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Mathematics
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Stochastic Processes and Random Fields
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Stochastic Processes and Random Fields

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

Deadline for manuscript submissions: closed (10 January 2022) | Viewed by 8581

Special Issue Editor

Prof. Dr. Leonid Piterbarg

E-Mail Website
Guest Editor
Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA
Interests: stochastic differential and partial differential equations and their applications; interpolation and extrapolation of random fields; physical processes in random media; median filtering of random processes and fields; estimating parameters in SDEs and SPDEs

Special Issue Information

Dear Colleagues,

Many practical problems in hydrodynamics, physical oceanography, physics of the atmosphere, and other areas of science are formulated in terms of stochastic differential equations (SDEs) or stochastic partial differential equations (SPDEs); for example, investigating passive scalar evolution in a random velocity field or modeling Lagrangian motion in stochastic flows. Not less important is the inverse problem: estimating parameters of velocity and dissipation under observations of a passive scalar or Lagrangian trajectories.

This Special Issue addresses forward and inverse problems in SDEs and SPDEs with possible applications in any area of science and engineering.

Prof. Dr. Leonid Piterbarg
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 Media
  • Stochastic Flows
  • Estimation in SDEs and SPDEs
  • Markov Random Fields

Published Papers (5 papers)

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Research

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19 pages, 759 KiB  
Article
Generalized Counting Processes in a Stochastic Environment
by Davide Cocco and Massimiliano Giona
Mathematics 2021, 9(20), 2573; https://0-doi-org.brum.beds.ac.uk/10.3390/math9202573 - 14 Oct 2021
Cited by 5 | Viewed by 2176
Abstract
This paper addresses the generalization of counting processes through the age formalism of Lévy Walks. Simple counting processes are introduced and their properties are analyzed: Poisson processes or fractional Poisson processes can be recovered as particular cases. The stationarity assumption in the renewal [...] Read more.
This paper addresses the generalization of counting processes through the age formalism of Lévy Walks. Simple counting processes are introduced and their properties are analyzed: Poisson processes or fractional Poisson processes can be recovered as particular cases. The stationarity assumption in the renewal mechanism characterizing simple counting processes can be modified in different ways, leading to the definition of generalized counting processes. In the case that the transition mechanism of a counting process depends on the environmental conditions—i.e., the parameters describing the occurrence of new events are themselves stochastic processes—the counting processes is said to be influenced by environmental stochasticity. The properties of this class of processes are analyzed, providing several examples and applications and showing the occurrence of new phenomena related to the modulation of the long-term scaling exponent by environmental noise. Full article
(This article belongs to the Special Issue Stochastic Processes and Random Fields)
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23 pages, 357 KiB  
Article
An Edgeworth Expansion for the Ratio of Two Functionals of Gaussian Fields and Optimal Berry–Esseen Bounds
by Yoon-Tae Kim and Hyun-Suk Park
Mathematics 2021, 9(18), 2223; https://0-doi-org.brum.beds.ac.uk/10.3390/math9182223 - 10 Sep 2021
Viewed by 1146
Abstract
This paper is concerned with the rate of convergence of the distribution of the sequence {Fn/Gn}, where Fn and Gn are each functionals of infinite-dimensional Gaussian fields. This form very frequently appears in the [...] Read more.
This paper is concerned with the rate of convergence of the distribution of the sequence {Fn/Gn}, where Fn and Gn are each functionals of infinite-dimensional Gaussian fields. This form very frequently appears in the estimation problem of parameters occurring in Stochastic Differential Equations (SDEs) and Stochastic Partial Differential Equations (SPDEs). We develop a new technique to compute the exact rate of convergence on the Kolmogorov distance for the normal approximation of Fn/Gn. As a tool for our work, an Edgeworth expansion for the distribution of Fn/Gn, with an explicitly expressed remainder, will be developed, and this remainder term will be controlled to obtain an optimal bound. As an application, we provide an optimal Berry–Esseen bound of the Maximum Likelihood Estimator (MLE) of an unknown parameter appearing in SDEs and SPDEs. Full article
