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Malliavin Derivative with Mathematica is it possible?

Posted 10 years ago

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}}]

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