Symmetry Analysis of the Stochastic Logistic Equation

We apply the recently developed theory of symmetry of stochastic differential equations to stochastic versions of the logistic equation; these may have environmental or demographical noise, or both—in which case we speak of the complete model. We study all these cases, both with constant and with non-constant noise amplitude, and show that the only one in which there are nontrivial symmetries is that of the stochastic logistic equation with (constant amplitude) environmental noise. In this case, the general theory of symmetry of stochastic differential equations is used to obtain an explicit integration, i.e., an explicit formula for the process in terms of any single realization of the driving Wiener process.

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Explicit Integration of Stiff Stochastic Differential Equations via an Efficient Implementation of Stochastic Computational Singular Perturbation

Communications in Computational Physics ◽

10.4208/cicp.oa-2018-0138 ◽

2019 ◽

Vol 25 (5) ◽

Author(s):

Lijin Wang ◽

Yanzhao Cao ◽

Sau-Hai Lam

Keyword(s):

Differential Equations ◽

Singular Perturbation ◽

Stochastic Differential Equations ◽

Efficient Implementation ◽

Explicit Integration ◽

Computational Singular Perturbation

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Numerical Study of Random Periodic Lipschitz Shadowing of Stochastic Differential Equations

Discrete Dynamics in Nature and Society ◽

10.1155/2018/1967508 ◽

2018 ◽

Vol 2018 ◽

pp. 1-10 ◽

Cited By ~ 1

Author(s):

Qingyi Zhan ◽

Xiangdong Xie

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Numerical Method ◽

Numerical Study ◽

Numerical Approach ◽

Logistic Equation ◽

Numerical Implementation ◽

Mean Square ◽

Numerical Computations

This paper is devoted to a new numerical approach for the possibility of(ω,Lδ)-periodic Lipschitz shadowing of a class of stochastic differential equations. The existence of(ω,Lδ)-periodic Lipschitz shadowing orbits and expression of shadowing distance are established. The numerical implementation approaches to the shadowing distance by the random Romberg algorithm are presented, and the convergence of this method is also proved to be mean-square. This ensures the feasibility of the numerical method. The practical use of these theorems and the associated algorithms is demonstrated in the numerical computations of the(ω,Lδ)-periodic Lipschitz shadowing orbits of the stochastic logistic equation.

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BSDEs with right upper-semicontinuous reflecting obstacle and stochastic Lipschitz coefficient

Random Operators and Stochastic Equations ◽

10.1515/rose-2019-2005 ◽

2019 ◽

Vol 27 (1) ◽

pp. 27-41 ◽

Cited By ~ 3

Author(s):

Mohamed Marzougue ◽

Mohamed El Otmani

Keyword(s):

General Theory ◽

Differential Equations ◽

Stochastic Differential Equations ◽

Optimal Stopping ◽

Existence And Uniqueness ◽

Backward Stochastic Differential Equations ◽

Upper Semicontinuous ◽

Optimal Stopping Theory ◽

Uniqueness Of A Solution

Abstract This paper proves the existence and uniqueness of a solution to reflected backward stochastic differential equations with a lower obstacle, which is assumed to be right upper-semicontinuous. The result is established where the coefficient is stochastic Lipschitz by using some tools from the general theory of processes such as Mertens decomposition of optional strong supermartingales and other tools from optimal stopping theory.

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Multi-parameter Singular Integrals. (AM-189)

10.23943/princeton/9780691162515.001.0001 ◽

2017 ◽

Author(s):

Brian Street

Keyword(s):

Partial Differential Equations ◽

Graduate Students ◽

General Theory ◽

Differential Equations ◽

Singular Integrals ◽

Main Tool ◽

Classical Theorem ◽

Quantitative Version ◽

Linear Partial Differential Equations

This book develops a new theory of multi-parameter singular integrals associated with Carnot–Carathéodory balls. The book first details the classical theory of Calderón–Zygmund singular integrals and applications to linear partial differential equations. It then outlines the theory of multi-parameter Carnot–Carathéodory geometry, where the main tool is a quantitative version of the classical theorem of Frobenius. The book then gives several examples of multi-parameter singular integrals arising naturally in various problems. The final chapter of the book develops a general theory of singular integrals that generalizes and unifies these examples. This is one of the first general theories of multi-parameter singular integrals that goes beyond the product theory of singular integrals and their analogs. This book will interest graduate students and researchers working in singular integrals and related fields.

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