Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 13E / b

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- Solution b

Solution

When jumps are small relative to a large population and moments exist, the Kramers-Moyal expansion through second order gives

P_{t} = - \partial_{x} (u P) + \partial_{x}^{2} (D P), u = \sum_{r} r W (x, r), D = \frac{1}{2} \sum_{r} r^{2} W (x, r) .

This Fokker-Planck equation is a conservation law with vanishing boundary flux;

u

is the drift and

D

the diffusivity. Equal constant rates for jumps

r = \pm 1

give pure diffusion.

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