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Mathematics
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Stochastic Processes: Theory, Simulation and Applications
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Stochastic Processes: Theory, Simulation and Applications

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

Deadline for manuscript submissions: closed (31 August 2024) | Viewed by 7691

Special Issue Editors

Prof. Dr. Antonio Di Crescenzo

E-Mail Website
Guest Editor
Department of Mathematics, University of Salerno, I-84100 Salerno, Italy
Interests: stochastic processes; applied probability; probability theory; stochastic models
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Virginia Giorno

E-Mail Website
Guest Editor
Dipartimento di Informatica/DI, University of Salerno, Salerno, Italy
Interests: diffusion processes for growth phenomena; theoretical studies on Markov and Gaussian processes; models to describe neuronal systems; first-passage-times
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

We are very pleased to invite you to contribute to this Special Issue of Mathematics, which is dedicated to collecting innovative results.

Contributions on the theory and simulation of stochastic processes and their applications are very welcome. The focus is also oriented toward, but not limited to, the design and analysis of probabilistic models, Markov and Gaussian stochastic processes, computational methods, first-passage-time problems, and applications to biomathematical modeling and queueing systems. Attention is also given to problems of both theoretical and computational nature related to the following themes: 

  • Probabilistic models for neuronal systems;
  • Adaptive service systems;
  • Population growth models in random environments;
  • Algorithms for the evaluation of first-crossing probability densities through suitable boundaries;
  • Asymptotic behavior of probability densities for Markov and Gauss processes.

Prof. Dr. Antonio Di Crescenzo
Prof. Dr. Virginia Giorno
Guest Editors

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

  • birth–death processes
  • computational methods for stochastic models
  • diffusion processes
  • first-passage-time problems
  • gauss–Markov processes
  • Markov chains
  • neuronal modeling
  • population dynamics
  • probability theory
  • queueing systems
  • random walks
  • simulation of stochastic processes
  • stochastic models in biology
  • stochastic processes and applications

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Published Papers (7 papers)

