Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 201 / 1

A standard Brownian motion is a real-valued process $(B_t{)}_{t\ge 0}$ with $B_0=0$, almost surely continuous paths, and independent increments satisfying $B_t-B_s\sim N(0,t-s)$ for $s<t$.

For a partition of $[a,b]$, every Riemann sum $S_n={\sum }_j{B}_{t_j}\Delta t_j$ is a centered Gaussian random variable, withPath continuity gives $S_n\to {\int }_a^b{B}_{s}ds$ almost surely, and the covariance bound $\mathbb{E}|B_s-B_t{|}^2=|s-t|$ also gives convergence in $L^2$. Hence the limit is Gaussian, centered, and passage to the limit in the displayed sums gives variance

Choose step functions $f_n\to f$ in $L^2[0,1]$. Part (i) shows that $\int B_s{f}_n(s)ds$ is centered Gaussian. Since the kernel $K(r,s)=\min (r,s)$ is bounded,The $L^2$ limit is therefore centered Gaussian, and its variance is

Writing $c_i(t)=⟨1_{[0,t]},f_i{⟩}_{L^2}$ gives $X_n(t)={\sum }_{i=1}^n{\alpha }_i{c}_i(t)$, henceBy Parseval identity, the variance tends to $\Vert 1_{[0,t]}{\Vert }_2^2=t$. Thus $X_n(t)\stackrel{d}{\to }N(0,t)$.

For fixed $t$, $(X_n(t))$ is a martingale with independent centered increments andThe $L^2$ Martingale convergence theorem gives convergence both almost surely and in $L^2$ to a random variable $X(t)$.

For real $u_1,\dots ,u_j$, the preceding series construction givesIt is centered Gaussian, and Parseval identity makes its varianceThis is the variance of ${\sum }_r{u}_r{B}_{t_r}$. The Cramer-Wold theorem proves equality of the two joint distributions.