Did someone extend stochastic calculus beyond time only? For example, immagine a variable $S$ exists in a $t-w$ plane. Then, the dynamics of this variable would follow a SDE of the like:
$dS(t,w) = \mu_t dt + \mu_w dw + \sigma_t dW(t) + \sigma_w dW(w)$
And if we picked a function of said variable, say $f(t,w,S(t,w))$, it would follow:
$df(t,w,S(t,w)) = \frac{\partial f}{\partial t} \mu_t dt + \frac{\partial f}{\partial w} \mu_w dw + \frac{1}{2}\frac{\partial^2 f}{\partial t^2} \sigma_t^2 dt + \frac{1}{2}\frac{\partial^2 f}{\partial w^2} \sigma_w^2 dw$ + etc.
Did anyone see this before and has papers to link?
Thanks.
Note: I'm not interested in multivariate Ito or when S is a vector or a matrix.