Positivity and lower bounds for the density of Wiener functionals

We consider a functional on the Wiener space which is smooth and not degenerated in Malliavin sense and we give a criterion of strict positivity of the density. We also give lower bounds for the density. These results are based on the representation of the density by means of the Riesz transform introduced by Malliavin and Thalmaier and on the estimates of the Riesz transform given Bally and Caramellino

Large deviation estimates of the crossing probability for pinned Gaussian processes

The paper deals with the asymptotic behavior of the bridge of a Gaussian process conditioned to stay in $n$ fixed points at $n$ fixed past instants. In particular, functional large deviation results are stated for small time. Several examples are considered: integrated or not fractional Brownian motion, $m$-fold integrated Brownian motion. As an application, the asymptotic behavior of the exit probability is studied and used for the practical purpose of the numerical computation, via Monte Carlo methods, of the hitting probability up to a given time.Comment: 33 pages. Keywords: conditioned Gaussian processes; reproducing kernel Hilbert spaces; large deviations; exit time probabilities; Monte Carlo method

Regularity of Wiener functionals under a H\"ormander type condition of order one

We study the local existence and regularity of the density of the law of a functional on the Wiener space which satisfies a criterion that generalizes the H\"ormander condition of order one (that is, involving the first order Lie brackets) for diffusion processes

A hybrid tree/finite-difference approach for Heston-Hull-White type models

We study a hybrid tree-finite difference method which permits to obtain efficient and accurate European and American option prices in the Heston Hull-White and Heston Hull-White2d models. Moreover, as a by-product, we provide a new simulation scheme to be used for Monte Carlo evaluations. Numerical results show the reliability and the efficiency of the proposed method

Convergence in Total Variation for nonlinear functionals of random hyperspherical harmonics

Random hyperspherical harmonics are Gaussian Laplace eigenfunctions on the unit d-dimensional sphere (d >= 2). We study the convergence in Total Variation distance for their nonlinear statistics in the high energy limit, i.e., for diverging sequences of Laplace eigenvalues. Our approach takes advantage of a recent result by Bally, Caramellino and Poly (2020): combining the Central Limit Theorem in Wasserstein distance obtained by Marinucci and Rossi (2015) for Hermite-rank 2 functionals with new results on the asymptotic behavior of their Malliavin-Sobolev norms, we are able to establish second order Gaussian fluctuations in this stronger probability metric as soon as the functional is regular enough. Our argument requires some novel estimates on moments of products of Gegenbauer polynomials that may be of independent interest, which we prove via the link between graph theory and diagram formulas. (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:/

On Sharp Large Deviations for the bridge of a general Diffusion

We provide sharp Large Deviation estimates for the probability of exit from a domain for the bridge of a $d$-dimensional general diffusion process $X$, as the conditioning time tends to $0$. This kind of results is motivated by applications to numerical simulation. In particular we investigate the influence of the drift $b$ of $X$. It turns out that the sharp asymptotics for the exit time probability are independent of the drift, provided $b$ enjoyes a simple condition that is always satisfied in dimension $1$. On the other hand, we show that the drift can be influential if this assumption is not satisfied.

Using moment approximations to study the density of jump driven SDEs

In order to study the regularity of the density of a solution of a infinite activity jump driven stochastic differential equation we consider the following two-step approximation method. First, we use the solution of the moment problem in order to approximate the small jumps by another whose Lévy measure has finite support. In a second step we replace the approximation of the first two moments by a small noise Brownian motion based on the Assmussen-Rosinski approach. This approximation needs to satisfy certain properties in order to apply the "balance" method which allows the study of densities for the solution process based on Malliavin Calculus for the Brownian motion. Our results apply to situations where the Lévy measure is absolutely continuous with respect to the Lebesgue measure or purely atomic measures or combinations of them

Temporal stability of stimulus representation increases along rodent visual cortical hierarchies

Cortical representations of brief, static stimuli become more invariant to identity-preserving transformations along the ventral stream. Likewise, increased invariance along the visual hierarchy should imply greater temporal persistence of temporally structured dynamic stimuli, possibly complemented by temporal broadening of neuronal receptive fields. However, such stimuli could engage adaptive and predictive processes, whose impact on neural coding dynamics is unknown. By probing the rat analog of the ventral stream with movies, we uncovered a hierarchy of temporal scales, with deeper areas encoding visual information more persistently. Furthermore, the impact of intrinsic dynamics on the stability of stimulus representations grew gradually along the hierarchy. A database of recordings from mouse showed similar trends, additionally revealing dependencies on the behavioral state. Overall, these findings show that visual representations become progressively more stable along rodent visual processing hierarchies, with an important contribution provided by intrinsic processing