Short observation times

If the relaxation time (the inverse of the smallest eigenvalue) is larger than the observation time the estimation of the unconditional momenta must be corrected. In the limit of continuous observation of the process

$$\mathsf{var}\!(\xi) = {1\over T}\,\mathsf{E}\,\int_0^T\!\ma... ...\xi_t \,\, \mathsf{E}\int_0^T\!\mathrm{d}\xi_t^\prime \quad .$$ (83)

The first term in the sum is just the instantaneous variance $\Sigma\Sigma^\prime\,\mathrm{d}t$. The correction to the instantaneous conditional variance is therefore (define the sample mean $\mu_{0T} = \mathsf{E}(\xi_T-\xi_0)/T$)

$$\mathsf{var}\!(\xi) - \Sigma\Sigma^\prime = -\mu_{0T}\,\mu_{0T}^\prime\quad ,$$ (84)

i.e. the unconditional process variance has to be adjusted down. The sample mean is

$$\mu_{0T} = \frac{\mathsf{E}(\xi_T-\xi_0)}{T} = {1\over T}\i... ... a\,\mathrm{d}\tau + \frac{e^{TA}-1}{T}\,\mathsf{E}\,\xi_0$$ (85)

In the limit $T\to\infty$ this simplifies to $\mu_{0\infty} = -A^-\, a$