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The law of iterated logarithm for solution of stochastic equations with periodic coefficients. (English. Russian original) Zbl 0879.60024
Math. Notes 59, No. 5, 557-559 (1996); translation from Mat. Zametki 59, No. 5, 771-773 (1996).
The author considers the $$d$$-dimensional stochastic equation $\xi(t)=\int_0^tb(s,\xi(s)) ds+\int_0^t\sigma(s,\xi(s)) dw(s),\;\;t\geq 0,$ where $$w$$ is an $$m$$-dimensional standard Wiener process and a $$d$$-dimensional vector function $$b$$ and a $$d\times m$$-matrix function are periodic functions with respect to each of the variables. It is proved under appropriate conditions that $P\Biggl\{\limsup_{t\to\infty}{\xi_i(t)-\lambda_it\over \sqrt{2H_it\ln\ln t}}=1 \Biggr\}=1,\quad P\Biggl\{\liminf_{t\to\infty}{\xi_i(t)-\lambda_it\over \sqrt{2H_it\ln\ln t}}=-1 \Biggr\}=1,\quad i=1,...,d,$ with the same numbers $$H_i,\lambda_i$$.
MSC:
 60F15 Strong limit theorems 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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References:
 [1] A. V. Skorokhod,Random Processes With Independent Increments [in Russian], Nauka, Moscow (1964). · Zbl 0132.12504 [2] A. Fridman,Trans. Amer. Math. Soc.,170, No. 8, 359–384, (1972). · doi:10.1090/S0002-9947-1972-0378118-9 [3] A. Bensoussan, J. L. Lions, and G. Papanicolaou,Asymptotic Analysis for Periodic Structures, North Holland Publ., Amsterdam (1978). · Zbl 0404.35001 [4] I. I. Gikhman and A. V. Skorokhod,The Theory of Random Processes [in Russian], Vol. 3, Nauka, Moscow (1975). · Zbl 0348.60042
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