Brownian Motion

Stochastic Processes
Gaussian Processes


(primarily from Brémaud (2020)) Definition Standard Brownian motion, \(W_t,t\in\mathbb{R}^+_0\) is a continuous centered Gaussian Process with independent increments, such that \(W_0=0\) and \(Var(W_t-W_s)=t-s\).

We have \[\mathbb{E}[W_t]=0\] and \[\mathbb{E}[W_tW_s]=\min(t,s).\]

Brownian motion arises in the limit of a scaled random walk

\[W_t=\lim_{N\rightarrow \infty} \frac{\sum_{k=1}^{\lfloor N t \rfloor} Z_k}{\sqrt{N}}\]

with \(Z_k\) being i.i.d. and uniformly distributed over \(\{-1,1\}\).

Simple, invariant operations on Brownian motion.

Theorem The following operations on standard Brownian Motion \(W_t\) result in \(X_t\) being standard Brownian motion as well:

  1. Symmetrization: \(X_t=-W_t\)
  2. Delay: \(X_t=W_{t+a}-W_t\) where \(a>0\)
  3. Scaling: \(X_t=\sqrt{c}W_{t/c}\) where \(c>0\)
  4. Time inversion: \(X_t=tW_{1/t}\)

Proof 1. \[-W_t=\lim_{N\rightarrow \infty} \frac{\sum_{k=1}^{\lfloor N t \rfloor} -Z_k}{\sqrt{N}}\equiv \lim_{N\rightarrow \infty} \frac{\sum_{k=1}^{\lfloor N t \rfloor} Z_k}{\sqrt{N}}=W_t\] as \(Z_k\) is uniformly distributed over \(\{-1,1\}\).

  1. For \(N\) finite, we have \[\frac{\sum_{k=1}^{\lfloor N (t+a) \rfloor}Z_k}{\sqrt{N}}-\frac{\sum_{k=1}^{\lfloor N t \rfloor}Z_k}{\sqrt{N}}=\frac{\sum_{k=\lfloor a\rfloor}^{\lfloor N (t+a) \rfloor}Z_k}{\sqrt{N}}\] Taking \(N\) to the limit, we get \[W_{t+a}=\lim_{N\rightarrow\infty}\frac{\sum_{k=\lfloor a \rfloor}^{\lfloor N (t+a) \rfloor}Z_k}{\sqrt{N}},\] which is, again, standard Brownian motion, with the time index shifted by \(a\).


Brémaud, Pierre. 2020. Probability Theory and Stochastic Processes. Springer.