Constructing Brownian Motion

This note answers the existence question. It first shows that the four defining properties force the law to be essentially unique (up to drift and variance). It then constructs standard Brownian motion explicitly on the interval $[0,1]$ using Lévy's dyadic-interpolation scheme: a countable family of independent normals is assembled step by step, always maintaining consistency with the joint normal structure. The construction is grounded in the linear-combinations-of-normals characterisation from the previous lesson — no technical measure theory beyond the multivariate CLT is required.

• State the four defining properties of Brownian motion and explain why each is physically and financially motivated.
• Derive that $\mathbb{E}[W_t] = 0$ and $\text{Var}(W_t) = t$ from the stationarity and independence of increments alone.
• Explain Kolmogorov's consistency theorem (at a conceptual level) and why it justifies assembling finite-dimensional joint normals into a bona-fide process.
• Describe the dyadic-interpolation construction: how $W_{k/2^n}$ is built from $W_{k/2^{n-1}}$ using a fresh independent normal at each step.
• Verify that consecutive increments at each dyadic level are independent standard normals, using Proposition 2.2.1.
• State the Kolmogorov–Chentsov criterion and use it to assert Hölder continuity of BM paths.

1. The four defining properties
2. Consistency and the existence problem
3. Lévy's dyadic construction
4. Brownian motion with drift
5. Path structure: nowhere differentiability and total variation