I have a set of Langevin equations given by

$${\mathbf{\dot{x}}} = \mathbf{-Q \,x} + \mathbf{\eta} \tag{1}$$

where $\eta$ is white Gaussian noise and $Q$ is not a function of $x$.

Using Euler's method for SDE, I generated a time series of $\bf{x}$, which (as expected) is noisy due to $\eta$. Using the time series of $\bf{x}$, generating $\langle(\bf{-Qx} + \bf{\eta})(\bf{-Qx} + \bf{\eta)^T}\rangle$ gave me a finite matrix.

However, for the case where I don't know $\bf{Q}$ and $\eta$, I need to recover $\bf\dot{x}$ from the noisy time series $\bf x$. How do I get $\langle\bf\dot{x}\dot{x}^T\rangle$ from the generated $\bf x$? I hope this can be done and would give me the same value for $\langle(\bf{-Qx} + \bf{\eta})(\bf{-Qx} + \bf{\eta)^T}\rangle$.

Equation 1 has the solution

$${x} (t) = e^{-Qt}x(0) +\int_{0}^{t}dt'e^{-(t-t)'Q}\eta(t')$$

but now I want to recover $\bf{\dot{x}}$ from the time series of $\bf{x}$. Any insight on this problem is highly appreciated. Thank you.