Sur la variation quadratique de certaines mesures vectorielles

Abstract

Soit B tun mouvement brownien r6el, /~t son temps local en a, et soit a =a 1 <a2< ... <a,=b une subdivision variable de ~ telle que a--* -o% b-~ + 00 et dont le pas tend vers z6ro, on a alors au sens L2:i Par ailleurs si 0<t 1 <.,.< t,,=t est une subdivision de [0, t] dont le pas tend vers z6ro, on a de m6me le r6sultat classique suivant:i iLe point de d6part de ce travail est la remarque que la coincidence entre (1) et (2) n'est pas fortuite mais tient au fait que la variation quadratique d'une semimartingale continue peut s'obtenir comme limite de sommes quadratiques selon une large classe de partitions de la tribu pr6visible. L'6tude nous conduit/t une d6finition de mesures vectorielles admettant une variation quadratique. Pour de telles mesures, la variation quadratique v6rifie une propri6t6 de stabilit6 par image analogue ~t la masse d'une mesure positive born6e.

An Extension of Ito’s Differentiation Formula

“…This corollary holds without the hypothesis that F(t, 0) = 0; suppose F(t, x) satisfies the hypotheses of the corollary except possibly the condition F(t, 0) = 0. Then G(t, x) = F(t, x) -F(t, 0) satisfies all the hypotheses, and so the result holds for G. However, Denote the three terms in the expectation by 7 (1) , / (2) and 7 (3) , respectively. [ 4 ] Yamada, T., On some representations concerning stochastic integrals, to appear.…”

“…A key step in formulae (1) and (2) is the definition of the integrals with respect to dJJl for fixed t > 0. Recall Tanaka's formula for the local time at a:…”

“…By initially considering step functions of the form and linear combinations of expression (3), Bouleau and Yor [1] show that if…”

“…Lf denotes the local time of a continuous semimartingale X at a Bouleau and Yor [1] have obtained a form of Ito's differentiation formula which states that for absolutely continuous functions F(x) (…”

The generalized Bouleau-Yor identity for a sub-fractional Brownian motion

“…For K = 1 and H = 1 2 , the process B is classical Brownian motion W and the above results first are studied by Bouleau-Yor [3] and Föllmer et al [9]. Moreover, these have also been extended to semimartingales by Bardina-Rovira [2], Eisenbaum [4,5], Elworthy et al [6], Feng-Zhao [8], Peskir [17], Rogers-Walsh [18], Yan-Yang [28].…”