Open Access. Powered by Scholars. Published by Universities.®
- Keyword
-
- Malliavin calculus (3)
- stochastic evolution equation (see) (2)
- Local characteristics (2)
- Lyapunov exponents (2)
- Stochastic flow (2)
-
- Stochastic semiflow (2)
- (perfect) cocycle (1)
- stochastic differential equation (SDE) (1)
- stochastic differential equation (s.d.e.) (1)
- <em>C</em><sup><em>k</em></sup> (1)
- <em>C<sup>k</sup></em> cocycle (1)
- anticipating stochastic partial differential equation (spde) (1)
- stochastic partial differential equation (spde) (1)
- Anticipating calculus (1)
- Asymptotic invariance (1)
- Brownian motion (1)
- Chen-Souriau calculus (1)
- Cocycle (1)
- Delay equation (1)
- Exponential growth rate (1)
- Hereditary delay systems (1)
- Hyperbolic stationary trajectory (1)
- Hyperbolicity (1)
- Itô’s formula (1)
- Local stable (unstable) manifolds (1)
- Local stable/unstable manifolds (1)
- Milstein scheme (1)
- Multiplicative ergodic theorem (1)
- Oseledec spectrum (1)
- Poisson process (1)
Articles 1 - 9 of 9
Full-Text Articles in Probability
The Weak Euler Scheme For Stochastic Delay Equations, Evelyn Buckwar, Rachel Kuske, Salah-Eldin A. Mohammed, Tony Shardlow
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.
The Substitution Theorem For Semilinear Stochastic Partial Differential Equations, Salah-Eldin A. Mohammed, Tusheng Zhang
The Substitution Theorem For Semilinear Stochastic Partial Differential Equations, Salah-Eldin A. Mohammed, Tusheng Zhang
Articles and Preprints
In this article we establish a substitution theorem for semilinear stochastic evolution equations (see's) depending on the initial condition as an infinite-dimensional parameter. Due to the infinitedimensionality of the initial conditions and of the stochastic dynamics, existing finite-dimensional results do not apply. The substitution theorem is proved using Malliavin calculus techniques together with new estimates on the underlying stochastic semiflow. Applications of the theorem include dynamic characterizations of solutions of stochastic partial differential equations (spde's) with anticipating initial conditions and non-ergodic stationary solutions. In particular, our result gives a new existence theorem for solutions of semilinear Stratonovich spde's with anticipating …
The Stable Manifold Theorem For Semilinear Stochastic Evolution Equations And Stochastic Partial Differential Equations, Salah-Eldin A. Mohammed, Tusheng Zhang, Huaizhong Zhao
The Stable Manifold Theorem For Semilinear Stochastic Evolution Equations And Stochastic Partial Differential Equations, Salah-Eldin A. Mohammed, Tusheng Zhang, Huaizhong Zhao
Articles and Preprints
The main objective of this paper is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see’s) and stochastic partial differential equations (spde’s) near stationary solutions. Such characterization is realized through the long-term behavior of the solution field near stationary points. The analysis falls in two parts 1, 2.
In Part 1, we prove general existence and compactness theorems for Ck-cocycles of semilinear see’s and spde’s. Our results cover a large class of semilinear see’s as well as certain semilinear spde’s with Lipschitz and non-Lipschitz terms such as stochastic reaction diffusion equations and the …
Discrete-Time Approximations Of Stochastic Delay Equations: The Milstein Scheme, Yaozhong Hu, Salah-Eldin A. Mohammed, Feng Yan
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.
Stochastic Functional Differential Equations On Manifolds, Rémi Léandre, Salah-Eldin A. Mohammed
Stochastic Functional Differential Equations On Manifolds, Rémi Léandre, Salah-Eldin A. Mohammed
Articles and Preprints
In this paper, we study stochastic functional differential equations (sfde's) whose solutions are constrained to live on a smooth compact Riemannian manifold. We prove the existence and uniqueness of solutions to such sfde's. We consider examples of geometrical sfde's and establish the smooth dependence of the solution on finite-dimensional parameters.
The Stable Manifold Theorem For Stochastic Differential Equations, Salah-Eldin A. Mohammed, Michael K. R. Scheutzow
The Stable Manifold Theorem For Stochastic Differential Equations, Salah-Eldin A. Mohammed, Michael K. R. Scheutzow
Articles and Preprints
We formulate and prove a local stable manifold theorem for stochastic differential equations (SDEs) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and Itô-type equations are treated. Starting with the existence of a stochastic flow for a SDE, we introduce the notion of a hyperbolic stationary trajectory. We prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution. For Stratonovich SDEs, the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating SDE. The proof of the stable manifold theorem is based …
Spatial Estimates For Stochastic Flows In Euclidean Space, Salah-Eldin A. Mohammed, Michael K. R. Scheutzow
Spatial Estimates For Stochastic Flows In Euclidean Space, Salah-Eldin A. Mohammed, Michael K. R. Scheutzow
Articles and Preprints
We study the behavior for large |x| of Kunita-type stochastic flows φ(t, ω, x) on Rd, driven by continuous spatial semimartingales. For this class of flows we prove new spatial estimates for large |x|, under very mild regularity conditions on the driving semimartingale random field. It is expected that the results would be of interest for the theory of stochastic flows on noncompact manifolds as well as in the study of nonlinear filtering, stochastic functional and partial differential equations. Some examples and counterexamples are given.
Lyapunov Exponents Of Linear Stochastic Functional-Differential Equations. Ii. Examples And Case Studies, Salah-Eldin A. Mohammed, Michael K. R. Scheutzow
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 …
Smooth Densities For Degenerate Stochastic Delay Equations With Hereditary Drift, Denis R. Bell, Salah-Eldin A. Mohammed
Smooth Densities For Degenerate Stochastic Delay Equations With Hereditary Drift, Denis R. Bell, Salah-Eldin A. Mohammed
Articles and Preprints
We establish the existence of smooth densities for solutions of Rd-valued stochastic hereditary differential systems of the form
dx(t) = H(t,x)dt + g(t, x(t - r))dW(t).
In the above equation, W is an n-dimensional Wiener process, r is a positive time delay, H is a nonanticipating functional defined on the space of paths in Rd and g is an n x d matrix-valued function defined on [0, ∞) x Rd, such that gg* has …