Intuitive explanation as to why we need a Brownian integral to model a stock.

Question

When I was presented the simple Black-Scholes model I was presented with the simple difference equation

$$\frac{\Delta S_t}{S_t}=\mu \Delta t+\sigma\Delta B_t.$$

It was explained that the relative change in the stocks risky asset would consist of a drift term, and a random term($\Delta B_t$).

A more advanced model is the SDE:

$$\frac{d S_t}{S_t}=\mu(t,\omega)dt+\sigma(t,\omega)dB_t.$$

Here $\mu$, and $\sigma$ are progressively measurable processes, and $\sigma$, must of course be such that the Itô-integral is well-defined. However, now that we have randomness in $\mu$, why do we still need the $d B_t$?

Answer 1

Because there is also a filtration with respect to which $\mu$ and $\sigma$ are predictable.

Think of a discrete process $$ \frac{S_{t+\Delta t} - S_t}{S_t} = \mu(t,\omega) + \sigma(t,\omega) \Delta B_t .$$ At time $t$, the quantities $\mu(t,\omega)$ and $\sigma(t,\omega)$ depend upon everything that is known at time $t$. At time $t = 0$, they are random variables, but at time $t$ they are completely known. All that is unknown is whether the stock will go up or down.