Brownian Motion

Brownian motion is the cornerstone process of quantitative finance. Once you internalize three properties — start at zero, independent Gaussian increments, continuous paths — most of stochastic calculus falls out cleanly.

Let $\{W_t{\}}_{t\ge 0}$ be a standard Brownian motion on $(\Omega ,\mathcal{F},\{{\mathcal{F}}_t\},\mathbb{P})$ with $W_0=0$, continuous paths, and independent increments satisfying $W_t-W_s\sim \mathcal{N}(0,t-s)$ for $s\le t$.

Part A

Prove that $W_t$ is a martingale with respect to its natural filtration: $\mathbb{E}[W_t\mid {\mathcal{F}}_s]=W_s$ for all $s\le t$.

Part B

Compute the conditional variance $\text{Var}(W_t\mid {\mathcal{F}}_s)$ and the unconditional covariance $\text{Cov}(W_s,W_t)$ for $s\le t$.

Part C

A junior trader claims that since each $W_t\sim \mathcal{N}(0,t)$, the joint distribution of $(W_1,W_2)$ is just two independent Gaussians. Identify the error and write down the correct joint distribution.