Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 13E / e / iii / Solution

The third Kramers-Moyal expansion coefficient is $F=\frac{1}{6}{\sum }_r{r}^{3}W=(a-b)/6$. At a first minimum the third derivative has no fixed sign, so the maximum-principle argument fails; finite truncations beyond second order need not preserve positivity.