I have a very simple question about a notation:
$$dW^1dW^2 = \rho dt\tag{1}$$
where $W^1, W^2$ are BM. Usually this is translated to "correlated Brownian motion". So my first question:
1. Is $(1)$ a short notation for $E[W^1W^2]=\rho t$?
Moreover, why does it follow that $d\langle W^1, W^2\rangle = \rho dt$? The bracket process attains a version which is the quadratic variation, so different from correlation. It would be appreciated if someone could clarify these different notation usually seen in Quantitative Finance papers. Many thanks
My bet would be that the definition is in fact through the quadratic covariation: $\langle w_t,\bar w_t\rangle = \rho t$ for all $t$. Then, as a result from Ito lemma you get: $$ \mathrm dw_t\bar w_t = w_t\mathrm d\bar w_t + \bar w_t\mathrm dw_t + \mathrm d\langle w_t,\bar w_t\rangle $$ which gives you that $\Bbb Ew_t\bar w_t = \rho t$. I am not sure whether it works the other way around. Personally, whenever I read such assumption, I think of it as $\bar w_t = \rho w_t + \sqrt{1 - \rho^2}b_t$ where $b$ is a Brownian motion independent from $w$.