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31 Brownian Motion Construction Using Hilbert Spaces

Last Step: Karhunen–Loève Expansion–Style Construction

We represent Brownian motion on $[0,1]$ as

\[B(t) = \sum_{k=0}^\infty S_k(t)\, Z_k,\]

where

  • $Z_k \overset{\mathrm{iid}}{\sim} N(0,1)$,
  • $S_k(t)$ are special deterministic functions forming an orthonormal system in $L^2[0,1]$,
  • The dyadic indexing uses
    \(2^n \le k < 2^{n+1},\qquad k' = k - 2^n,\quad k'=0,1,\ldots,2^n-1.\)

For each dyadic block:

\[0 \le t \le 1,\qquad 0 \le S_k(t) \le \frac{2^{-n/2}}{2}.\]

Covariance Check

\(\sum_{k=0}^\infty S_k(t_1) S_k(t_2) = t_1 \wedge t_2 = \operatorname{Cov}\big(B_{t_1}, B_{t_2}\big),\) and
\(E(B_t)=0.\)

Thus the finite-dimensional distributions agree with standard Brownian motion.

Need to Prove the Series is Continuous

We need to show:

\[B(t)=\sum_{k=0}^{\infty} S_k(t) Z_k \quad \text{has continuous sample paths}.\]

Basic Uniform Convergence Criterion

Suppose:

  • $f_k \in C[0,1]$,
  • $\max_{0\le x\le 1} f_k(x) \le a_k$,
  • $\sum_{k=1}^\infty a_k < \infty$.

Then the series $\sum_{k=1}^\infty f_k(x)$ converges uniformly and defines a continuous function.

Apply to partial block sums:

\[\sum_{k=2^n}^{2^{n+1}-1} S_k(t) Z_k \le \frac{C(\omega)}{2} \sqrt{n+1}\, 2^{-n/2} < \infty,\]

because, from last time, for $\omega\in\Omega$,

\(|Z_k(\omega)| \le C(\omega)\sqrt{\log k},\qquad k\ge 2,\) and if $2^n \le k < 2^{n+1}$,

\[|Z_k(\omega)|\le C(\omega)\sqrt{n+1}.\]

Since

\[\sqrt{n+1}\,2^{-n/2}\]

is absolutely summable, the series converges uniformly on $[0,1]$, giving continuous sample paths.

Thus the Hilbert–space expansion constructs Brownian motion.

Next Section: Markov Processes

Markov Property for Brownian Motion

Let $t_0 \ge 0$. Consider the increment process

\[X(h) = B(t_0 + h) - B(t_0), \qquad h\ge 0.\]

Distributional Properties

  • $X(h) \sim N(0,h)$,
  • $E[X(h)] = 0$,
  • $\operatorname{Cov}(X(h_1), X(h_2)) = h_1 \wedge h_2$,
  • ${X(h)}_{h\ge 0}$ has continuous sample paths.

Hence ${X(h)}{h\ge 0}$ is standard Brownian motion, independent of $\mathcal{F}{t_0}$.

Thus

\[B(t_0 + h) = B(t_0) + X(h).\]

Canonical Setup

\[\Omega = C[0,\infty), \qquad B_t(\omega)=\omega(t).\]

Let

\(\mathcal{F}^0_t = \sigma\{B_s : 0\le s\le t\},\) and let $W = P_0$ be Wiener measure.

For any $x\in\mathbb{R}$, define a shifted measure $P_x$ by

\[P_x(A) = P_0(A - x),\qquad A\in \mathcal{F}.\]

Then

\[\text{Law}\{B(t_0 + h)\mid\mathcal{F}^0_{t_0}\} = P_{B(t_0)}.\]

Shift Operator

For $s\ge 0$,

\[\theta_s(\omega)(t)=\omega(s+t), \qquad t\ge 0.\]

Then under $P_x$,

\[\text{Law of } \{B(t_0+h)\}_{h\ge 0} = \text{Law of } \theta_{t_0}(\omega) = P_{B(t_0)}.\]

Filtration Regularity

Define the right-continuous modification

\[\mathcal{F}^+_t = \bigcap_{s>t} \mathcal{F}^0_s.\]

Then

  • $\mathcal{F}^0_t \subset \mathcal{F}^+_t$,
  • $\mathcal{F}^+_t$ is right-continuous,
  • Typically one works with $\mathcal{F}_t = \mathcal{F}^+_t$.

π–λ System Argument (Monotone Class Theorem)

Let $\mathcal{A}$ be a $\pi$-system of sets in $\Omega$.
Let $\mathcal{H}$ be a class of random variables satisfying:

  1. $A\in\mathcal{A} \Rightarrow 1_A\in\mathcal{H}$,
  2. $f,g\in\mathcal{H} \Rightarrow f+g\in\mathcal{H}$,
  3. If $f_n\in\mathcal{H}$, $f_n\ge 0$, $f_n\uparrow f$, and $f_n$ bounded, then $f\in\mathcal{H}$,
  4. $1\in\mathcal{H}$.

Then

\[\mathcal{H} \supset \{f\text{ bounded, measurable w.r.t. }\sigma(\mathcal{A})\}.\]

This proves extensions of identities from simple function classes to all bounded measurable functions.

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