We understand SDEs like "$dX_t = b(t,X_t)dt + \sigma(t,X_t)dB_t$" for Brownian process $B$ to be formally the same as "$\frac{dX_t}{dt} = b(t,X_t) + \sigma(t,X_t)W_t$" where $W$ is a ``white noise" which is not a stochastic process in the usual sense of the term.

I wanted to know as to how far does this go : Like can I define a product of derivatives of two stochastic processes $X_t$ above and another $Y_t$ which is suppose given as, $dY_t = b_1(X_t,t)dt + \sigma_1(X_t,t)dB_t$?

- Like is there meaning to writing something like,

$$ \frac{dX_t}{dt} \cdot \frac{dY_t}{dt} = (b(t,X_t) + \sigma(t,X_t)W_t )\cdot (b_1(t,X_t) + \sigma_1(t,X_t)W_t ) $$

- Further, is there a way to define $\mathbb{E} \left [ \frac{dX_t}{dt} \cdot \frac{dY_t}{dt}\right ]$ from the above and compute it? If yes, can someone kindly show/reference it?

define$dX_t/dt$ in the above way, then of course you can multiply these random random variables. Usually the Itô integral is a fundamental object, and "$dX_t = U_t dt + V_t dB_t$" is understood as $X_t = X_0 + \int_0^t U_t dt + \int_0^t V_t dB_t$. $\endgroup$