Basic Stochastic Calculus & the Black-Scholes PDE

A quasi-formal definition of a Brownian Motion (BM)

An $R^{n}$ - valued stochastic process ${W_{t} : t \in [0, + \infty)}$ on a probability space $(Ω, F, P)$ is an n-dimensional Brownian motion when:

$W_{0} = 0$ (not strictly necessary, but we can always shift the process to make it start at 0)
the functions $t \to W_{t} (ω)$ are continuous for each $ω \in Ω$ (i.e., the paths are continuous)
for each $s, t \in [0, + \infty)$ with $s < t$ , the random variable $W_{t} - W_{s}$ is dependent of all $W_{u}$ for $u \in [0, s]$ (i.e., the process has independent increments)
$W_{t} - W_{s}$ has a normal distribution with mean $0$ and covariance matrix $(t - s) I_{n}$ where $I_{n}$ is the $n \times n$ identity matrix

Grape