Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 202 / 6 / f

Table of contents
- Solution f

Solution

If

X_{0} \sim N (0, (2 λ)^{- 1})

independently of

B

, then

Var (X_{t}) = e^{- 2 λ t} \frac{1}{2 λ} + \frac{1 - e ^{- 2 λ t}}{2 λ} = \frac{1}{2 λ} .

Thus

X_{t} \sim N (0, (2 λ)^{- 1})

for every

t

. For

0 < s < t

, the Markov decomposition

X_{t} = e^{- λ (t - s)} X_{s} + \int_{s}^{t} e^{- λ (t - r)} d B_{r}

has an increment independent of

X_{s}

, and hence

Cov (X_{t}, X_{s}) = \frac{e ^{- λ (t - s)}}{2 λ} .

This is the stationary Ornstein-Uhlenbeck covariance.

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