Write $\mathbb E X_1=\mu$ and $\operatorname{Var}(X_1)=\sigma^2<\infty$. For rational $t=m/n$, stationarity and independence of the $n$ increments over intervals of length $1/n$ give
$$
\mathbb E X_t=t\mu,
\qquad
\operatorname{Var}(X_t)=t\sigma^2,
$$
using <variance additivity for independent random variables>. <Stochastic continuity> extends both identities from rational to real $t$. In the centered case $\mu=0$, this becomes $\mathbb E X_t=0$ and $\mathbb E X_t^2=t\sigma^2$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 201 / 6 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-201/6/b/solution