Two-parameter p, q-variation Paths and Integrations of Local Times

Abstract

Summary. In this paper, we prove two main results. The first one is to give a new condition for the existence of two-parameter p, q-variation path integrals. Our condition of locally bounded p, q-variation is more natural and easy to verify than those of Young. This result can be easily generalized to multi-parameter case. The second result is to define the integral of local time

Rough path integral of local time

“…We call Z the canonical geometric rough path associated with Z. About the first level path Z(m) 1 a,b , the method and results are similar to those in Chapter 4 in [3]. We can prove sup m sup D l |Z(m) 1…”

“…Let [x , x ] be any interval in R. From the proof of Lemma 2.1 in [1], for any p 2, we know that there exists a constant c > 0 such that E|L b t −L a t | p c|b − a| p/2 , i.e.L x t satisfies the Hölder condition in [3] with exponent 1 2 . Denote by w a control of g(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]. For K = 1 and H = 1 2 , the process B is a standard fractional Brownian motion B H with Hurst index H. Yan et al [25,27] studied the integration with respect to local time of fractional Brownian motion, and the weighted quadratic covariation [f (B H ), B H ] (W ) of f (B H ) and B H .…”

Integration with Respect to Fractional Local Time with Hurst Index 1/2 < H < 1

“…Still there are always some new problems requiring the use of "Itô formula" under lighter conditions. One approach of extending Itô's formula is by using local timespace calculus to absolutely continuous function F with locally bounded measurable derivative F (see Bouleau and Yor [3], Eisenbaum [5,6], Feng and Zhao [7], Föllmer et al [8], Peskir [23], Russo and Vallois [24], Yan and Yang [26], and the references therein). Moreover, the backward integral and quadratic covariation are fundamental tools in these discussions.…”