Local Time-Space Calculus and Extensions of Itô’s Formula

The fundamental result of stochastic calculus is Itô’s formula (2.1) firstly established by Itô [5] for a standard Brownian motion and then later extended to continuous semimartingales by Kunita and Watanabe [7]. [For simplicity in this article we will not consider semimartingales with jumps.] The function F appearing in Itô’s formula is C2 in the space variable, and the correction to the classic Leibnitz-Newton formula (the final term in (2.1) below) is expressed by means of the quadratic variation (2.2). Various extensions of the Itô formula have been established for functions F which are not C2 in the space variable. The best know of these extensions is the Itô-Tanaka formula (2.3) firstly derived by Tanaka [13] for F (x) = jxj and then extended to absolutely continuous F with F 0 of bounded variation by Meyer [9] and Wang [14]. The correction term appearing in this formula is expressed by means of the local time (2.4) which goes back to Lévy [8] (see e.g. [6] or [12]). A different extension to absolutely continuous F with locally bounded F 0 due to Bouleau and Yor [1] is given in (2.5) below. The correction term appearing in this formula is also expressed by means of the local time (2.4), however, in a different manner which suggests a formal integration by parts. Both formulas (2.3) and (2.5) are derived only in dimension one. Motivated by applications in free-boundary problems of optimal stopping [10] we have recently derived an extension of Itô’s formula [11] stated in (2.11) below. The most interesting in this