\(dX_t = \mu_t dt + \sigma_t dB_t\)

\(df(t, X_t) = \left(\frac{\partial f}{\partial t} + \mu_{t}\frac{\partial f}{\partial x} + \frac{1}{2} \sigma^2_t\frac{\partial^2 f}{\partial x^2}\right) dt + \sigma_t\frac{\partial f}{\partial x} dB_t\)

This is an equation called Ito's Lemma which is very commonly used in mathematical finance.

The first line says that you have a new function over time that has a "non-wiggly" part added to a "wiggly" part. Typically this is a stock price which has a "non-wiggly" part and a "wiggly" part.

So the first line just defines the process.

Now I want to calculate a new function that is based on the random process. A trivial example is if I have the equation for one share of stock and I want to calculate the behavior of two shares of stock. A less trivial example would be if I have a basket of stocks that is changing over time, or if I have something like a stock option which whose value isn't a trivial function of that stock.

The second line says that if I perform a function on a random process (i.e a function with a "non-wiggly" and a "wiggly" part) that I get a new random process with a different "non-wiggly" and a "wiggly" part.