On an estimate of the convergence rate (Q2777848)

The author investigates properties of solutions of the following Cauchy problems NEWLINE\[NEWLINE\frac{dX_{\varepsilon}}{dt}=a(t,X_{\varepsilon}(t))+ \sigma(t,X_{\varepsilon}(t))\eta(t/\varepsilon), \quad X_{\varepsilon}(0)=x_0; NEWLINE\]NEWLINE NEWLINE\[NEWLINE\frac{dY_{\varepsilon}}{dt}=a(t,Y_{\varepsilon}(t))+ \sigma(t,X_0(t))\eta(t/\varepsilon), \quad Y_{\varepsilon}(0)=x_0, NEWLINE\]NEWLINE where \(X_0(t)\) is a solution of the problem \(dX_0/dt=a(t,X_0(t))\), \(X_0(0)=x_0\) and \(\eta(t)\) is a strictly stationary stochastic process with zero mean, and the corresponding equations NEWLINE\[NEWLINEdZ_{\varepsilon}(t)=a(t,Z_{\varepsilon}(t)) dt+ \sqrt{\varepsilon}\sigma(t,X_0(t)) dw_{\varepsilon}(t), \quad Z_{\varepsilon}(0)=x_0; NEWLINE\]NEWLINE NEWLINE\[NEWLINEd\psi_{\varepsilon}(t)=a(t,X_0(t)) dt+ \sqrt{\varepsilon}\sigma(t,X_0(t)) dw_{\varepsilon}(t), \quad \psi_{\varepsilon}(0)=x_0. NEWLINE\]NEWLINE Estimates for probabilities of deviations of solutions to these equations: \(P\{\sup_{0\leq t\leq T} |X_{\varepsilon}(t)-Z_{\varepsilon}(t)|/\sqrt{\varepsilon}>R\};\) \(P\{\sup_{0\leq t\leq T} |X_{\varepsilon}(t)-\psi_{\varepsilon}(t)|/\sqrt{\varepsilon}>R\};\) \(P\{\sup_{0\leq t\leq T} |X_{\varepsilon}(t)-Y_{\varepsilon}(t)|/\sqrt{\varepsilon}>R\}\) are proposed under some conditions on the coefficients \(a(t,x),\sigma(t,x)\).