I've just learned about integrating a formula of the form: $$\int_{0}^t X_u\mathrm{d}W_u$$
Where the integrator $W_u$ is the brownian motion. For me, this is new territory.
I am only aware of integrating with respect to time. But now it's with respect to the brownian motion. What exactly does this mean?
I read somewhere $X_u$ represents a numerical value. Where $X:\Omega \to \mathbb{R} $. With $\Omega$ representing all possible outcomes.
So more concrete, consider a game of coin tossing where you win 1\$ each time it's heads and you conversely lose 1\$ when it's not. $X_u$ could then in one instance represent 2$ (a possible outcome) after a certain time t. The probability for this event happening after time t, is taken from that Brownian motion?
So as I see it now, the formula: $$\int_{0}^t X_u\mathrm{d}W_u$$ considers all those possible outcomes $X_u$ weighted with their respective probabilities taken from $W_u$ and integrates over that?
