Brownian motion

Brownian motion: the continuous analogue of a random walk and the building block of continuous stochastic models.

It is a continuous stochastic process with independent increments that are normally distributed.

• $W(0)=0$
• $W(t)\sim N(0,t)$
  • so $E\left[W(t{)}^2\right]=t$
• $W\left(t_{i+1}\right)-W\left(t_i\right)\sim N\left(0,t_{i+1}-t_i\right)$

For all $0<\mathrm{s}<\mathrm{t}$,

$$\begin{array}{rl}\mathrm{Cov}\left(W_s,W_t\right)& =\mathrm{Cov}\left(W_s,W_s+\left(W_t-W_s\right)\right)\\ & =\mathrm{Cov}\left(W_s,W_s\right)+\mathrm{Cov}\left(W_s\left(W_t-W_s\right)\right)\\ & =\mathrm{Var}\left(W_s\right)\\ & =s\end{array}$$

Properties:

• martingale property: $\mathbb{E}[W(t+s)|W(t)]=W(t)$
  • a BM without drift is a martingale
• markov property: for $u<s$, $P(W(t)|W(s),W(u))=P(W(t))$