Question : Consider the Stochastic process $\dot{x}$ = $\begin{bmatrix}0&1\\-1&0\end{bmatrix}x$ ;
where the initial state is normal with zero mean value and the covariance
$cov[x(0),x(0)]$ = $\begin{bmatrix}1&0\\0&1\end{bmatrix}$
Give a predictor for the process which predicts x(t+h) based on observation of { $x_{1}$(s), $t_{0}<s<t$} .
This is a question from 2nd Chapter of Introduction to Stochastic Control theory by Astrom. Im new to this Stochastic optimal control andhaving tough time to even start approaching this problem.
Could you please help. Thanks in Advance
This is essentially a simplified case of the continuous time Kalman filter. This filter is also called the Bucy-Kalman filter, which in your case can be reduced to
$$ \dot{P}(t) = A\,P(t) + P(t)\,A^\top, \tag{1} $$
with $P(t)$ the covariance matrix of $x(t)$, so $P(0)$ would be the identity matrix, and $A$ the skew-symmetric matrix used to define the dynamics of $x(t)$.
In general, one could solve $(1)$ by integrating the ordinary differential equation numerically or rewrite it a little such that it can be formulated similar to how Gramians are defined. However, in this case with the given $P(0)$ and $A$ it follows that $\dot{P}(0)=0$ and therefore $P(t) = P(0)$.