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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-219/3/i/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 219 / 3 / i / a / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-05

(a) ⟷ P 2

. The exponential covariance function gives a stationary Ornstein-Uhlenbeck process. At a small lag

h

E [(f (t + h) - f (t))^{2}] = 2 A^{2} (1 - e^{- ∣ h ∣/ τ}) \sim \frac{2 A ^{2}}{τ} ∣ h ∣.

(1)

The variance of a difference quotient therefore diverges as

h \to 0

, so the process has no mean-square derivative. Its continuous version has rough, nondifferentiable paths, matching the jagged P2 trace in the original PDF.

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