An approximate Taylor method for Stochastic Functional Differential Equations via polynomial condition

Abstract The subject of this paper is an analytic approximate method for a class of stochastic functional differential equations with coefficients that do not necessarily satisfy the Lipschitz condition nor linear growth condition but they satisfy some polynomial conditions. Also, equations from the observed class have unique solutions with bounded moments. Approximate equations are defined on partitions of the time interval and their drift and diffusion coefficients are Taylor approximations of the coefficients of the initial equation. Taylor approximations require Fréchet derivatives since the coefficients of the initial equation are functionals. The main results of this paper are the Lp and almost sure convergence of the sequence of the approximate solutions to the exact solution of the initial equation. An example that illustrates the theoretical results and contains the proof of the existence, uniqueness and moment boundedness of the approximate solution is displayed.

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Asymptotic Behavior of Densities for Stochastic Functional Differential Equations

International Journal of Stochastic Analysis ◽

10.1155/2013/537023 ◽

2013 ◽

Vol 2013 ◽

pp. 1-17 ◽

Cited By ~ 2

Author(s):

Akihiro Kitagawa ◽

Atsushi Takeuchi

Keyword(s):

Differential Equations ◽

Functional Differential Equations ◽

Solution Process ◽

Time Interval ◽

Stochastic Functional Differential Equations ◽

Finite Time Interval ◽

Markovian Processes ◽

The Family ◽

Functional Differential ◽

Stochastic Functional

Consider stochastic functional differential equations depending on whole past histories in a finite time interval, which determine non-Markovian processes. Under the uniformly elliptic condition on the coefficients of the diffusion terms, the solution admits a smooth density with respect to the Lebesgue measure. In the present paper, we will study the large deviations for the family of the solution process and the asymptotic behaviors of the density. The Malliavin calculus plays a crucial role in our argument.

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Numerical Solutions of Stochastic Functional Differential Equations

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157000000425 ◽

2003 ◽

Vol 6 ◽

pp. 141-161 ◽

Cited By ~ 50

Author(s):

Xuerong Mao

Keyword(s):

Differential Equations ◽

Numerical Solutions ◽

Functional Differential Equations ◽

Convergence Theory ◽

Stochastic Functional Differential Equations ◽

True Solution ◽

Linear Growth Condition ◽

Mean Square Convergence ◽

Functional Differential ◽

Stochastic Functional

AbstractIn this paper, the strong mean square convergence theory is established for the numerical solutions of stochastic functional differential equations (SFDEs) under the local Lipschitz condition and the linear growth condition. These two conditions are generally imposed to guarantee the existence and uniqueness of the true solution, so the numerical results given here were obtained under quite general conditions.

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Stability of Nonlinear Neutral Stochastic Functional Differential Equations

Journal of Applied Mathematics ◽

10.1155/2010/425762 ◽

2010 ◽

Vol 2010 ◽

pp. 1-26 ◽

Cited By ~ 11

Author(s):

Minggao Xue ◽

Shaobo Zhou ◽

Shigeng Hu

Keyword(s):

Differential Equations ◽

Growth Condition ◽

Linear Growth ◽

Functional Differential Equations ◽

Stochastic Functional Differential Equations ◽

Linear Growth Condition ◽

Highly Nonlinear ◽

Functional Differential ◽

New Criteria ◽

Stochastic Functional

Neutral stochastic functional differential equations (NSFDEs) have recently been studied intensively. The well-known conditions imposed for the existence and uniqueness and exponential stability of the global solution are the local Lipschitz condition and the linear growth condition. Therefore, the existing results cannot be applied to many important nonlinear NSFDEs. The main aim of this paper is to remove the linear growth condition and establish a Khasminskii-type test for nonlinear NSFDEs. New criteria not only cover a wide class of highly nonlinear NSFDEs but they can also be verified much more easily than the classical criteria. Finally, several examples are given to illustrate main results.

