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: 


1. 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 $t$.
2. 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 $t$.



Does there exist a stochastic time derivative?
The Malliavin derivative satisfies the requisite chain rule. However, if $Z\_t$ is no longer a function of the Wiener process, then the Malliavin derivative of $f(Z\_t)$ is identically zero. 


To see this, we briefly recall the nice link between the Girsanov theorem and Malliavin calculus. Let $\epsilon>0$ be a perturbation parameter and let $\phi \in L^2([0,T]; \mathbb{R}^d)$. Recall that Girsanov theorem says that $Z\_t$ (given by the OP) on the probability space $(\Omega, \mathcal{F}, P)$ is a *weak* solution to $$
dZ\_t = \mu(t,Z\_t) dt + \sigma(t,Z\_t) \left( d W\_t + \epsilon \phi(t) dt \right)
$$ 
on the probability space $(\Omega, \mathcal{F}, \tilde P)$ where
$$
 \frac{d \tilde P}{dP}= \exp\left( -\frac{\epsilon^2}{2} \int\_0^T | \phi(s)|^2 ds + \epsilon \int\_0^T \phi(s) dW(s) \right) 
$$
More to the point, the variations taken in the Malliavin derivative are precisely of this form: the Wiener process is varied arbitrarily in the direction of the deterministic $\phi(\cdot)$. From this link we immediately see that if $\sigma=0$, then these variations have no effect on $f(Z\_t)$.


**Reference**


Nualart, D. (2006). **The Malliavin calculus and related topics.** Springer.


