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Unifying the Dynkin and Lebesgue–Stieltjes formulae

Part of: Special processes Markov processes Stochastic processes

Published online by Cambridge University Press:  04 April 2017

Offer Kella  and
Marc Yor
Offer Kella*
Affiliation:
The Hebrew University of Jerusalem
*
* Postal address: Department of Statistics, The Hebrew University of Jerusalem, Jerusalem 9190501, Israel. Email address: [email protected]
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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)=∑k=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.

Keywords

Lévy systemMarkov processjump diffusionlocal martingaleDynkin’s formula

MSC classification

Primary: 60G44: Martingales with continuous parameter
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60G51: Processes with independent increments; L'evy processes 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 1 , March 2017 , pp. 252 - 266
Copyright
Copyright © Applied Probability Trust 2017 

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