Malliavin Derivatives and Derivatives of Functionals of the Wiener Process with Respect to a Scale Parameter

Abstract

Let $F(c_0w)$ be a functional of the Wiener process with variance parameter $c^2_0$ and let $F(cw)$ be an extension of $F(c_0w)$ to $F(cw), c \in (0, c_0)$. Relations are derived between the Malliavin derivatives, between the derivatives with respect to the scale parameter $(\partial F(\rho cw)/\partial\rho)_{p = 1}$ and `noncoherent derivatives' such as $(dE(F(cw + \sqrt\varepsilon c\tilde{w}) \mid w)/d\varepsilon)_{\varepsilon = 0}$ where $\tilde{w}$ is another Wiener process independent of $w$ and between the generator of the nontime-homogeneous Ornstein-Uhlenbeck process.