(This article belongs to the Special Issue Stochastic Processes and Random Fields)
10 pages, 317 KiB  
Article
Cauchy Problem for a Stochastic Fractional Differential Equation with Caputo-Itô Derivative
by Jorge Sanchez-Ortiz, Omar U. Lopez-Cresencio, Francisco J. Ariza-Hernandez and Martin P. Arciga-Alejandre
Mathematics 2021, 9(13), 1479; https://0-doi-org.brum.beds.ac.uk/10.3390/math9131479 - 24 Jun 2021
Cited by 1 | Viewed by 1425
Abstract
In this note, we define an operator on a space of Itô processes, which we call Caputo-Itô derivative, then we considerer a Cauchy problem for a stochastic fractional differential equation with this derivative. We demonstrate the existence and uniqueness by a contraction mapping [...] Read more.
In this note, we define an operator on a space of Itô processes, which we call Caputo-Itô derivative, then we considerer a Cauchy problem for a stochastic fractional differential equation with this derivative. We demonstrate the existence and uniqueness by a contraction mapping argument and some examples are given. Full article
(This article belongs to the Special Issue Stochastic Processes and Random Fields)
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16 pages, 438 KiB  
Article
Caustic Frequency in 2D Stochastic Flows Modeling Turbulence
by Leonid I. Piterbarg
Mathematics 2021, 9(8), 797; https://0-doi-org.brum.beds.ac.uk/10.3390/math9080797 - 07 Apr 2021
Viewed by 1162
Abstract
Stochastic flows mimicking 2D turbulence in compressible media are considered. Particles driven by such flows can collide and we study the collision (caustic) frequency. Caustics occur when the Jacobian of a flow vanishes. First, a system of nonlinear stochastic differential equations involving the [...] Read more.
Stochastic flows mimicking 2D turbulence in compressible media are considered. Particles driven by such flows can collide and we study the collision (caustic) frequency. Caustics occur when the Jacobian of a flow vanishes. First, a system of nonlinear stochastic differential equations involving the Jacobian is derived and reduced to a smaller number of unknowns. Then, for special cases of the stochastic forcing, upper and lower bounds are found for the mean number of caustics as a function of Stokes number. The bounds yield an exact asymptotic for small Stokes numbers. The efficiency of the bounds is verified numerically. As auxiliary results we give rigorous proofs of the well known expressions for the caustic frequency and Lyapunov exponent in the one-dimensional model. Our findings may also be used for estimating the mean time when a 2D Riemann type partial differential equation with a stochastic forcing loses uniqueness of solutions. Full article
(This article belongs to the Special Issue Stochastic Processes and Random Fields)
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Review

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12 pages, 9242 KiB  
Review
Generation of Two Correlated Stationary Gaussian Processes
by Guo-Qiang Cai, Ronghua Huan and Weiqiu Zhu
Mathematics 2021, 9(21), 2687; https://0-doi-org.brum.beds.ac.uk/10.3390/math9212687 - 22 Oct 2021
Cited by 1 | Viewed by 1096
Abstract
Since correlated stochastic processes are often presented in practical problems, feasible methods to model and generate correlated processes appropriately are needed for analysis and simulation. The present paper systematically presents three methods to generate two correlated stationary Gaussian processes. They are (1) the [...] Read more.
Since correlated stochastic processes are often presented in practical problems, feasible methods to model and generate correlated processes appropriately are needed for analysis and simulation. The present paper systematically presents three methods to generate two correlated stationary Gaussian processes. They are (1) the method of linear filters, (2) the method of series expansion with random amplitudes, and (3) the method of series expansion with random phases. All three methods intend to match the power spectral density for each process but use information of different levels of correlation. The advantages and disadvantages of each method are discussed. Full article
(This article belongs to the Special Issue Stochastic Processes and Random Fields)
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