The increment law immediately gives
$$
\boxed{\mathbb E(W_{t+h}-W_t)^2=h}.
$$
The difference quotient has second moment
$$
\mathbb E\left(\frac{W_{t+h}-W_t}{h}\right)^2=\frac1h\to\infty.
$$
Thus increments scale as $\sqrt h$, rather than $h$, and no finite derivative is suggested. In fact this heuristic is strengthened by the theorem on <nowhere differentiability of Brownian motion>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 353 / 4 / b / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-353/4/b/ii/solution