The law of iterated logarithm for solution of stochastic equations with periodic coefficients (Q1360807)

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 \(H_i,\lambda_i\).