Abstract.
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 Itô'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.
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Received: 1 March 1999 / Revised version: 15 April 2001 / Published online: 11 December 2001
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Errami, M., Russo, F. & Vallois, P. Itô's formula for C 1,λ-functions of a càdlàg process and related calculus. Probab Theory Relat Fields 122, 191–221 (2002). https://doi.org/10.1007/s004400100168
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DOI: https://doi.org/10.1007/s004400100168
Keywords
- Variation Process
- Quadratic Variation
- Stochastic Calculus
- Smoothness Property
- Related Calculus