Quasi-maximal likelihood estimator of the unknown parameter in system with ``physical'' white noise (Q2755168)

This article deals with estimation of the unknown parameter \(\theta_0\in \Theta\subset R^{k}\) in the Cauchy problem NEWLINE\[NEWLINEdX_{\epsilon}^{\theta_0}(t)/dt=a(t,X_{\epsilon}^{\theta_0}(t),\theta_0)+\sigma(t)\eta(t/\epsilon),\quad X_{\epsilon}^{\theta_0}(t)=x_0,NEWLINE\]NEWLINE where \(\eta(t),\;t\geq 0\), is a stationary process satisfying the uniformly strong mixing condition. The perturbations generated by such processes are called ``physical'' white noise. Let \(\mu_{\epsilon}^{\theta},\;mu_{\epsilon}^{\theta_0}\) be measures which generate solutions of the equations NEWLINE\[NEWLINEd\xi_{\epsilon}^{\theta}(t)=a(t,X_{0}^{\theta},\theta) dt+\sqrt{\epsilon}\sigma(t) dw(t),\quad \xi_{\epsilon}^{\theta}(0)=x_0,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\text{and}\quad d\xi_{\epsilon}^{\theta_0}(t)=a(t,X_{0}^{\theta_0},\theta_0) dt+\sqrt{\epsilon}\sigma(t) dw(t),\quad \xi_{\epsilon}^{\theta_0}(0)=x_0,NEWLINE\]NEWLINE respectively, where \(X_{0}^{\theta}\) is a solution to the problem \(dX_{0}^{\theta}(t)/dt=a(t,X_{0}^{\theta}(t),\theta)\), \(X_{0}^{\theta}(0)=x_0\), and \(w(t)\) is a Wiener process. Let \(\rho_{\epsilon}(X(\cdot),\theta,\theta_0)=d\mu_{\epsilon}^{\theta}/d\mu_{\epsilon}^{\theta_0}(X(\cdot))\). The estimator \(\theta_{\epsilon}\) for which NEWLINE\[NEWLINE\rho_{\epsilon}(X_{\epsilon}^{\theta_0}(\cdot),\theta_{\epsilon},\theta_0)=\max\limits_{\theta\in\Theta} \rho_{\epsilon}(X_{\epsilon}^{\theta_0}(\cdot),\theta,\theta_0)NEWLINE\]NEWLINE is called quasi-maximal likelihood estimator. The author obtains the inequality NEWLINE\[NEWLINEP\{\phi(\epsilon)|\theta_{\epsilon}-\theta_0|>R\}\leq C_1\exp\{-C_2R\}+p(\epsilon),\quad \text{where} \phi(\epsilon)\to+\infty,\;p(\epsilon)\to 0,\;\text{as} \epsilon\to 0.NEWLINE\]NEWLINE The functions \(\phi(\epsilon),\;p(\epsilon)\) are presented in explicit form.