A latent variable evolves as $x_t = x_{t-1} + e_t$ where $e_t$ has a gaussian distribution with 0 mean and a variance which defines the volatility of the overall process. Let $o_t$ be the noisy observation of $x$ — i.e. a gaussian processes.
I'm trying to work out the covariance of $x_t$ and $x_{t-1}$ i.e. $\operatorname{cov}(x_t, x_{t-1}) $ given $o_t$ for all $t$, using the Kalman filter. Kalman filter estimates the distribution of $x_t$ as a normal with mean $m_t$ and variance $w_t$ (i.e. the classic Kalman filter).
My current estimate is $\operatorname{cov}(x_t, x_{t-1}) = w_t -m_t * m_{t-1}$, however I've found a paper which defines it very differently:
I was wondering if anyone had any idea how this derivation was taken, or where I'm going wrong?
Paper is here: Piray and Daw - A simple model for learning in volatile environments.