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Mathematics
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Stochastic Processes and Their Applications
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Stochastic Processes and Their Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 August 2022) | Viewed by 10173

Special Issue Editor

Prof. Dr. Xavier Bardina

E-Mail Website
Guest Editor
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193-Bellaterra, Spain
Interests: Stochastic processes; Weak convergence of Gaussian processes; Fractional Brownian motion; stochastic models for biological systems

Special Issue Information

Dear Colleagues,

The purpose of this Special Issue is to gather the latest contributions on the recent advances in the theory and applications of stochastic processes and stochastic models. You are kindly invited to contribute to this Special Issue on “Stochastic Processes and Their Applications” with an original research article or comprehensive review. The focus is mainly on the latest innovations in the field of stochastic theory and its practical applications in terms of concepts and techniques, and is oriented towards a broad spectrum of mathematical, scientific and engineering interests. Besides the main topics of stochastic analysis theory, the subjects of interests include applications to mathematical statistical physics, ergodic theory, mathematical biology, mathematical statistics, telecommunications modelling, inventories and dams, reliability, storage, queueing theory, mathematical finance, operations research and theoretical computer science.

Prof. Dr. Xavier Bardina
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

  • Stochastic processes
  • Stochastic models
  • Markov models
  • Weak convergence of processes
  • Approximation methods
  • Stochastic differential equations
  • Stochastic partial differential equations
  • Stochastic models for biological systems
  • Stochastic models applied to epidemiology.

Published Papers (5 papers)

