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: 31 March 2024 | Viewed by 4496

Special Issue Editors

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
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

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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

Published Papers (6 papers)

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Research

15 pages, 312 KiB  
Article
A Matrix-Multiplicative Solution for Multi-Dimensional QBD Processes
Mathematics 2024, 12(3), 444; https://doi.org/10.3390/math12030444 - 30 Jan 2024
Viewed by 340
Abstract
We consider an irreducible positive-recurrent discrete-time Markov process on the state space X=+M×J, where + 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=+M×J, where + 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 nk 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
Mathematics 2024, 12(1), 67; https://doi.org/10.3390/math12010067 - 24 Dec 2023
Viewed by 526
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
Mathematics 2023, 11(21), 4521; https://doi.org/10.3390/math11214521 - 02 Nov 2023
Viewed by 739
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
Mathematics 2023, 11(20), 4265; https://doi.org/10.3390/math11204265 - 12 Oct 2023
Viewed by 444
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
Mathematics 2023, 11(13), 2919; https://doi.org/10.3390/math11132919 - 29 Jun 2023
Cited by 1 | Viewed by 644
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
Mathematics 2023, 11(9), 2118; https://doi.org/10.3390/math11092118 - 29 Apr 2023
Viewed by 964
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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