Abstract Sufficient Conditions for Large and Moderate Deviations in the Small Noise Limit

2019 ◽  
pp. 245-260
Author(s):  
Amarjit Budhiraja ◽  
Paul Dupuis
Keyword(s):  
Sufficient Conditions ◽  
Moderate Deviations ◽  
Small Noise ◽  
Noise Limit

On the rate of convergence of probabilities of moderate deviations

1970 ◽  
Vol 11 (1) ◽  
pp. 91-94 ◽  
Author(s):  
V. K. Rohatgi
Keyword(s):  
Sufficient Conditions ◽  
Moderate Deviations ◽  
Independent Random Variables ◽  
Standard Normal Distribution ◽  
Positive Real ◽  
Moment Conditions ◽  
Standard Normal ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Asymptotic Forms

Let {Xn: n ≧ 1} be a sequence of independent random variables and write Letand let . Suppose that converges in law to the standard normal distribution (see [5, 280] for necessary and sufficient conditions). Let {xn} be a monotonic sequence of positive real numbers such that xn → ∞ as n → ∞. Then as n → ∞ for all ε > 0. [6] Rubin and Sethuraman call probabilities of the form probabilities of moderate deviations and obtain asymptotic forms for such probabilities under appropriate moment conditions.

scholarly journals Convergence of invariant densities in the small-noise limit

Nonlinearity ◽  
2004 ◽  
Vol 18 (2) ◽  
pp. 659-683 ◽  
Author(s):  
Kevin K Lin
Keyword(s):  
Small Noise ◽  
Noise Limit ◽  
Invariant Densities

Multiple-objective risk-sensitive control and its small noise limit

Automatica ◽  
2003 ◽  
Vol 39 (3) ◽  
pp. 533-541 ◽  
Author(s):  
Andrew E.B. Lim ◽  
Xun Yu Zhou ◽  
John B. Moore
Keyword(s):  
Multiple Objective ◽  
Small Noise ◽  
Risk Sensitive ◽  
Objective Risk ◽  
Risk Sensitive Control ◽  
Noise Limit

scholarly journals Homogenization of Coupled Fast-Slow Systems via Intermediate Stochastic Regularization

2021 ◽  
Vol 183 (2) ◽  
Author(s):  
Maximilian Engel ◽  
Marios Antonios Gkogkas ◽  
Christian Kuehn
Keyword(s):  
Diffusion Coefficients ◽  
Sufficient Conditions ◽  
Analytical Techniques ◽  
Slow Variable ◽  
Small Time ◽  
Separation Parameter ◽  
Time Scale Separation ◽  
Weakly Coupled ◽  
Noise Limit ◽  
And Diffusion

AbstractIn this paper we study coupled fast-slow ordinary differential equations (ODEs) with small time scale separation parameter $$\varepsilon $$ ε such that, for every fixed value of the slow variable, the fast dynamics are sufficiently chaotic with ergodic invariant measure. Convergence of the slow process to the solution of a homogenized stochastic differential equation (SDE) in the limit $$\varepsilon $$ ε to zero, with explicit formulas for drift and diffusion coefficients, has so far only been obtained for the case that the fast dynamics evolve independently. In this paper we give sufficient conditions for the convergence of the first moments of the slow variable in the coupled case. Our proof is based upon a new method of stochastic regularization and functional-analytical techniques combined via a double limit procedure involving a zero-noise limit as well as considering $$\varepsilon $$ ε to zero. We also give exact formulas for the drift and diffusion coefficients for the limiting SDE. As a main application of our theory, we study weakly-coupled systems, where the coupling only occurs in lower time scales.

scholarly journals Impact of Correlated Noises on Additive Dynamical Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Chujin Li ◽  
Jinqiao Duan
Keyword(s):  
Dynamical Systems ◽  
White Noise ◽  
Probability Density ◽  
Probability Density Functions ◽  
Hurst Parameter ◽  
Random Solution ◽  
Small Noise ◽  
Fractional Brownian Motions ◽  
Noise Limit ◽  
Correlated Noises

Impact of correlated noises on dynamical systems is investigated by considering Fokker-Planck type equations under the fractional white noise measure, which correspond to stochastic differential equations driven by fractional Brownian motions with the Hurst parameterH>1/2. Firstly, by constructing the fractional white noise framework, one small noise limit theorem is proved, which provides an estimate for the deviation of random solution orbits from the corresponding deterministic orbits. Secondly, numerical experiments are conducted to examine the probability density evolutions of two special dynamical systems, as the Hurst parameterHvaries. Certain behaviors of the probability density functions are observed.

Sign in / Sign up

Export Citation Format

Share Document