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

Table of contents
- Solution iii

Solution

For this two-dimensional Ornstein-Uhlenbeck process, the covariance matrix

C (t) = E [x (t) x (t)^{T}]

satisfies

\frac{d C}{d t} = a C + C a^{T} + b .

Thus the stationary covariance solves the continuous Lyapunov equation

a C + C a^{T} + b = 0 .

Write

C = (u v v w) .

Substitution gives

- 2 u + 2 v + 1 = 0, w - 2 u - 2 v = 0, - 4 v - 2 w + 1 = 0.

Solving,

u = \frac{5}{12}, v = - \frac{1}{12}, w = \frac{2}{3} .

Therefore

C = (5/12 - 1/12 - 1/12 2/3) .

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