A law of the iterated logarithm for stochastic processes defined by differential equations with a small parameter
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M.A. Kouritzin and A.J. Heunis, "A law of the iterated logarithm for stochastic processes defined by differential equations with a small parameter'', Annals of Probability, 22(2) (1994) pp. 659-679.
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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≤1E|Xϵ(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].
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http://purl.org/coar/resource_type/c_6501 http://purl.org/coar/version/c_970fb48d4fbd8a85
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en