1 Brownian Motion
1.1 What is Brownian Motion?
Consider a continuous-time and continuous state stochastic process \(B:=(B_{t})_{t>0}\) taking values in \(\mathbb{R}\) with initial value \(B_{0}=0\) almost surely (i.e. \(\mathbb{P}(B_{0}\neq 0)=0\)).
We define general \(n\)-increments for \(0\leq t<t+n\) by
\[ \Delta_{n}B_{t}=B_{t+n}-B_{t}. \]
We define the increments for subsequence of times \(0<t_{1}<\dots<t_{n}\) by
\[ \Delta B_{t_{i}}=B_{t_{i+1}}-B_{t_{i}}. \]
The process \(B\) is said have independent increments if for a sequence of times \(0 < t_{1}< \dots < t_{n}\) the increments \(\Delta B_{t_{i}}\) are mutually independent random variables, that is, for any \(i\neq j\) \[ \Delta B_{t_{j}}\perp \Delta B_{t_{i}}. \]
Note: The joint distribution factors into a product of distributions. The key point to understand is that intervals that do not overlap contain no information about each other.
The process \(B\) has normal stationary increments if for all \(0\leq s < t\) we have that \[ \Delta_{n} B_{t}\sim \text{Normal}(0, n\sigma^2). \]
The process \(B\) is said to have continuous paths if \(\omega \mapsto B_{t}(\omega)\) is a continuous function of \(t\) on \(\mathbb{R}\) almost surely.
The process \(B\) is said to be a Wiener process (follow a Brownian motion) if it satisfies the following:
- \(B\) has independent increments.
- \(B\) has normal stationary increments.
- \(B\) has continuous paths almost surely.
1.2 Properties of Brownian Motion
A Brownian motion (BM) or a Wiener process with variance parameter \(\sigma^2>0\) is a stochastic process \((B_{t})_{t \geq 0}\) taking values in \(\mathbb{R}\) that satisfies: 4. \(B_{0}=0~a.s.\) (starts at zero); 5. \(\forall~0 < t_{1}<\dots<t_{n},~B_{t_{1}}-B_{0}, B_{t_{2}}-B_{t_{1}}, \dots, B_{t_{n}}-B_{t_{n-1}}\) are independent (independent increments); 6. \(\forall s < t, B_{t}-B_{s}\sim\mathcal{N}(0,\sigma^2(t-s))\) (normal stationary increments); 7. \(\omega\mapsto B_{t}(\omega)\) is a continuous function of \(t\) on \(\mathbb{R}_{+}\) almost surely (continuous paths no jumps).