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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:
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