Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 219 / 3 / a

Solution

Put

ρ = e^{- Δ t / τ} = e^{- x}

and retain

R = k (0, 0)

(y_{1}, y_{2}, y_{3})

R 1 ρ ρ^{2} ρ 1 ρ ρ^{2} ρ 1 .

Conditioning a multivariate normal distribution and simplifying gives

y_{3} ∣ y_{1}, y_{2} \sim N (μ + ρ (y_{2} - m u), R (1 - ρ^{2})) .

The absence of

y_{1}

from both conditional moments shows that

y_{3}

and

y_{1}

are conditionally independent given

y_{2}

. Equivalently, the exponential-kernel Gaussian process is the stationary Ornstein-Uhlenbeck process, which is Markov.

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