Lets look at brownian motion now. And yes, its the same as what our high school teachers taught about the particles moving in random motion. Here, we attempt to give it a proper structure and definition to work with.

A Brownian Motion is a random process \{ X_t : t \ge  0 \} with parameters (\mu, \sigma) if
For 0 \textless t_1 \textless t_2 \textless \ldots \textless t_{n-1} \textless t_n, (X_{t_2} - X_{t_1}), (X_{t_3} - X_{t_2}), \ldots, (X_{t_n} - X_{t_{n-1}}) are mutually independent. This is often called the independent increments property.
For s > 0, X_{t+s} - X_t \sim \mathrm{N} ( \mu s, \sigma^2 s)
X_t is a continuous function of t.
We say that X_t is a \mathrm{B} (\mu, \sigma) Brownian motion with drift \mu and volatility \sigma.

For the special case of \mu = 0 and \sigma = 1, we have a standard Brownian motion. We can denote it with W_t and assume that W_0 = 0
If X_t \sim \mathrm{B}(\mu, \sigma) and X_0 = x then X_t = x + \mu t + \sigma W_t where W_t is a standard brownian motion. Thus, X_t \sim \mathrm{N}(x+\mu t, \sigma^2 t)

Random Paths of Brownian Motion. Source: Columbia University
Random Paths of Brownian Motion. Source: Columbia University

The next concept is important in finance, that is, Information Filtrations.
For any random process, we will use \mathcal{F}_t to denote the information available at time t.
– the set \{\mathcal{F}_t\}_{t \ge 0} is then the information filtration.
\mathbb{E}[.|\mathcal{F}_t] denotes an expectation conditional on time t information available.

Note: The independent increment property of Brownian Motion implies that any function of W_{t+s} - W_t is independent of \mathcal{F}_t and that (W_{t+s}-W_t) \sim \mathrm{N}(0,s).

So let us do a bit of math to obtain \mathbb{E}_0[W_{t+s}W_s] for instance.

Using condition expectation identity, we have
\mathbb{E}_0 [W_{t+s}W_s]
= \mathbb{E}_0 [(W_{t+s} - W_s + W_s)Ws]
= \mathbb{E}_0 [(W_{t+s}-W_s)W_s] + \mathbb{E}_0 [{W_s}^2]
= \mathbb{E}_0 [\mathbb{E}_s[(W_{t+s} - W_s)W_s]] + s
= \mathbb{E}_0 [W_s \mathbb{E}_s[(W_{t+s} - W_s)]] + s
= \mathbb{E}_0 [W_s 0] + s
= 0 + s
= s

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