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The Boundedness and Exponential Stability Criterions for Nonlinear Hybrid Neutral Stochastic Functional Differential Equations

Abstract and Applied Analysis ◽

10.1155/2013/138031 ◽

2013 ◽

Vol 2013 ◽

pp. 1-12 ◽

Cited By ~ 2

Author(s):

Xiaofeng Zong ◽

Fuke Wu ◽

Chengming Huang

Keyword(s):

Differential Equations ◽

Exponential Stability ◽

Linear Growth ◽

Functional Differential Equations ◽

Stochastic Functional Differential Equations ◽

Neutral Differential Equations ◽

Linear Growth Condition ◽

Functional Differential ◽

Theoretical Results ◽

Stochastic Functional

Neutral differential equations have been used to describe the systems that not only depend on the present and past states but also involve derivatives with delays. This paper considers hybrid nonlinear neutral stochastic functional differential equations (HNSFDEs) without the linear growth condition and examines the boundedness and exponential stability. Two illustrative examples are given to show the effectiveness of our theoretical results.

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Approximations of Numerical Method for Neutral Stochastic Functional Differential Equations with Markovian Switching

Journal of Applied Mathematics ◽

10.1155/2012/675651 ◽

2012 ◽

Vol 2012 ◽

pp. 1-32

Author(s):

Hua Yang ◽

Feng Jiang

Keyword(s):

Differential Equations ◽

Stochastic Systems ◽

Numerical Solutions ◽

Functional Differential Equations ◽

Approximate Solutions ◽

Markovian Switching ◽

Stochastic Functional Differential Equations ◽

Em Method ◽

Functional Differential ◽

Stochastic Functional

Stochastic systems with Markovian switching have been used in a variety of application areas, including biology, epidemiology, mechanics, economics, and finance. In this paper, we study the Euler-Maruyama (EM) method for neutral stochastic functional differential equations with Markovian switching. The main aim is to show that the numerical solutions will converge to the true solutions. Moreover, we obtain the convergence order of the approximate solutions.

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The Almost Sure Asymptotic Stability and Boundedness of Stochastic Functional Differential Equations with Polynomial Growth Condition

Abstract and Applied Analysis ◽

10.1155/2014/629426 ◽

2014 ◽

Vol 2014 ◽

pp. 1-11 ◽

Cited By ~ 1

Author(s):

Lichao Feng ◽

Shoumei Li

Keyword(s):

Asymptotic Stability ◽

Differential Equations ◽

Growth Condition ◽

Polynomial Growth ◽

Linear Growth ◽

Functional Differential Equations ◽

Stochastic Functional Differential Equations ◽

Linear Growth Condition ◽

Functional Differential ◽

Stochastic Functional

Stability and boundedness are two of the most important topics in the study of stochastic functional differential equations (SFDEs). This paper mainly discusses the almost sure asymptotic stability and the boundedness of nonlinear SFDEs satisfying the local Lipschitz condition but not the linear growth condition. Here we assume that the coefficients of SFDEs are polynomial or dominated by polynomial functions. We give sufficient criteria on the almost sure asymptotic stability and the boundedness for this kind of nonlinear SFDEs. Some nontrivial examples are provided to illustrate our results.

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Existence and Uniqueness of Solutions to Neutral Stochastic Functional Differential Equations with Poisson Jumps

Abstract and Applied Analysis ◽

10.1155/2012/371239 ◽

2012 ◽

Vol 2012 ◽

pp. 1-20 ◽

Cited By ~ 6

Author(s):

Jianguo Tan ◽

Hongli Wang ◽

Yongfeng Guo

Keyword(s):

Differential Equations ◽

Lipschitz Condition ◽

Existence And Uniqueness ◽

Functional Differential Equations ◽

Stochastic Functional Differential Equations ◽

Poisson Jumps ◽

Uniqueness Of Solutions ◽

Linear Growth Condition ◽

Functional Differential ◽

Stochastic Functional

A class of neutral stochastic functional differential equations with Poisson jumps (NSFDEwPJs),d[x(t)-G(xt)]=f(xt,t)dt+g(xt,t)dW(t)+h(xt,t)dN(t),t∈[t0,T], with initial valuext0=ξ={ξ(θ):-τ≤θ≤0}, is investigated. First, we consider the existence and uniqueness of solutions to NSFDEwPJs under the uniform Lipschitz condition, the linear growth condition, and the contractive mapping. Then, the uniform Lipschitz condition is replaced by the local Lipschitz condition, and the existence and uniqueness theorem for NSFDEwPJs is also derived.

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