Quadratic variation of a càdlàg semimartingale as a.s. limit of the normalized truncated variations

ABSTRACT

For a real càdlàg path x, we define sequence of semi-explicit quantities, which do not depend on any partitions and such that whenever x is a path of a càdlàg semimartingale then these quantities tend a.s. to the continuous part of the quadratic variation of the semimartingale. Next, we derive several consequences of this result and propose a new approach to define Föllmer's pathwise integral.

KEYWORDS:

• Quadratic variation
• truncated variation
• càdlàg semimartingales
• Föllmer's pathwise integral

MATHEMATICS SUBJECT CLASSIFICATION (2010):

• 60G48
• 60H05

Acknowledgments

The author would like to thank the anonymous referees for their valuable comments which led to improvement of this work.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was partially founded by the National Science Centre, Poland [grant no. 2016/21/B/ST1/01489].