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Unifying the Dynkin and Lebesgue–Stieltjes formulae

Part of: Markov processes Special 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: offer.kella@gmail.com
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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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References

[1] Andersen, L. N. et al. (2015).Lévy Matters (Lecture Notes Math. 2149).Springer,Cham.CrossRefGoogle Scholar
[2] Applebaum, D. (2009).Lévy Processes and Stochastic Calculus, 2nd edn.Cambridge University Press.Google Scholar
[3] Asmussen, S. (2003).Applied Probability and Queues, 2nd edn.Springer,New York.Google Scholar
[4] Asmussen, S. and Albrecher, H. (2010).Ruin Probabilities, 2nd edn.World Scientific,River Edge.Google Scholar
[5] Asmussen, S. and Kella, O. (2000).A multi-dimensional martingale for Markov additive processes and its applications.Adv. Appl. Prob. 32,376–393.Google Scholar
[6] Asmussen, S. and Kella, O. (2001).On optional stopping of some exponential martingales for Lévy processes with or without reflection.Stoch. Process. Appl. 91,47–55.Google Scholar
[7] Asmussen, S., Avram, F. and Pistorius, M. R. (2004).Russian and American put options under exponential phase-type Lévy models.Stoch. Process. Appl. 109,79–111.Google Scholar
[8] Asmussen, S. and Pihlsgård, M. (2007).Loss rates for Lévy processes with two reflecting barriers.Math. Operat. Res. 32,308–321.Google Scholar
[9] Boxma, O. and Kella, O. (2014).Decomposition results for stochastic storage processes and queues with alternating Lévy inputs.Queueing Systems 77,97–112.Google Scholar
[10] Boxma, O.,Perry, D. and Stadje, W. (2001). Clearing models for M∕G∕1 queues.Queueing Systems 38,287–306.Google Scholar
[11] Dębicki, K. and Mandjes, M. (2015).Queues and Lévy Fluctuation Theory.Springer,Cham CrossRefGoogle Scholar
[12] Frostig, E. (2005).The expected time to ruin in a risk process with constant barrier via martingales.Insurance Math. Econom. 37,216–228.Google Scholar
[13] Kella, O. and Boxma, O. (2013).Useful martingales for stochastic storage processes with Lévy-type input.J. Appl. Prob. 50,439–449.Google Scholar
[14] Kella, O. and Whitt, W. (1992).Useful martingales for stochastic storage processes with Lévy input.J. Appl. Prob. 29,396–403.Google Scholar
[15] Kyprianou, A. E. (2006).Fluctuations of Lévy Processes with Applications.Springer,Berlin.Google Scholar
[16] Kyprianou, A. E. (2013).Gerber–Shiu Risk Theory.Springer,Cham.Google Scholar
[17] Kyprianou, A. E. and Palmowski, Z. (2004).A Martingale review of some fluctuation theory for spectrally negative Lévy processes Springer, (Lecture Notes Math. 1857).Berlin, pp.16–29.Google Scholar
[18] Nguyen-Ngoc, L. and Yor, M. (2005).Some Martingales Associated to Reflected Lévy Processes, (Lecture Notes Math. 1857).Springer,Berlin, pp.42–69.Google Scholar
[19] Palmowski, Z. and Rolski, T. (2002).A technique for exponential change of measure for Markov processes.Bernoulli 8,767–785.Google Scholar
[20] Perry, D., Stadje, W. and Yosef, R. (2003).Annuities with controlled random interest rates.Insurance Math. Econom. 32,245–253.Google Scholar
[21] Protter, P. E. (2004).Stochastic Integration and Differential Equations, 2nd edn.Springer,Berlin.Google Scholar
[22] Rüdiger, B. (2004). Stochastic integration with respect to compensated Poisson random measures on separable Banach spaces.Stoch. Stoch. Rep. 76,213–242.Google Scholar
[23] Whitt, W. (2002).Stochastic Process Limits,Springer,New York CrossRefGoogle Scholar