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I am studying two stochastic timeseries $X(t)$ and $Y(t)$, and I have reasons to believe that they are related by the following equation

$X(t) = Y(t-\Delta t) + dW(t)$

where $dW$ is the increment of a Wiener process, i.e. white gaussian additive noise, with a certain variance $\sigma^2$. I am trying to estimate $\Delta t$. In this case $\sigma^2$ is known, the system is stationary and ergodic, and the equation applies to detrended $X,Y$. Otherwise, I can also assume that the expected values are zero, i.e. $E(X)=E(Y)=0$.

I have some timeseries of $X,Y$ at my disposal, and have attempted the problem by estimating the covariance $C_{X,Y}(t)$ of $X$ and $Y$, but failed to observe a drop in the covariance $C_{X,Y}(\Delta t)$ as I was expecting.

Are there any established techniques to estimate the delay $\Delta t$? This seems to me a rather common problem, and I am struggling to find good references on this.

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