Non-linear functionals of the Brownian bridge and some applications

Abstract

International audienceLet $\{b^{F}(t); t in [0; 1]\}$ be an F-Brownian bridge process. We study the asymptotic behaviour of non-linear functionals of regularizations by convolution of this process and apply these results to the estimation of the variance of a non-homogeneous diffusion and to the convergence of the number of crossings of a level by the regularized process to a modification of the local time of the Brownian bridge as the regularization parameter goes to 0

Asymptotics for theLp-deviation of the variance estimator under diffusion

“…. By using stable convergence, as in [2], we can deduce our results from the case b ≡ 0. In this particular case our process is a time-changed Brownian motion.…”

“…is proved in [2], where Σ 2 (t) is defined by equation (11). Alternative estimation techniques and some CLT are proposed in Soulier [17], Genon-Catalot et al [6] and in Brugière [3] under close settings.…”

“…where Σ 2 (t) is defined in equation (11) and it is also the limiting variance in the CLT as proved in [2].…”

Convergence of non-linear functionals of smoothed empirical processes and kernel density estimates

“…Using this viewpoint we can look at the asymptotics for a variant of the Kullback deviation. The results are obtained passing to functionals of the Brownian Bridge via the strong approximation theorem and then studying the asymptotic behaviour of these functionals using methods developed in a previous article [4]. With these same methods we can also study the number of crossings of a level by the Smoothed Empirical Process, defined by Yukich [17], as we explain in the next section, where we describe more precisely our results.…”

Deviation of orderpfor estimators of the variance in first-order stochastic differential equation (SDE)

“…In the first article, optimal rates of convergence are established, whereas in the latter a central limit theorem (CLT) is shown to hold and the L 2 deviation is studied. More recently, Berzin et al [8] estimate the variance of a non-homogeneous diffusion process using a smoothing by convolutions process. That is, one observes,…”