Directional Malliavin Derivatives: A Characterisation of Independence and a Generalised Chain Rule

We define a directional Malliavin derivative connected to a continuous linear operator. We show that this directional Malliavin derivative being zero is equivalent to some measurability or independence condition on the random variable. Using this, we obtain that two random variables, whose classical Malliavin derivatives live in orthogonal subspaces, are independent. We also extend the chain rule to directional Malliavin derivatives and a broader class of functions with weaker regularity assumptions.