Theorem For Stochastic Differential Equations Research Articles

Almost Sure Comparisons for First Passage Times of Diffusion Processes through Boundaries

Conditions on the boundary and parameters that produce ordering in the first passage time distributions of two different diffusion processes are proved making use of comparison theorems for stochastic differential equations. Three applications of interest in stochastic modeling are presented: a sensitivity analysis for diffusion models characterized by means of first passage times, the comparison of different diffusion models where first passage times represent an important feature and the determination of upper and lower bounds for first passage time distributions.

STOCHASTIC EQUATIONS AND DIRICHLET OPERATORS ON INFINITE PRODUCT MANIFOLDS

We discuss elements of stochastic analysis on product manifolds (infinite products of compact Riemannian manifolds). We introduce differentiable structures on product manifolds and prove the existence and uniqueness theorem for stochastic differential equations on them. This result is applied to the construction of Glauber dynamics for classical lattice models with compact spin spaces.

Approximation theorem for stochastic differential equations with interaction1

Approximation theorem for stochastic differential equations with interaction1

Approximation theorem for stochastic differential equations with interaction

The approximation theorem for an infinite system of interacting stochastic differential equations is proved. This result is applied then for describing the support of the distribution of the solution of this system.

Stochastic analysis, rough path analysis and fractional Brownian motions

In this paper we show, by using dyadic approximations, the existence of a geometric rough path associated with a fractional Brownian motion with Hurst parameter greater than 1/4. Using the integral representation of fractional Brownian motions, we furthermore obtain a Skohorod integral representation of the geometric rough path we constructed. By the results in [Ly1], a stochastic integration theory may be established for fractional Brownian motions, and strong solutions and a Wong-Zakai type limit theorem for stochastic differential equations driven by fractional Brownian motions can be deduced accordingly. The method can actually be applied to a larger class of Gaussian processes with covariance functions satisfying a simple decay condition.

Quasi-sure analysis of two-parameter stochastic differential equations

We prove a Stroock-Varadhan type quasi-sure limit theorem for stochastic differential equations in the plane.

On Stochastic Differential Equations with Locally Unbounded Drift

We study the regularizing effect of the noise on differential equations with irregular coefficients. We present existence and uniqueness theorems for stochastic differential equations with locally unbounded drift.

The Stable Manifold Theorem for Stochastic Differential Equations

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 on Ruelle–Oseledec multiplicative ergodic theory.

On the integrability condition in the multiplicative ergodic theorem for stochastic differential equations

The multiplicative ergodic theorem is valid under an integrability condition on the linearized flow with respect to an invariant measure. We investigate the case were the flow is generated by a stochastic differential equation and give a criterion in terms of the vector fields and the (generally non-adapted) invariant measure assuring the validity of the integrability condition.

Weak limit theorems for stochastic differential equations driven by martingale measures

Let Mn be a sequence of martingale measures and let S'(Rd ) be the dual of Schwartz space. If in , we proved that it implies . Using this result, we shall extend a theorem by Kurtz and Protter. Let Yn be an S'(Rd )-valued process satisfying .We show the weak convergence of solutions of the sequence of equations to a solution of a limiting equation . We also consider more general stochastic differential equations with respect to Poisson point processes and show a result of propogation of chaos as an application of our results

Localization and mode conversion for elastic waves in randomly layered media II

This paper is Part II of a two part work in which we derive localization theory for elastic waves in plane-stratified media, a multimode problem complicated by the interconversion of shear and compressional waves, both in propagation and in backscatter. We consider the low frequency limit, i.e., when the randomness constitutes a microstructure. In Part I, we set up the general suite of problems and derived the probability density and moments for the fraction of reflected energy which remains in the same mode (shear or compressional) as the incident field. Our main mathematical tool was a limit theorem for stochastic differential equations with a small parameter. In this part we use the limit theorem of Part I and the Oseledec theorem, which establishes the existence of the localization length and other structural information, to compute: the localization length and another deterministic length, called the equilibration length, which gives the scale for equilibration of shear and compressional energy in propagation; and the probability density of the ratio of shear to compressional energy in transmission through a large slab. This last quantity is shown to be asymptotically independent of the incident field. We also extend the results to the small fluctuation, rather than the low frequency case.

