On stochastic derivative: an apparent contradiction(?)

Question

If I understood it correctly, for any given stochastic process $Y_t$, generally depending upon position $x$ and time $t$, a function $f(Y_t,t)$ is said to be differentiable with respect to $Y_t$ if its derivative $$\tag{1}\frac{\partial f(y,t)}{\partial y}$$ should exist for all possible realizations $y(x,t)$ of $Y_t$. Isn't this contradictory? For suppose $f$ is the identity function. It's clearly everywhere differentiable, but the Wiener process, which can be seen as the identity function applied to it, is not.

Answer 1

The natural derivative in stochastic calculus is the Malliavin Derivative. Take a look in these posts: Wiki, Malliavin derivative

Let $\mathcal H$ be a separable Hilbert. Let $W_t$ is the $1$-dimensional Wiener process. Consider $$ F=f(W(h_1),W(h_2),\ldots,W(h_n)),\ h_1, h_2,\ldots,h_n\in\mathcal{H}, \, n\ge 1, $$ where $f$ is a smooth function with their partial derivatives have polynomial growth. We define Malliavin's derivative of $F$ as the random variable value: $$ \displaystyle DF=\sum_{k=1}^n\frac{\partial f}{\partial x_k}(W(h_1), W(h_2),\ldots,W(h_n))h_k. $$ It satisfies the following properties:

1. $\displaystyle D(FG)=FDG+GDF.$
2. $\displaystyle \mathbf{E}(\langle DF,h\rangle_{\mathcal{H}})=\mathbf{E}(FW(h))$
3. $\displaystyle \mathbf{E}(G\langle DF,h\rangle_{\mathcal{H}})=\mathbf{E}(-F\langle DG,h \rangle_{\mathcal{H}}+FGW(h))$