Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 13A / d

Table of contents
- Solution d

Solution

The diffusion approximation of a birth-death process uses the first two jump moments. For these unit upward and downward jumps they give

u (x) = α + β x - γ x (x - 1), D (x) = \frac{1}{2} {α + β x + γ x (x - 1)} .

The factor

1/2

in

D

is required because the equation is written with

+ \partial_{x}^{2} (D P)

, rather than

\frac{1}{2} \partial_{x}^{2} (BP)

.

Assuming the boundary terms vanish, integration by parts gives

\frac{d}{d t} ⟨ x ⟩ = \int x [- \frac{\partial ( u P )}{\partial x} + \frac{\partial ^{2} ( D P )}{\partial x ^{2}}] d x = \int u (x) P (x, t) d x = ⟨ u (x)⟩ .

Consequently

\frac{d}{d t} ⟨ x ⟩ = α + β ⟨ x ⟩ - γ ⟨ x (x - 1)⟩,

which is precisely the discrete first-moment equation in part (b).

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