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
MalliavinDer[F_ + G_] := MalliavinDer[F]+MalliavinDer[G]
;
MalliavinDer[F_G_]:=MalliavinDer[F]G+ MalliavinDer[G]F
MalliavinDer[f(F)]:=f'[F]MalliavinDer[F]
.
On the other hand, we have for example:
MalliavinDer[\int_0^T h(t)dWt] := h(t)
MalliavinDer[W_s] := Piecewise[{{1, s <t}, {0, s >t}}]
MalliavinDer[f(W_s] := f'[W_s])Piecewise[{{1, s <t}, {0, s >t}}]