Generalised Clark-Ocone formulae for differential forms

Generalised Clark-ocone Formulae for Differential Forms

We generalise the Clark-Ocone formula for functions to give analogous representations for differential forms on the classical Wiener space. Such formulae provide explicit expressions for closed and co-closed differential forms and, as a by-product, a new proof of the triviality of the L de Rham cohomology groups on the Wiener space, alternative to Shigekawa’s approach [15] and the chaos-theoret...

Weitzenböck and Clark-Ocone decompositions for differential forms on the space of normal martingales∗

We present a framework for the construction of Weitzenböck and ClarkOcone formulae for differential forms on the probability space of a normal martingale. This approach covers existing constructions based on Brownian motion, and extends them to other normal martingales such as compensated Poisson processes. It also applies to the path space of Brownian motion on a Lie group and to other geometr...

Probability approximation by Clark-Ocone covariance representation

Based on the Stein method and a general integration by parts framework we derive various bounds on the distance between probability measures. We show that this framework can be implemented on the Poisson space by covariance identities obtained from the Clark-Ocone representation formula and derivation operators. Our approach avoids the use of the inverse of the Ornstein-Uhlenbeck operator as in...

An extension of the Clark-Ocone formula

The Clark-Haussmann-Ocone theorem WHITE NOISE GENERALIZATIONS OF THE CLARK-HAUSSMANN-OCONE THEOREM, WITH APPLICATION TO MATHEMATICAL FINANCE

We use a white noise approach to Malliavin calculus to prove the following white noise generalization of the Clark-Haussmann-Ocone formula F (ω) = E[F ] + T 0 E[D t F |F t ] ⋄ W (t)dt Here E[F ] denotes the generalized expectation, D t F (ω) = dF dω is the (generalized) Malliavin derivative,⋄ is the Wick product and W (t) is 1-dimensional Gaussian white noise. The formula holds for all f ∈ G * ...