28 — Brownian Motion (March 28)
Definition of Standard Brownian Motion (SBM)
A process ${B(t)}_{t \ge 0}$ on $(\Omega, \mathcal{F}, P)$ is a standard Brownian motion if:
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Initial condition:
\(B(0) = 0.\) - Independent and stationary increments:
- For all $t,h \ge 0$,
\(B(t+h) - B(t) \stackrel{d}{=} B(h).\) - For $0 \le s < t$,
\(B(t) - B(s) \perp \sigma(B(u) : u \le s).\)
- For all $t,h \ge 0$,
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Gaussian increments:
\(B(t+h) - B(t) \sim N(0,h).\) - Continuous sample paths:
For every $\omega$, the map $t \mapsto B_t(\omega)$ is continuous on $\mathbb{R}^+$.
Finite-Dimensional Distributions
Given times $0 < t_1 < t_2 < \cdots < t_k$:
Define increments: \(U_1 = B_{t_1}, \qquad U_2 = B_{t_2} - B_{t_1}, \quad \ldots, \quad U_k = B_{t_k} - B_{t_{k-1}}.\)
By independence of increments: \((U_1, \ldots, U_k) \text{ are independent}.\)
Each increment has density: \(p_t(x) = \frac{1}{\sqrt{2\pi t}} e^{-x^2/(2t)}, \quad t>0.\)
Therefore the joint density is: \(f_{U_1,\ldots,U_k}(u_1,\ldots,u_k) = p_{t_1}(u_1)\, p_{t_2 - t_1}(u_2)\cdots p_{t_k - t_{k-1}}(u_k).\)
Changing variables back to $(B_{t_1},\ldots,B_{t_k})$ involves a triangular Jacobian with determinant $1$. Thus: \(f_{B_{t_1},\ldots,B_{t_k}}(x_1,\ldots,x_k) = \prod_{i=1}^k p_{t_i - t_{i-1}}(x_i - x_{i-1}),\) where $t_0 = 0$ and $x_0 = 0$.
Alternative Definition (Gaussian Process Characterization)
A process ${B_t}_{t\ge 0}$ is SBM if:
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Gaussian process:
For any $t_1<\cdots<t_n$,
\((B_{t_1},\ldots,B_{t_n}) \sim \text{MVN}\left(0,\;\Sigma\right), \quad \Sigma_{ij}= t_i \wedge t_j.\) -
Mean and covariance:
\(E[B_t]=0, \qquad E[B_s B_t] = s \wedge t.\) -
Continuous sample paths as before.
Kolmogorov Extension and Continuity
- The covariance matrix $[t_i \wedge t_j]$ is consistent, therefore by Kolmogorov’s Extension Theorem, a process with these finite-dimensional distributions exists.
- But continuity is not automatic. We need Kolmogorov’s continuity criterion.
Kolmogorov Continuity Criterion
If a process ${X_t}_{0 \le t \le 1}$ satisfies
\(E\lvert X_t - X_s\rvert^{\beta} \le C |t-s|^{1+\alpha},\)
for some $\beta > 0$ and $\alpha>0$, then $X_t$ has Hölder continuous paths with exponent
\(\gamma < \frac{\alpha}{\beta}.\)
Apply to Brownian Motion
For $0 \le s < t$:
\[B_t - B_s \stackrel{d}{=} \sqrt{t-s}\, Z, \quad Z\sim N(0,1).\]Take $\beta = 2m$. Then: \(E|B_t - B_s|^{2m} = |t-s|^{m} E|Z|^{2m}\) where \(E|Z|^{2m} = 1\cdot 3\cdot 5 \cdots (2m-1).\)
Thus: \(m = \frac{\beta}{2},\qquad \alpha = m - 1.\)
So the Hölder exponent is: \(\gamma < \frac{m-1}{2m} \longrightarrow \frac12 \text{ as } m\to\infty.\)
Conclusion: Brownian motion is Hölder continuous for every $\gamma < \tfrac12$.
Remarks on Brownian Paths
- Brownian motion is continuous but nowhere differentiable.
- Its sample paths have no points of increase (illustrated on page 3).
- It has extremely irregular behavior despite continuity.
Preview
- Dvoretzky’s CLT for martingales.
- Work of Erdős and Kakutani.
- As a warm-up, simple random walk converges to Brownian motion when rescaled, with scaling constant $C = 1/2$.
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