Strong Convergence in the Stochastic Averaging Principle

In this note we consider the almost sure convergence (as ϵ→0) of solution Xϵ(•), defined over the interval 0 ≤ τ ≤ 1, of the random ordinary differential equation Ẋϵ(τ) = F(Xϵ(τ), τ/ϵ) subject to Xϵ(0) = x0. Here {F(x, t, ω), t ≥ 0} is a strong mixing process for each x and (x, t) → F(x, t, ω) is subject to regularity conditions which ensure the existence of a unique solution over 0 ≤ τ ≤ 1 for all ϵ > 0. Under rather weak conditions it is shown that the function Xϵ(•, ω) converges a.s. to the solution x0(•) of a non-random averaged differential equation ẋ0(τ) = F(x0(τ)) subject to x0(0) = x0, the convergence being uniform over 0 ≤ τ ≤ 1.