## The Setup

Suppose I have a stochastic process $f(Z_t)$ where $Z_t$ solve the $d$-dimensional SDE $$ dZ_t = \mu(t,Z_t)dt + \sigma(t,Z_t)dW_t $$ and $f$ is a smooth function.

## My Question

Is there a notion of *time-derivative* "$d_t$" of the process $f(Z_t)$ which satisfies:

- Some sort of chain rule like $$ d_tf(Z_t) = \partial_t f(Z_t) d_t(Z_t), $$ where $\partial_t$ is the usual derivative wrt $t$.
- If $Z_t$ is deterministic (ie: $\sigma(t,Z_t)=0$) and $\mu(t,z)$ is $C^1$ in $t$ then $$d_t=\partial_t,$$ ie: $d_t$ reduces to the usual derivative when $f(Z_t)$ is a smooth function of $t$.