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Ordinary Differential Equations and Applied Dynamics Commons™

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Southern Illinois University Carbondale

Articles 1 - 4 of 4

Full-Text Articles in Ordinary Differential Equations and Applied Dynamics

The Weak Euler Scheme For Stochastic Delay Equations, Evelyn Buckwar, Rachel Kuske, Salah-Eldin A. Mohammed, Tony Shardlow May 2008

The Weak Euler Scheme For Stochastic Delay Equations, Evelyn Buckwar, Rachel Kuske, Salah-Eldin A. Mohammed, Tony Shardlow

Articles and Preprints

We study weak convergence of an Euler scheme for non-linear stochastic delay differential equations (SDDEs) driven by multidimensional Brownian motion. The Euler scheme has weak order of convergence 1, as in the case of stochastic ordinary differential equations (SODEs) (i.e., without delay). The result holds for SDDEs with multiple finite fixed delays in the drift and diffusion terms. Although the set-up is non-anticipating, our approach uses the Malliavin calculus and the anticipating stochastic analysis techniques of Nualart and Pardoux.

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Hartman-Grobman Theorems Along Hyperbolic Stationary Trajectories, Edson A. Coayla-Teran, Salah-Eldin A. Mohammed, Paulo Régis C. Ruffino Feb 2007

Hartman-Grobman Theorems Along Hyperbolic Stationary Trajectories, Edson A. Coayla-Teran, Salah-Eldin A. Mohammed, Paulo Régis C. Ruffino

Articles and Preprints

We extend the Hartman-Grobman theorems on discrete random dynamical systems (RDS), proved in [7], in two directions: For continuous RDS and for hyperbolic stationary trajectories. In this last case there exists a conjugacy between traveling neighbourhoods of trajectories and neighbourhoods of the origin in the corresponding tangent bundle. We present applications to deterministic dynamical systems.

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Discrete-Time Approximations Of Stochastic Delay Equations: The Milstein Scheme, Yaozhong Hu, Salah-Eldin A. Mohammed, Feng Yan Jan 2004

Discrete-Time Approximations Of Stochastic Delay Equations: The Milstein Scheme, Yaozhong Hu, Salah-Eldin A. Mohammed, Feng Yan

Articles and Preprints

In this paper, we develop a strong Milstein approximation scheme for solving stochastic delay differential equations (SDDE's). The scheme has convergence order 1. In order to establish the scheme, we prove an infinite-dimensional Itô formula for "tame" functions acting on the segment process of the solution of an SDDE. It is interesting to note that the presence of the memory in the SDDE requires the use of the Malliavin calculus and the anticipating stochastic analysis of Nualart and Pardoux. Given the non-anticipating nature of the SDDE, the use of anticipating calculus methods appears to be novel.

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Lyapunov Exponents Of Linear Stochastic Functional-Differential Equations. Ii. Examples And Case Studies, Salah-Eldin A. Mohammed, Michael K. R. Scheutzow Jan 1997

Lyapunov Exponents Of Linear Stochastic Functional-Differential Equations. Ii. Examples And Case Studies, Salah-Eldin A. Mohammed, Michael K. R. Scheutzow

Articles and Preprints

We give several examples and examine case studies of linear stochastic functional differential equations. The examples fall into two broad classes: regular and singular, according to whether an underlying stochastic semi-flow exists or not. In the singular case, we obtain upper and lower bounds on the maximal exponential growth rate $\overlineλ1$(σ) of the trajectories expressed in terms of the noise variance σ . Roughly speaking we show that for small σ, $\overlineλ1$(σ) behaves like -σ2 /2, while for large σ, it grows like logσ. In the regular case, it is shown that a discrete Oseledec …

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