Strong convergence in averaging principle for stochastic hyperbolic–parabolic equations with two time-scales

Averaging principle for slow-fast stochastic differential equations with time dependent locally Lipschitz coefficients

In this paper, we are concerned with the stochastic averaging principle for stochastic differential equations (SDEs) with non-Lipschitz coefficients driven by fractional Brownian motion (fBm) of the Hurst parameter H ∈ (1 2, 1). We define the stochastic integrals with respect to the fBm in the integral formulation of the SDEs as pathwise integrals and we adopt the non-Lipschitz condition proposed by Taniguchi (1992) which is a much weaker condition with wider range of applications. The averaged SDEs are established. We then use their corresponding solutions to approximate the solutions of the original SDEs both in the sense of mean square and of probability. One can find that the similar asymptotic results are suitable for those non-Lipschitz SDEs with fBm under different types of stochastic integrals.

Stochastic averaging principle for differential equations with non-Lipschitz coefficients driven by fractional Brownian motion

https://cronfa.swan.ac.uk/Record/cronfa51701/Download/51701__15342__8ca6dc37fd9d48feba54001da03aa48f.pdf

Stochastic averaging for stochastic differential equations driven by fractional Brownian motion and standard Brownian motion

https://cronfa.swan.ac.uk/Record/cronfa32037/Download/32037__17861__8a75377a24f84f9a9d15cda877d5283b.pdf

Two-time-scales hyperbolic–parabolic equations driven by Poisson random measures: Existence, uniqueness and averaging principles

This paper investigates the stochastic averaging of slow-fast dynamical systems driven by fractional Brownian motion with the Hurst parameter $H$ in the interval $(\frac{1}{2},1)$. We establish an averaging principle by which the obtained simplified systems (the so-called averaged systems) will be applied to replace the original systems approximately through their solutions. Here, the solutions to averaged equations of slow variables which are unrelated to fast variables can converge to the solutions of slow variables to the original slow-fast dynamical systems in the sense of mean square. Therefore, the dimension reduction is realized since the solutions of uncoupled averaged equations can substitute that of coupled equations of the original slow-fast dynamical systems, namely, the asymptotic solutions dynamics will be obtained by the proposed stochastic averaging approach.

Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion

A limit theorem for the solutions of slow-fast systems with fractional Brownian motion

The approximate transient response of a nonlinear vibro-impact system under Gaussian white noise excitation is investigated by the methods of stochastic averaging and the Mellin transform. The Zhuravlev nonsmooth transformation is utilized to convert the nonlinear vibro-impact system into an equivalent system without velocity jumps by introducing an impulsive damping term. The Ito stochastic differential equation with respect to the amplitude response and the related Fokker–Plank–Kolmogorov (FPK) equation governing the amplitude response probability density of the vibro-impact system are derived with the stochastic averaging method. The Mellin transform is introduced to solve the FPK equation. The differential relations of complex fractional moments are obtained. The probability density function for transient response of this system constructed by solving a set of differential equations yields complex fractional moments. Two illustrative examples are examined to evaluate the effectiveness of the proposed solution procedure. The effects of restitution factors are investigated on the transient probability density distribution of the vibro-impact systems. At the same time, the convergence and the error analyses for different restitution factors are illustrated. Influences of the truncate term on the convergence and the error are further illustrated. The results obtained from the proposed procedure agree well with the results from Monte Carlo simulations.

Transient response of nonlinear vibro-impact system under Gaussian white noise excitation through complex fractional moments

The purpose of this article is to investigate an averaging principle for multi-valued stochastic differential equations (MSDEs) driven by Poisson point processes. The solutions to MSDEs driven by Poisson point processes can be approximated by solutions to averaged MSDEs in the sense of both convergence in mean square and convergence in probability. Finally, an example is presented to illustrate the averaging principle.

Rong Guo

Papers

Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion

A limit theorem for the solutions of slow-fast systems with fractional Brownian motion

Transient response of nonlinear vibro-impact system under Gaussian white noise excitation through complex fractional moments

Stochastic averaging principles for multi-valued stochastic differential equations driven by poisson point Processes