Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 6C / c / Solution

The deterministic drift vanishes at the positive stable point

n_{*} = \frac{λ}{2 β}, A^{'} (n_{*}) = - 4 β n_{*} < 0.

When

n_{*}

is large, fluctuations sample only a narrow range around it, so the linear noise approximation replaces

A (n) ≃ A^{'} (n_{*}) (n - n_{*}), B (n) ≃ B (n_{*}) = 3 λ .

The resulting Ornstein-Uhlenbeck stationary density is the stated normal distribution with

μ = n_{*} = \frac{λ}{2 β},

and

σ^{2} = - \frac{B ( n _{*} )}{2 A ^{'} ( n _{*} )} = \frac{3 λ}{8 β n _{*}} = \frac{3}{4} \frac{λ}{2 β} = \frac{3 μ}{4} .

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