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Research

15 pages, 366 KiB  
Article
Optimal Harvesting of Stochastically Fluctuating Populations Driven by a Generalized Logistic SDE Growth Model
by Nuno M. Brites
Mathematics 2022, 10(17), 3098; https://0-doi-org.brum.beds.ac.uk/10.3390/math10173098 - 29 Aug 2022
Viewed by 1176
Abstract
We describe the growth dynamics of a stock using stochastic differential equations with a generalized logistic growth model which encompasses several well-known growth functions as special cases. For each model, we compute the optimal variable effort policy and compare the expected net present [...] Read more.
We describe the growth dynamics of a stock using stochastic differential equations with a generalized logistic growth model which encompasses several well-known growth functions as special cases. For each model, we compute the optimal variable effort policy and compare the expected net present value of the total profit earned by the harvester among policies. In addition, we further extend the study to include parameters sensitivity, such as the costs and volatility, and present an explicitly Crank–Nicolson discretization scheme necessary to obtain optimal policies. Full article
(This article belongs to the Special Issue Stochastic Processes and Their Applications)
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32 pages, 507 KiB  
Article
State and Control Path-Dependent Stochastic Zero-Sum Differential Games: Viscosity Solutions of Path-Dependent Hamilton–Jacobi–Isaacs Equations
by Jun Moon
Mathematics 2022, 10(10), 1766; https://0-doi-org.brum.beds.ac.uk/10.3390/math10101766 - 22 May 2022
Cited by 1 | Viewed by 1652
Abstract
In this paper, we consider the two-player state and control path-dependent stochastic zero-sum differential game. In our problem setup, the state process, which is controlled by the players, is dependent on (current and past) paths of state and control processes of the players. [...] Read more.
In this paper, we consider the two-player state and control path-dependent stochastic zero-sum differential game. In our problem setup, the state process, which is controlled by the players, is dependent on (current and past) paths of state and control processes of the players. Furthermore, the running cost of the objective functional depends on both state and control paths of the players. We use the notion of non-anticipative strategies to define lower and upper value functionals of the game, where unlike the existing literature, these value functions are dependent on the initial states and control paths of the players. In the first main result of this paper, we prove that the (lower and upper) value functionals satisfy the dynamic programming principle (DPP), for which unlike the existing literature, the Skorohod metric is necessary to maintain the separability of càdlàg (state and control) spaces. We introduce the lower and upper Hamilton–Jacobi–Isaacs (HJI) equations from the DPP, which correspond to the state and control path-dependent nonlinear second-order partial differential equations. In the second main result of this paper, we show that by using the functional Itô calculus, the lower and upper value functionals are viscosity solutions of (lower and upper) state and control path-dependent HJI equations, where the notion of viscosity solutions is defined on a compact κ-Hölder space to use several important estimates and to guarantee the existence of minimum and maximum points between the (lower and upper) value functionals and the test functions. Based on these two main results, we also show that the Isaacs condition and the uniqueness of viscosity solutions imply the existence of the game value. Finally, we prove the uniqueness of classical solutions for the (state path-dependent) HJI equations in the state path-dependent case, where its proof requires establishing an equivalent classical solution structure as well as an appropriate contradiction argument. Full article
(This article belongs to the Special Issue Stochastic Processes and Their Applications)
16 pages, 1041 KiB  
Article
Optimal Stabilization of Linear Stochastic System with Statistically Uncertain Piecewise Constant Drift
by Andrey Borisov, Alexey Bosov and Gregory Miller
Mathematics 2022, 10(2), 184; https://0-doi-org.brum.beds.ac.uk/10.3390/math10020184 - 07 Jan 2022
Cited by 4 | Viewed by 1335
Abstract
The paper presents an optimal control problem for the partially observable stochastic differential system driven by an external Markov jump process. The available controlled observations are indirect and corrupted by some Wiener noise. The goal is to optimize a linear function of the [...] Read more.
The paper presents an optimal control problem for the partially observable stochastic differential system driven by an external Markov jump process. The available controlled observations are indirect and corrupted by some Wiener noise. The goal is to optimize a linear function of the state (output) given a general quadratic criterion. The separation principle, verified for the system at hand, allows examination of the control problem apart from the filter optimization. The solution to the latter problem is provided by the Wonham filter. The solution to the former control problem is obtained by formulating an equivalent control problem with a linear drift/nonlinear diffusion stochastic process and with complete information. This problem, in turn, is immediately solved by the application of the dynamic programming method. The applicability of the obtained theoretical results is illustrated by a numerical example, where an optimal amplification/stabilization problem is solved for an unstable externally controlled step-wise mechanical actuator. Full article
(This article belongs to the Special Issue Stochastic Processes and Their Applications)
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13 pages, 912 KiB  
Article
SiCaSMA: An Alternative Stochastic Description via Concatenation of Markov Processes for a Class of Catalytic Systems
by Vincent Wagner and Nicole Erika Radde
Mathematics 2021, 9(10), 1074; https://0-doi-org.brum.beds.ac.uk/10.3390/math9101074 - 11 May 2021
Cited by 2 | Viewed by 1616
Abstract
The Chemical Master Equation is a standard approach to model biochemical reaction networks. It consists of a system of linear differential equations, in which each state corresponds to a possible configuration of the reaction system, and the solution describes a time-dependent probability distribution [...] Read more.
The Chemical Master Equation is a standard approach to model biochemical reaction networks. It consists of a system of linear differential equations, in which each state corresponds to a possible configuration of the reaction system, and the solution describes a time-dependent probability distribution over all configurations. The Stochastic Simulation Algorithm (SSA) is a method to simulate sample paths from this stochastic process. Both approaches are only applicable for small systems, characterized by few reactions and small numbers of molecules. For larger systems, the CME is computationally intractable due to a large number of possible configurations, and the SSA suffers from large reaction propensities. In our study, we focus on catalytic reaction systems, in which substrates are converted by catalytic molecules. We present an alternative description of these systems, called SiCaSMA, in which the full system is subdivided into smaller subsystems with one catalyst molecule each. These single catalyst subsystems can be analyzed individually, and their solutions are concatenated to give the solution of the full system. We show the validity of our approach by applying it to two test-bed reaction systems, a reversible switch of a molecule and methyltransferase-mediated DNA methylation. Full article
(This article belongs to the Special Issue Stochastic Processes and Their Applications)
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17 pages, 487 KiB  
Article
On the Discretization of Continuous Probability Distributions Using a Probabilistic Rounding Mechanism
by Chénangnon Frédéric Tovissodé, Sèwanou Hermann Honfo, Jonas Têlé Doumatè and Romain Glèlè Kakaï
Mathematics 2021, 9(5), 555; https://0-doi-org.brum.beds.ac.uk/10.3390/math9050555 - 06 Mar 2021
Cited by 4 | Viewed by 2782
Abstract
Most existing flexible count distributions allow only approximate inference when used in a regression context. This work proposes a new framework to provide an exact and flexible alternative for modeling and simulating count data with various types of dispersion (equi-, under-, and over-dispersion). [...] Read more.
Most existing flexible count distributions allow only approximate inference when used in a regression context. This work proposes a new framework to provide an exact and flexible alternative for modeling and simulating count data with various types of dispersion (equi-, under-, and over-dispersion). The new method, referred to as “balanced discretization”, consists of discretizing continuous probability distributions while preserving expectations. It is easy to generate pseudo random variates from the resulting balanced discrete distribution since it has a simple stochastic representation (probabilistic rounding) in terms of the continuous distribution. For illustrative purposes, we develop the family of balanced discrete gamma distributions that can model equi-, under-, and over-dispersed count data. This family of count distributions is appropriate for building flexible count regression models because the expectation of the distribution has a simple expression in terms of the parameters of the distribution. Using the Jensen–Shannon divergence measure, we show that under the equidispersion restriction, the family of balanced discrete gamma distributions is similar to the Poisson distribution. Based on this, we conjecture that while covering all types of dispersions, a count regression model based on the balanced discrete gamma distribution will allow recovering a near Poisson distribution model fit when the data are Poisson distributed. Full article
(This article belongs to the Special Issue Stochastic Processes and Their Applications)
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