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Research

17 pages, 731 KiB  
Article
New Computer Experiment Designs with Area-Interaction Point Processes
by Ahmed Ait Ameur, Hichem Elmossaoui and Nadia Oukid
Mathematics 2024, 12(15), 2397; https://doi.org/10.3390/math12152397 - 31 Jul 2024
Viewed by 548
Abstract
This article presents a novel method for constructing computer experiment designs based on the theory of area-interaction point processes. This method is essential for capturing the interactions between different elements within a modeled system, offering a more flexible and adaptable approach compared with [...] Read more.
This article presents a novel method for constructing computer experiment designs based on the theory of area-interaction point processes. This method is essential for capturing the interactions between different elements within a modeled system, offering a more flexible and adaptable approach compared with traditional mathematical modeling. Unlike conventional rough models that rely on simplified equations, our method employs the Markov Chain Monte Carlo (MCMC) method and the Metropolis–Hastings algorithm combined with Voronoi tessellations. It uses a new dynamic called homogeneous birth and death dynamics of a set of points to generate the designs. This approach does not require the development of specific mathematical models for each system under study, making it universally applicable while achieving comparable results. Furthermore, we provide an in-depth analysis of the convergence properties of the Markov Chain to ensure the reliability of the generated designs. An expanded literature review situates our work within the context of existing research, highlighting its unique contributions and advancements. A comparison between our approach and other existing computer experiment designs has been performed. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory, Simulation and Applications)
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15 pages, 312 KiB  
Article
A Matrix-Multiplicative Solution for Multi-Dimensional QBD Processes
by Valeriy Naumov
Mathematics 2024, 12(3), 444; https://doi.org/10.3390/math12030444 - 30 Jan 2024
Viewed by 706
Abstract
We consider an irreducible positive-recurrent discrete-time Markov process on the state space X=ℤ+M×J, where ℤ+ is the set of non-negative integers and J={1,2,…,n}. The [...] Read more.
We consider an irreducible positive-recurrent discrete-time Markov process on the state space X=ℤ+M×J, where ℤ+ is the set of non-negative integers and J={1,2,…,n}. The number of states in J may be either finite or infinite. We assume that the process is a homogeneous quasi-birth-and-death process (QBD). It means that the one-step transition probability between non-boundary states (k,i) and (n,j) may depend on i,j, and n−k but not on the specific values of k and n. It is shown that the stationary probability vector of the process is expressed through square matrices of order n, which are the minimal non-negative solutions to nonlinear matrix equations. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory, Simulation and Applications)
15 pages, 651 KiB  
Article
Wasserstein Dissimilarity for Copula-Based Clustering of Time Series with Spatial Information
by Alessia Benevento and Fabrizio Durante
Mathematics 2024, 12(1), 67; https://doi.org/10.3390/math12010067 - 24 Dec 2023
Cited by 1 | Viewed by 995
Abstract
The clustering of time series with geo-referenced data requires a suitable dissimilarity matrix interpreting the comovements of the time series and taking into account the spatial constraints. In this paper, we propose a new way to compute the dissimilarity matrix, merging both types [...] Read more.
The clustering of time series with geo-referenced data requires a suitable dissimilarity matrix interpreting the comovements of the time series and taking into account the spatial constraints. In this paper, we propose a new way to compute the dissimilarity matrix, merging both types of information, which leverages on the Wasserstein distance. We then make a quasi-Gaussian assumption that yields more convenient formulas in terms of the joint correlation matrix. The method is illustrated in a case study involving climatological data. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory, Simulation and Applications)
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31 pages, 1302 KiB  
Article
Time-Inhomogeneous Finite Birth Processes with Applications in Epidemic Models
by Virginia Giorno and Amelia G. Nobile
Mathematics 2023, 11(21), 4521; https://doi.org/10.3390/math11214521 - 2 Nov 2023
Cited by 1 | Viewed by 1078
Abstract
We consider the evolution of a finite population constituted by susceptible and infectious individuals and compare several time-inhomogeneous deterministic models with their stochastic counterpart based on finite birth processes. For these processes, we determine the explicit expressions of the transition probabilities and of [...] Read more.
We consider the evolution of a finite population constituted by susceptible and infectious individuals and compare several time-inhomogeneous deterministic models with their stochastic counterpart based on finite birth processes. For these processes, we determine the explicit expressions of the transition probabilities and of the first-passage time densities. For time-homogeneous finite birth processes, the behavior of the mean and the variance of the first-passage time density is also analyzed. Moreover, the approximate duration until the entire population is infected is obtained for a large population size. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory, Simulation and Applications)
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12 pages, 1109 KiB  
Article
Numerical Computation of Distributions in Finite-State Inhomogeneous Continuous Time Markov Chains, Based on Ergodicity Bounds and Piecewise Constant Approximation
by Yacov Satin, Rostislav Razumchik, Ilya Usov and Alexander Zeifman
Mathematics 2023, 11(20), 4265; https://doi.org/10.3390/math11204265 - 12 Oct 2023
Viewed by 740
Abstract
In this paper it is shown, that if a possibly inhomogeneous Markov chain with continuous time and finite state space is weakly ergodic and all the entries of its intensity matrix are locally integrable, then, using available results from the perturbation theory, its [...] Read more.
In this paper it is shown, that if a possibly inhomogeneous Markov chain with continuous time and finite state space is weakly ergodic and all the entries of its intensity matrix are locally integrable, then, using available results from the perturbation theory, its time-dependent probability characteristics can be approximately obtained from another Markov chain, having piecewise constant intensities and the same state space. The approximation error (the taxicab distance between the state probability distributions) is provided. It is shown how the Cauchy operator and the state probability distribution for an arbitrary initial condition can be calculated. The findings are illustrated with the numerical examples. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory, Simulation and Applications)
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14 pages, 368 KiB  
Article
Optimizing Air Pollution Modeling with a Highly-Convergent Quasi-Monte Carlo Method: A Case Study on the UNI-DEM Framework
by Venelin Todorov, Slavi Georgiev, Ivan Georgiev, Snezhinka Zaharieva and Ivan Dimov
Mathematics 2023, 11(13), 2919; https://doi.org/10.3390/math11132919 - 29 Jun 2023
Cited by 2 | Viewed by 1032
Abstract
In this study, we present the development of an advanced air pollution modeling approach, which incorporates cutting-edge stochastic techniques for large-scale simulations of long-range air pollutant transportation. The Unified Danish Eulerian Model (UNI-DEM) serves as a crucial mathematical framework with numerous applications in [...] Read more.
In this study, we present the development of an advanced air pollution modeling approach, which incorporates cutting-edge stochastic techniques for large-scale simulations of long-range air pollutant transportation. The Unified Danish Eulerian Model (UNI-DEM) serves as a crucial mathematical framework with numerous applications in studies concerning the detrimental effects of heightened air pollution levels. We employ the UNI-DEM model in our research to obtain trustworthy insights into critical questions pertaining to environmental preservation. Our proposed methodology is a highly convergent quasi-Monte Carlo technique that relies on a unique symmetrization lattice rule. By fusing the concepts of special functions and optimal generating vectors, we create a novel algorithm grounded in the component-by-component construction method, which has been recently introduced. This amalgamation yields particularly impressive outcomes for lower-dimensional cases, substantially enhancing the performance of the most advanced existing methods for calculating the Sobol sensitivity indices of the UNI-DEM model. This improvement is vital, as these indices form an essential component of the digital ecosystem for environmental analysis. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory, Simulation and Applications)
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13 pages, 1383 KiB  
Article
Solving Stochastic Nonlinear Poisson-Boltzmann Equations Using a Collocation Method Based on RBFs
by Samaneh Mokhtari, Ali Mesforush, Reza Mokhtari, Rahman Akbari and Clemens Heitzinger
Mathematics 2023, 11(9), 2118; https://doi.org/10.3390/math11092118 - 29 Apr 2023
Viewed by 1339
Abstract
In this paper, we present a numerical scheme based on a collocation method to solve stochastic non-linear Poisson–Boltzmann equations (PBE). This equation is a generalized version of the non-linear Poisson–Boltzmann equations arising from a form of biomolecular modeling to the stochastic case. Applying [...] Read more.
In this paper, we present a numerical scheme based on a collocation method to solve stochastic non-linear Poisson–Boltzmann equations (PBE). This equation is a generalized version of the non-linear Poisson–Boltzmann equations arising from a form of biomolecular modeling to the stochastic case. Applying the collocation method based on radial basis functions (RBFs) allows us to deal with the difficulties arising from the complexity of the domain. To indicate the accuracy of the RBF method, we present numerical results for two-dimensional models, we also study the stability of this method numerically. We examine our results with the RBF-reference value and the Chebyshev Spectral Collocation (CSC) method. Furthermore, we discuss finding the appropriate shape parameter to obtain an accurate numerical solution besides greatest stability. We have exerted the Newton–Raphson approach for solving the system of non-linear equations resulting from discretization by the RBF technique. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory, Simulation and Applications)
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