Stein's method for Brownian approximations

Motivated by a theorem of Barbour, we revisit some of the classical limit theorems in probability from the viewpoint of the Stein method. We setup the framework to bound Wasserstein distances between some distributions on infinite dimensional spaces. We show that the convergence rate for the Poisson approximation of the Brownian motion is as expected proportional to $\lambda^{-1/2}$ where $\lambda$ is the intensity of the Poisson process. We also exhibit the speed of convergence for the Donsker Theorem and for the linear interpolation of the Brownian motion. By iterating the procedure, we give Edgeworth expansions with precise error bounds.Comment: Communications on Stochastic Analysis (2013

Enquête sur l’imaginaire du roman pornographique (1739‑1789) : les bibliothèques

Contrairement à ce que prétend traditionnellement la critique, il y a des bibliothèques dans le roman pornographique. En distinguant la bibliothèque comme lieu et la bibliothèque imaginaire, il s'agit de montrer comment la topographie de ce roman s'oppose à celle de l'utopie classique : il y a des objets dans l'oeuvre pornographique, mais dont le libertin se défait progressivement, histoire de rester seul avec sa victime et son désir. C'est ce que montre l'analyse d'une dizaine de romans publiés durant le demi-siècle qui précède la Révolution

Good rough path sequences and applications to anticipating stochastic calculus

We consider anticipative Stratonovich stochastic differential equations driven by some stochastic process lifted to a rough path. Neither adaptedness of initial point and vector fields nor commuting conditions between vector field is assumed. Under a simple condition on the stochastic process, we show that the unique solution of the above SDE understood in the rough path sense is actually a Stratonovich solution. We then show that this condition is satisfied by the Brownian motion. As application, we obtain rather flexible results such as support theorems, large deviation principles and Wong--Zakai approximations for SDEs driven by Brownian motion along anticipating vectorfields. In particular, this unifies many results on anticipative SDEs.Comment: Published at http://dx.doi.org/10.1214/009117906000000827 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Perturbed linear rough differential equations

International audienceWe study linear rough differential equations and we solve perturbed linear rough differential equation using the Duhamel principle. These results provide us with the key technical point to study the regularity of the differential of the Itô map in a subsequent article. Also, the notion of linear rough differential equations leads to consider multiplicative functionals with values in Banach algebra more general than tensor algebra and to consider extensions of classical results such as the Magnus and the Chen-Strichartz formula