Unifying the Dynkin and Lebesgue-Stieltjes formulae

Offer Kella*, Marc Yor

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

We establish a local martingale M associate with f(X,Y) under some restrictions on f, where Y is a process of bounded variation (on compact intervals) and either X is a jump diffusion (a special case being a Lévy process) or X is some general (càdlàg metric-space valued) Markov process. In the latter case, f is restricted to the form f(x,y)=Σk=1Kξk(x)ηk(y). This local martingale unifies both Dynkin's formula for Markov processes and the Lebesgue-Stieltjes integration (change of variable) formula for (right-continuous) functions of bounded variation. For the jump diffusion case, when further relatively easily verifiable conditions are assumed, then this local martingale becomes an L2-martingale. Convergence of the product of this Martingale with some deterministic function (of time) to 0 both in L2 and almost sure is also considered and sufficient conditions for functions for which this happens are identified.

Original languageAmerican English
Pages (from-to)252-266
Number of pages15
JournalJournal of Applied Probability
Volume54
Issue number1
DOIs
StatePublished - 1 Mar 2017

Bibliographical note

Publisher Copyright:
Copyright © 2017 Applied Probability Trust.

Keywords

  • Dynkin's formula
  • Lévy system
  • Markov process
  • jump diffusion
  • local martingale

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