Itô's formula for C 1,λ -functions of a càdlàg process and related calculus

 This article develops a framework of stochastic calculus with respect to a càdlàg finite quadratic variation process. We apply it to the study of a generalization of a semimartingale driven SDE studied by Kurtz, Pardoux and Protter KPP]. We prove an It?'s formula for functions f(X) of a semimartingale with jumps when f has weak smoothness properties. Examples of X for which this formula is valid are time reversible semimartingales and solutions of KPP] equations driven by Lévy processes, provided the sum of the absolute values of the jumps, raised to the power 1 + λ, is a.s. finite, where λ takes values between 0 and 1. Received: 1 March 1999 / Revised version: 15 April 2001 / Published online: 11 December 2001