Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 219 / 3 / a

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Solution

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y = (y_{1}^{⊤}, y_{2}^{⊤})^{⊤}

. After integrating out the Ornstein-Uhlenbeck process,

y ∣ t, θ \sim N_{2 N} (μ_{θ}, Σ_{θ}), μ_{θ} = (c 1 (c + Δ m) 1) .

k (a, b) = A^{2} e^{- ∣ a - b ∣/ τ}

, the covariance blocks are

(Σ_{11})_{jk} = k (t_{j}, t_{k}) + σ_{1, j}^{2} 1_{{j = k}},

(Σ_{22})_{jk} = k (t_{j} - Δ t, t_{k} - Δ t) + σ_{2, j}^{2} 1_{{j = k}},

and

(Σ_{12})_{jk} = k (t_{j}, t_{k} - Δ t)

, with

Σ_{21} = Σ_{12}^{⊤}

. Therefore

p (y_{1}, y_{2} ∣ t, θ) = (2 π)^{- N} ∣ Σ_{θ} ∣^{- 1/2} exp [- \frac{1}{2} (y - μ_{θ})^{⊤} Σ_{θ}^{- 1} (y - μ_{θ})] .

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