Malliavin Derivative with Mathematica is it possible?

Questions
1

Is it possible to define a Malliavin calculus with *Mathematica?

Consider a random variables on the Wiener-space \Omega=\mathcal{C}([0,1]) of the form

F=F(\omega)=\displaystyle\int_{0}^{T}h_{t}dW_{t}

for some deterministic h(.) in L^{2}[0,T].
Here, \omega is a path in the Wiener-space, \omega \in \Omega.

We define the Malliavin derivative by

DF=h,\text{ and }D_{t}F=h_{t}.

Let say MalliavinDer is this derivative. We have the following rules

  1. MalliavinDer[F_ + G_] := MalliavinDer[F]+MalliavinDer[G];

  2. MalliavinDer[F_G_]:=MalliavinDer[F]G+ MalliavinDer[G]F

  3. MalliavinDer[f(F)]:=f'[F]MalliavinDer[F].

On the other hand, we have for example:

  1. MalliavinDer[\int_0^T h(t)dWt] := h(t)

  2. MalliavinDer[W_s] := Piecewise[{{1, s <t}, {0, s >t}}]

  3. MalliavinDer[f(W_s] := f'[W_s])Piecewise[{{1, s <t}, {0, s >t}}]