# Brownian Motion

Stochastic Processes
Gaussian Processes

## Construction

(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 $$a$$.

## References

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