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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.… (More)
Abstract We prove a moment identity on the Wiener space that extends the Skorohod isometry to arbitrary powers of the Skorohod integral on the Wiener space. As simple consequences of this identity we obtain sufficient conditions for the Gaussianity of the law of the Skorohod integral and a recurrence relation for the moments of second order Wiener… (More)
We show that two multiple stochastic integrals I n (f n), I m (g m) with respect to the solution (M t) t2I R + of a deterministic structure equation are independent if and only if two contractions of f n and g m , denoted as f n 0 1 g m , f n 1 1 g m , vanish almost everywhere.
We study the absolute continuity of transformations defined by anticipative flows on Poisson space, and show that the process of densities associated to those transformations allows to solve anticipative linear stochastic differential equations on the Poisson space. Mathematics Subject Classification: 60H05, 60H07, 60J75.
These notes survey some aspects of discrete-time chaotic calculus and its applications, based on the chaos representation property for i.i.d. sequences of random variables. The topics covered include the Clark formula and predictable representation, anticipating calculus, covariance identities and functional inequalities (such as deviation and logarithmic… (More)
Concentration inequalities are obtained on Poisson space, for random functionals with finite or infinite variance. In particular, dimension free tail estimates and exponential integrability results are given for the Euclidean norm of vectors of independent functionals. In the finite variance case these results are applied to infinitely divisible random… (More)
We study a new interpretation of the Poisson space as a triplet (H,B, P ) where H is a Hilbert space, B is the completion of H and P is the extension to the Borel σ−algebra of B of a cylindrical measure on B. A discrete chaotic decomposition of L2(B,P ) is defined, along with multiple stochastic integrals of elements of Hon. It turns out that the… (More)
The advance in RF energy transfer and harvesting technique over the past decade has enabled wireless energy replenishment for electronic devices, which is deemed as a promising alternative to address the energy bottleneck of conventional battery-powered devices. In this paper, by using a stochastic geometry approach, we aim to analyze the performance of an… (More)
This paper aims to construct adaptedness and stochastic integration on Poisson space in the abstract setting of Hilbert spaces with minimal hypothesis, in particular without use of any notion of time or ordering on index sets. In this framework, several types of stochastic integrals are considered on simple processes and extended to larger domains. The… (More)
Given (Mt)t∈R+ and (M ∗ t )t∈R+ respectively a forward and a backward martingale with jumps and continuous parts, we prove that E[φ(Mt + M ∗ t )] is nonincreasing in t when φ is a convex function, provided the local characteristics of (Mt)t∈R+ and (M ∗ t )t∈R+ satisfy some comparison inequalities. We deduce convex concentration inequalities and deviation… (More)