TL;DR
This paper extends the functional Itô formula by establishing a Meyer-Tanaka formula for functionals of semimartingales, analyzing mollification techniques, and applying results to running maximum and max-martingales.
Contribution
It introduces a functional Meyer-Tanaka formula for semimartingales, expanding the functional Itô calculus framework with new convergence analysis methods.
Findings
Proved the functional Meyer-Tanaka formula for semimartingales.
Analyzed mollification and convergence properties of functionals.
Applied the theory to running maximum and max-martingales.
Abstract
The functional Ito formula, firstly introduced by Bruno Dupire for continuous semimartingales, might be extended in two directions: different dynamics for the underlying process and/or weaker assumptions on the regularity of the functional. In this paper, we pursue the former type by proving the functional version of the Meyer-Tanaka Formula. Following the idea of the proof of the classical time-dependent Meyer-Tanaka formula, we study the mollification of functionals and its convergence properties. As an example, we study the running maximum and the max-martingales of Yor and Obloj.
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