Delay estimation for some stationary diffusion-type processes (Q2742754)

The authors consider the following linear stochastic differential equation: NEWLINE\[NEWLINEdX_t = bX_{t-\theta} dt + dW_t,\quad 0\leq t \leq T,NEWLINE\]NEWLINE where \(b\) is a known negative parameter and the delay parameter \(\theta\) is unknown to the observer. The problem is to estimate \(\theta\) by the observations \(\{ X_t,\;0\leq t \leq T \}.\) Let \(\tilde{\theta}_T\) be a Bayesian estimator. To introduce \(\tilde{\theta}_T\), the authors suppose that \(\theta\) is a random variable with a prior density \(\pi(y)\). Then \(\theta_T = \int_\theta yp(y |X^T) dy\), where \(p(y |X^T)\) is the posterior density. Let \(\hat{\theta}_T\) be the maximum likelihood estimator. The following assertions are proved.NEWLINENEWLINENEWLINEFor any \(\theta_0 \in (0, (eb)^{-1})\) (under some assumptions): NEWLINE\[NEWLINE\lim_{\delta \to 0} \underset{T\to 0} {\underline\lim} \inf_{\hat\theta_T} \sup_{|\theta -\theta_0|< \delta} T^2 E_{\theta} (\theta -\theta_0)^2 \geq E\zeta^2,NEWLINE\]NEWLINE where NEWLINE\[NEWLINE\zeta = \bigl (\int_{-\infty}^\infty u Z(u) du\bigr)\bigl ( \int_{-\infty}^\infty Z(u) du\bigr)^{-1}, \quad Z(u) =\exp\{ bW_T (u) - 2^{-1} |u|b^2\}.NEWLINE\]NEWLINE For any \(p>0\), NEWLINE\[NEWLINE\lim_{T\to \infty} E|T(\tilde{\theta}_T - \theta)|^p = E \zeta^p.NEWLINE\]NEWLINE The authors write that the results presented in this paper show that in a certain sense the problem of delay estimation considered here is similar to disorder type problems, i.e., to problems of parameter estimation of trend coefficients in situations when the trend coefficient is a discontinuous function of the parameter.