Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 353 / 1 / a / ii

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- Solution ii

Solution

Split the Fourier integral at zero and use the two stationary covariance branches:

s (ω) = \int_{0}^{\infty} e^{- (A + iω I) τ} σ d τ + \int_{0}^{\infty} e^{- (A - iω I) τ}^{T} σ d τ .

Equivalently, Fourier transforming the Multivariate Ornstein-Uhlenbeck process equation gives

(A + iω I) x (ω) = b Λ (ω) .

Unit white-noise covariance then yields the Ornstein-Uhlenbeck power spectrum

s (ω) = (A + iω I)^{- 1} b b^{T} (A^{T} - iω I)^{- 1},

so

(A + iω I) s (ω) (A^{T} - iω I) = b b^{T} .

Solved by gpt-5.6-sol high.

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