reducible random media (random environment) are obtained. Namely, averaging, diffusion approximation and normal deviation or diffusion approximation with equilibrium by martingale weak convergence method are obtained. Applications of the above results to the additive functionals and dynamical systems in discrete-time produce the above tree types of asymptotic results. Full article
(This article belongs to the Special Issue New Trends in Random Evolutions and Their Applications)
13 pages, 291 KiB  
Article
Stability Estimates for Finite-Dimensional Distributions of Time-Inhomogeneous Markov Chains
by Vitaliy Golomoziy and Yuliya Mishura
Mathematics 2020, 8(2), 174; https://doi.org/10.3390/math8020174 - 2 Feb 2020
Cited by 5 | Viewed by 1828
Abstract
This paper is devoted to the study of the stability of finite-dimensional distribution of time-inhomogeneous, discrete-time Markov chains on a general state space. The main result of the paper provides an estimate for the absolute difference of finite-dimensional distributions of a given time-inhomogeneous [...] Read more.
This paper is devoted to the study of the stability of finite-dimensional distribution of time-inhomogeneous, discrete-time Markov chains on a general state space. The main result of the paper provides an estimate for the absolute difference of finite-dimensional distributions of a given time-inhomogeneous Markov chain and its perturbed version. By perturbation, we mean here small changes in the transition probabilities. Stability estimates are obtained using the coupling method. Full article
(This article belongs to the Special Issue New Trends in Random Evolutions and Their Applications)
62 pages, 616 KiB  
Article
Inhomogeneous Random Evolutions: Limit Theorems and Financial Applications
by Nelson Vadori and Anatoliy Swishchuk
Mathematics 2019, 7(5), 447; https://doi.org/10.3390/math7050447 - 19 May 2019
Cited by 6 | Viewed by 2467
Abstract
The paper is devoted to the inhomogeneous random evolutions (IHRE) and their applications in finance. We introduce and present some properties of IHRE. Then, we prove weak law of large numbers and central limit theorems for IHRE. Financial applications are given to illiquidity [...] Read more.
The paper is devoted to the inhomogeneous random evolutions (IHRE) and their applications in finance. We introduce and present some properties of IHRE. Then, we prove weak law of large numbers and central limit theorems for IHRE. Financial applications are given to illiquidity modeling using regime-switching time-inhomogeneous Levy price dynamics, to regime-switching Levy driven diffusion based price dynamics, and to a generalized version of the multi-asset model of price impact from distress selling, for which we retrieve and generalize their diffusion limit result for the price process. Full article
(This article belongs to the Special Issue New Trends in Random Evolutions and Their Applications)
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