Maximum likelihood estimator for the drift of a Brownian flow (Q2756665)

The author considers a temporally homogeneous Brownian flow \(U.\) A Brownian motion \(U\) takes values in \(C(\mathbb R^d \to \mathbb R^d)\), the continuous functions from \(\mathbb R^d\) to \(\mathbb R^d.\) \(U\) is defined as a stochastic process whose increments are Gaussian random fields on \(R^d\) and are independent. The distribution law of the Brownian motion is characterized by its drift and covariance function \(u_0(x)t=EU(x,t)\) and NEWLINE\[NEWLINECov(U^i(x,s),U^j(x,t)=a^{ij}(x,y)(s\wedge t),\quad i,j=1,\ldots,d,\quad x,y\in R^d,\quad s,t\in R^+.NEWLINE\]NEWLINE The author assumes that the covariance function is known. The problem is to estimate \(u_0.\) The estimator for \(u_0\) which maximizes the likelihood function is found. This estimator has the form \({\hat u}_0=\hat {\theta}\hat {\psi}\), where \(\hat {\psi}=(\psi_1,\psi_2,\ldots,\psi_M).\) The functions \(\psi_m\), \(m=1, \ldots, M\), are from an orthonormal basis. The author found the assumptions under which the following assertions hold true: NEWLINE\[NEWLINE\hat{\theta}(t){\buildrel P \over \longrightarrow}\theta^0,\quad (\hat{\theta}(t)-\theta^0)^\top A(t)(\hat{\theta}(t)-\theta^0){\buildrel d \over \longrightarrow} Y\quad\text{as}\quad t\to \infty,NEWLINE\]NEWLINE where \(Y\) is a random vector with \(\chi^2\) distribution with \(M\) degrees of freedom, and \(A(t)\) is an observed random process.