An inequality for the probability of large deviations of an estimate of an unknown parameter in the Cauchy problem for a first-order partial differential equation with rapid random oscillations (Q2737018)

Let us consider the Cauchy problem in the band \([0,T]\times R^1\): NEWLINE\[NEWLINE\partial X^{\theta_{0}}_{\epsilon}/\partial t+ a(t,x)\partial X^{\theta_{0}}_{\epsilon}/\partial x= f(t,x,X^{\theta_{0}}_{\epsilon}(t,x),\theta_{0})+\sigma(t,x)\eta(t/\epsilon),\quad X^{\theta_{0}}_{\epsilon}(t,x)\mid_{t=0}=\psi(x),NEWLINE\]NEWLINE where \(\eta(t)\), \(t\geq 0,\) is a stationary stochastic process with zero mean which satisfies the uniformly strong mixing condition with mixing coefficient \(\phi(t),\) \(\epsilon>0\) is a small parameter, \(\theta_{0}\in \Theta\subset R^1\) is the unknown parameter to be estimated by observation of a realization of the solution \(X\) of the equation. An exponential inequality for the probability of large deviations for the quasimaximum likelihood estimation \(\theta_{\epsilon}\) of the value of the parameter \(\theta_{0}\) is constructed. The method for construction of the estimators was given by \textit{I.A. Ibragimov} and \textit{R.Z. Khas'minskij} [Statistical estimation. Asymptotic theory. (1981; Zbl 0467.62026)].