Generalized Itǒ Formulae and Space-Time Lebesgue–Stieltjes Integrals of Local Times

Abstract

Generalised Itô formulae are proved for time dependent functions of continuous real valued semi-martingales.The conditions involve left space and time first derivatives, with the left space derivative required to have locally bounded 2-dimensional variation. In particular a class of functions with discontinuous first derivative is included. An estimate of Krylov allows further weakening of these conditions when the semi-martingale is a diffusion.

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

“…We need the following simple inequalities: Let f be a nonnegative and nondecreasing function, then 5) and for any v ≥ 1, 6) if the series…”

“…In fact investigations already began in Tanaka [28] with a beautiful use of local times introduced in Lévy [17]. The generalized Itô's formula in one-dimension for time independent convex functions was developed in Meyer [22] and for superharmonic functions in multidimensions in Brosamler [5] and for distance function in Kendall [15] and more recently for time dependent functions in Peskir [25], Ghomrasni and Peskir [12] and Elworthy, Truman and Zhao [6]. Meyer [22] proved if f is a convex function (or difference of two convex functions), then…”

“…We remark that the striking fact that L t (x) is of bounded quadratic variation in x in the sense of Revuz and Yor [26] and increasing in t did not play a significant role in the proof of (1.4). It is therefore reasonable to conjecture that the conditions in [6] defining the integrals of local times pathwise can be weakened. Inevitably, we have to go beyond Lebesgue-Stieltjes integral as it seems to us that Elworthy-Truman-Zhao's formula has achieved the best in the Lebesgue-Stieltjes integral framework.…”

“…Then if there exist monotone increasing functions ̺ and σ subject to ̺(u)σ(u) = u such that 6) then the integral…”

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 .…”