This is really important for anyone interested in Finance Modelling. As what the movie Wolf on Wall Street says:

They are referring to a geometric brownian motion.

Firstly, we will begin with the definitions.

We say that a random process, $X_t$, is a geometric Brownian motion (GBM) if for all $t \ge 0$
$X_t = e^{(\mu - \frac{\sigma^2}{2})t} + \sigma W_t$
where $W_t$ is a Standard Brownian Motion
Here $\mu$ is the drift and $\sigma$ is the volatility. We write $X_t \sim GBM (\mu, \sigma)$

Also note that
$X_{t+s}$
$= X_0 e^{(\mu - \frac{\sigma^2}{2})(t+s) + \sigma W_{t+s}}$
$= X_0 e^{(\mu - \frac{\sigma^2}{2})(t+s) + \sigma W_{t} + (\mu - \frac{\sigma^2}{2})s + \sigma (W_{t+s} - W_t)}$; This is a common technique for solving expectations.
$= X_t e^{(\mu - \frac{\sigma^2}{2})(s) + \sigma (W_{t+s} - W_t)}$. This is very useful for simulating security prices.

Consider $\mathbb{E}_t [X_{t+s}]$

$\mathbb{E}_t [X_{t+s}]$
$= \mathbb{E}_t [X_{t}e^{(\mu - \frac{\sigma^2}{2})s + \sigma(W_{t+s} - W_t)}]$; Notice this expansion is similar to before.
$= X_t e^{(\mu - \frac{\sigma^2}{2})s} \mathbb{E}^t [e^{\sigma (W_{t+s} - W_t)}]$
$= X_t e^{(\mu - \frac{\sigma^2}{2})s} e^{\frac{\sigma^2}{2}s}$
$= e^{\mu s} X_t$
This result tells us that the expected growth rate of $X_t$ is $\mu$.

From the definitions of Brownian Motion introduced earlier, we extend them to Geometric Brownian motion.
1. Fix $t_1, t_2, \ldots , t_n$. Then $\frac{X_{t_2}}{X_{t_1}}, \frac{X_{t_3}}{X_{t_2}}, \ldots \frac{X_{t_n}}{X_{t_{n-1}}}$ are mutually independent.
2. Paths of $X_t$ are continuous as function of $t$, meaning they do not jump.
3. For $s > 0$, $\mathrm{log}(\frac{X_{t+s}}{X_t}) \sim \mathrm{N}((\mu - \frac{\sigma^2}{2})s, \sigma^2 s)$

So now lets try to do some modelling of stock prices as a geometric brownian motion.

Suppose $X_t \sim GBM(\mu, \sigma)$. Clearly
1. $X_t > 0 \Rightarrow X_{t+s} > 0$ for any $s > 0$
This tells us that the limited liability of stock price is not violated.
2. The distribution of $\frac{X_{t+s}}{X_t}$ only depends on s and not on \$latex X_t.
We will look at the Black-Scholes option formula next time and will come back to review the geometric brownian motion for the underlying model.