A note on the Malliavin derivative operator under change of variable

Abstract

The Malliavin derivative operator is classically defined with respect to the standard Brownian motion on the Wiener space C 0 [ 0 , T ] . We define the Malliavin derivative with respect to arbitrary Brownian motions on general probability spaces and compute how the Malliavin derivative of a functional on the Wiener space changes when the functional is composed with transformation by a process which is sufficiently smooth. We then use this result to derive a formula which says how the Malliavin derivatives with respect to different Brownian motions on the same state space are related to each other. This has applications in many situations in Mathematical Finance, where Malliavin calculus is used.