Constructing Brownian Motion

A first undergraduate course in stochastic processes will simply assert the existence of Brownian motion and its properties. Let’s pretend it’s 1900 and prove existence of Brownian Motion via Kolmogorov and Lévy constructions.

There are multiple ways to show existence of Brownian Motion. Two popular approaches are the Kolmogorov construction of the Wiener measure and the Lévy construction with convergence.

Kolmogorov & Wiener Measure Construction

Statement of Wiener Measure

What does this mean?

Put simply, finite dimensional projections of paths $C([0, \infty))$ are multivariate Gaussian distributions with the specified BM covariance structure. As someone on MathOverflow writes, “This definition is analogous to describing a duck as the animal whose shadows look like 2-dimensional ducks,” and it’s kind of confusing.

The law of a stochastic process $X: \Omega \rightarrow S^T$ is the the pushforward measure $\mu = P \circ X^{-1}$, where $P$ a probability measure, and $X^{-1}$ the preimage of the $S^T$ valued random variable $X$.

In 1-D BM, we are thinking of $X: C([0,\infty)) \rightarrow \R$. Consider a measurable subset $B \subset \R$, i.e., some set of “positions” on the real line. The preimage $X^{-1}(B)$ is the set of continuous paths that cross into $B$. In particular, for times $t_1, ..., t_n$ the projection mapping is having us look at the set of continuous paths that are realized in $B$ at those times. The finite-dimensional distribution is the joint distribution of this random vector, and is a “projection” of the law $\mu$ onto a finite subset of times in $T$.

The above is kind of circular (using Brownian Motion to explain the measure which gives Brownian motion), but just for intuition.

How do we know this measure actually exists?

Kolmogorov Strategy

Want to prove existence of Wiener measure with Kolmogorov-Centsov approach:

• Let $D \subset [0,1]$ (later consider $[0, \infty))$ be a countable dense set. We know we can construct a Gaussian process on $D$ with mean $0$ and $\Cov(W_s, W_t) = s \wedge t$.
• Show a uniformly continuous function on $D$ can be extended to a uniformly continuous function on $[0,1]$.
• Show $\P$ a.s. in $w$ the function $t \mapsto W_t(w)$ is uniformly continuous on $t \in D$.
• Show limits of Gaussians are Gaussian.

Pf. to come later because I don’t want to LaTeX it

Wiener Measure to Brownian Motion

Above shows existence of Wiener measure/Wiener process, and in particular, $\{W_t\}_{t \in [0,1]}$ is Brownian motion.

Extend Brownian motion from $[0,1]$ to $[0, \infty)$:

Let $W^{(k)} = (W^{(k)}(t))_{t \in [0,1]}$ for $k = 1, 2, 3, ...$. Define

$$W(t) = \sum_{k=1}^{m-1} W^{(k)}(1) + W^{(m)}(t-m)$$

if $m \leq t < m+1.$

You can verify that $\{W(t)\}_{t\in[0,\infty)}$ is BM.

Lévy Construction

The key idea of the Lévy construction is to produce a sequence of continuous processes that converges to Brownian Motion.

Two necessary technical facts from real analysis:

Lévy Strategy

• Define a sequence $\{W^n_t\}_{t\in[0,1]}$ of continuous processes with the property that $\sum_n \sup_{t\in[0,1]}|W^{n+1}_t - W_t | < \infty$ a.s.
• This implies $\{W^n_t\} \rightarrow \{W_t\}$ a.s. with $W_t$ having continuous paths.
• Check $\{W_t\}$ has the correct Gaussian finite dimensional distributions.

pf later because i also don’t want to LaTeX it