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A law of the iterated logarithm for stochastic processes defined by differential equations with a small parameter

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Consider the following random ordinary differential equation: X˙ϵ(τ)=F(Xϵ(τ),τ/ϵ,ω)subject toXϵ(0)=x0, where {F(x,t,ω),t≥0} are stochastic processes indexed by x in Rd, and the dependence on x is sufficiently regular to ensure that the equation has a unique solution Xϵ(τ,ω) over the interval 0≤τ≤1 for each ϵ>0. Under rather general conditions one can associate with the preceding equation a nonrandom averaged equation: x˙0(τ)=F¯¯¯(x0(τ))subject tox0(0)=x0, such that limϵ→0sup0≤τ≤1EXϵ(τ)−x0(τ)=0. In this article we show that as ϵ→0 the random function (Xϵ(⋅)−x0(⋅))/2ϵloglogϵ−1−−−−−−−−−−√ almost surely converges to and clusters throughout a compact set K of C[0,1].

 Date created
 1994

 Type of Item
 Article (Published)