arXiv subjects

Nicolas Privault

Publications and source records attributed to Nicolas Privault.

62 records · Page 4Linked to original sources

arXiv2013

Feynman-Kac formula for Levy processes and semiclassical (Euclidean) momentum representation

We prove a version of the Feynman-Kac formula for Levy processes and integro-differential operators, with application to the momentum representation of suitable quantum (Euclidean) systems whose Hamiltonians involve Lévy-type potentials. Large deviation techniques are used to obtain the limiting behavior of the systems as the Planck constant approaches zero. It turns out that the limiting behavior coincides with fresh aspects of the semiclassical limit of (Euclidean) quantum mechanics. Non-trivial examples of Levy processes are considered as illustrations and precise asymptotics are given for the terms in both configuration and momentum representations.

math.PR↗

arXiv2013

Hedging in bond markets by the Clark-Ocone formula

Hedging strategies in bond markets are computed by martingale representation and the Clark-Ocone formula under the choice of a suitable of numeraire, in a model driven by the dynamics of bond prices. Applications are given to the hedging of swaptions and other interest rate derivatives, and our approach is compared to delta hedging when the underlying swap rate is modeled by a diffusion process.

q-fin.PR↗

arXiv2012

Girsanov identities for Poisson measures under quasi-nilpotent transformations

We prove a Girsanov identity on the Poisson space for anticipating transformations that satisfy a strong quasi-nilpotence condition. Applications are given to the Girsanov theorem and to the invariance of Poisson measures under random transformations. The proofs use combinatorial identities for the central moments of Poisson stochastic integrals.

math.PR↗

arXiv2012

Moments of Poisson stochastic integrals with random integrands

We compute the moment of order n of the Poisson stochastic integral of a random process u over a metric space X as a sum that runs over all partitions of {1,...,n} and involves the addition of points to Poisson configurations. This formula recovers known results in case u is a deterministic function on X.

math.PR↗

arXiv2011

Invariance of Poisson measures under random transformations

We prove that Poisson measures are invariant under (random) intensity preserving transformations whose finite difference gradient satisfies a cyclic vanishing condition. The proof relies on moment identities of independent interest for adapted and anticipating Poisson stochastic integrals, and is inspired by the method applied in [22] on the Wiener space, although the corresponding algebra is more complex than in the Wiener case. The examples of application include transformations conditioned by random sets such as the convex hull of a Poisson random measure.

math.PR↗

arXiv2009

SURE shrinkage of Gaussian paths and signal identification

Using integration by parts on Gaussian space we construct a Stein Unbiased Risk Estimator (SURE) for the drift of Gaussian processes using their local and occupation times. By almost-sure minimization of the SURE risk of shrinkage estimators we derive an estimation and de-noising procedure for an input signal perturbed by a continuous-time Gaussian noise.

math.ST↗

arXiv2008

Stein estimation for the drift of Gaussian processes using the Malliavin calculus

We consider the nonparametric functional estimation of the drift of a Gaussian process via minimax and Bayes estimators. In this context, we construct superefficient estimators of Stein type for such drifts using the Malliavin integration by parts formula and superharmonic functionals on Gaussian space. Our results are illustrated by numerical simulations and extend the construction of James--Stein type estimators for Gaussian processes by Berger and Wolpert [J. Multivariate Anal. 13 (1983) 401--424].

math.ST↗

arXiv2008

Convex ordering for random vectors using predictable representation

We prove convex ordering results for random vectors admitting a predictable representation in terms of a Brownian motion and a non-necessarily independent jump component. Our method uses forward-backward stochastic calculus and extends previous results in the one-dimensional case. We also study a geometric interpretation of convex ordering for discrete measures in connection with the conditions set on the jump heights and intensities of the considered processes.

math.PR↗