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Past exam of the mathematics course of the University of Cambridge

2020

Paper 1

For this two-dimensional <Ornstein-Uhlenbeck process>, the covariance matrix
$$
C(t)=\mathbb E[x(t)x(t)^T]
$$
satisfies
$$
\frac{dC}{dt}=aC+Ca^T+b.
$$
Thus the stationary covariance solves the continuous <Lyapunov equation>
$$
\boxed{aC+Ca^T+b=0}.
$$
Write
$$
C=
\begin{pmatrix}
u&v\\
v&w
\end{pmatrix}.
$$
Substitution gives
$$
-2u+2v+1=0,
\qquad
w-2u-2v=0,
\qquad
-4v-2w+1=0.
$$
Solving,
$$
u=\frac5{12},
\qquad
v=-\frac1{12},
\qquad
w=\frac23.
$$
Therefore
$$
\boxed{
C=
\begin{pmatrix}
5/12&-1/12\\
-1/12&2/3
\end{pmatrix}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 6B / iii / Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 6B / iii / Solution

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Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 6B / iii / Solution

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