Localization and mode conversion for elastic waves in randomly layered media I

This paper is Part I of a two-part work in which we derive localization theory for elastic waves in plane-stratified media, a multimode problem complicated by the interconversion of shear and compressional waves, both in propagation and in backscatter. We consider the low frequency limit, i.e., when the randomness constitutes a microstructure. In this part, we set up the general suite of problems and derive the probability density and moments for the fraction of reflected energy which remains in the same mode (shear or compressional) as the incident field. Our main mathematical tool is a limit theorem for stochastic differential equations with a small parameter. In Part II we will use the limit theorem of Part I and the Oseledec Theorem, which establishes the existence of the localization length and other structural information, to compute: the localization length and another deterministic length, called the equilibration length, which gives the scale for equilibration of shear and compressional energy in propagation; and the probability density of the ratio of shear to compressional energy in transmission through a large slab. This last quantity is shown to be asymptotically independent of the incident field. We also extend the results to the small fluctuation, rather than the low frequency case.

A uniform diffusion limit for random wave propagation with turning point

A random wave propagation problem with turning point is considered for a refractive, layered random medium. The variations of the medium structure are assumed to have two spatial scales; microscopic random fluctuations are superposed upon slowly varying macroscopic variations. An extension of a limit theorem for stochastic differential equations with multiple spatial scales is derived and proved to obtain a uniformly valid diffusion limit for random multiple scattering up to the turning point region. The scale dependence of the infinitesimal generator of the backward Kolmogorov equation provides an insight into the interplay of internal refraction and random scattering as one approaches the turning point.

Limit Theorems for Stochastic Differential Equations without After-Effect

The paper studies the limit behavior of a solution of a stochastic differential equation (SDE) without after-effect when the dependence of the equation coefficients on a parameter is nonregular and an unbounded growth in the parameter is accepted at some points of ${\bf R}^m $.

Approximation and support theorems in modulus spaces

We prove an approximation theorem for stochastic differential equations, under rather weak smoothness conditions on the coefficients, when the driving semimartingales are approximated by continuous semimartingales, in probability, and the solutions are considered in several Banach spaces, defined in terms of different types of the modulus of continuity. Hence Stroock-Varadhan's support theorem is obtained in these spaces, in particular, in appropriate Besov and Holder spaces.

Comparison theorems for stochastic differential equations in finite and infinite dimensions

We prove comparison theorems for systems of ordinary stochastic differential equations as well as for stochastic partial differential equations.

SINGULAR PERTURBATIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS

The main result is an analogue of Tikhonov's classical theorem for stochastic differential equations.

A note on comparison theorems for stochastic differential equations with respect to semimartingales

In this paper we employ Lyapunov-like function to establish some new comparison theorems for stochastic differential equations with different diffusion terms. Lyapunov-like function serves as a vehicle to transform a given complicated stochastic differential system into a relatively simpler system and therefore it is enough to investigate the properties of this simpler stochastic differential system. Our results generalize those of O'Brien [7] and Gal'cuk and Davis [5]. It is new to apply Lyapunov-like function to study the comparison theorems within the context of stochastic differential equations

Stability theorem for stochastic differential equations with jumps

Convergence in law of solutions of SDE having jumps is discussed assuming suitable convergence of the coefficients under a situation where the point process approaches a Poisson point process. As an application the asymptotic behavior of certain stochastic processes such as storage processes and random walks is also discussed.

A remark on Kunita's decomposition theorem

We use Michel Emery's stability theorem for stochastic differential equations to give a short proof for explicit solutions to linear stochastic differential equations over a solvable Lie group.