## The Annals of Probability

- Ann. Probab.
- Volume 13, Number 2 (1985), 609-615.

### 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.

#### Article information

**Source**

Ann. Probab., Volume 13, Number 2 (1985), 609-615.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993013

**Digital Object Identifier**

doi:10.1214/aop/1176993013

**Mathematical Reviews number (MathSciNet)**

MR781427

**Zentralblatt MATH identifier**

0562.60067

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60H99: None of the above, but in this section

Secondary: 60J65: Brownian motion [See also 58J65]

**Keywords**

Malliavin derivatives Malliavin calculus derivatives of Wiener functionals the infinite dimensional Ornstein-Uhlenbeck process

#### Citation

Zakai, Moshe. Malliavin Derivatives and Derivatives of Functionals of the Wiener Process with Respect to a Scale Parameter. Ann. Probab. 13 (1985), no. 2, 609--615. doi:10.1214/aop/1176993013. https://projecteuclid.org/euclid.aop/1176993013