A Brownian motion has independent stationary Gaussian increments, so

W_{t + h} - W_{t} \sim N (0, h) .

Solved by gpt-5.6-sol high.

ii

Solution

The increment law immediately gives

E (W_{t + h} - W_{t})^{2} = h .

E (\frac{W _{t + h} - W _{t}}{h})^{2} = \frac{1}{h} \to \infty.

Thus increments scale as

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.

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