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November 2016 The chaotic representation property of compensated-covariation stable families of martingales
Paolo Di Tella, Hans-Jürgen Engelbert
Ann. Probab. 44(6): 3965-4005 (November 2016). DOI: 10.1214/15-AOP1066


In the present paper, we study the chaotic representation property for certain families ${\mathscr{X}}$ of square integrable martingales on a finite time interval $[0,T]$. For this purpose, we introduce the notion of compensated-covariation stability of such families. The chaotic representation property will be defined using iterated integrals with respect to a given family ${\mathscr{X}}$ of square integrable martingales having deterministic mutual predictable covariation $\langle X,Y\rangle$ for all $X,Y\in{\mathscr{X}}$. The main result of the present paper is stated in Theorem 5.8 below: If ${\mathscr{X}}$ is a compensated-covariation stable family of square integrable martingales such that $\langle X,Y\rangle$ is deterministic for all $X,Y\in{\mathscr{X}}$ and, furthermore, the system of monomials generated by ${\mathscr{X}}$ is total in $L^{2}(\Omega,\mathscr{F}^{\mathscr{X}}_{T},\mathbb{P})$, then ${\mathscr{X}}$ possesses the chaotic representation property with respect to the $\sigma$-field $\mathscr{F}^{\mathscr{X}}_{T}$. We shall apply this result to the case of Lévy processes. Relative to the filtration $\mathbb{F}^{L}$ generated by a Lévy process $L$, we construct families of martingales which possess the chaotic representation property. As an illustration of the general results, we will also discuss applications to continuous Gaussian families of martingales and independent families of compensated Poisson processes. We conclude the paper by giving, for the case of Lévy processes, several examples of concrete families ${\mathscr{X}}$ of martingales including Teugels martingales.


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Paolo Di Tella. Hans-Jürgen Engelbert. "The chaotic representation property of compensated-covariation stable families of martingales." Ann. Probab. 44 (6) 3965 - 4005, November 2016.


Received: 1 June 2014; Revised: 1 September 2015; Published: November 2016
First available in Project Euclid: 14 November 2016

zbMATH: 1337.60080
MathSciNet: MR3572329
Digital Object Identifier: 10.1214/15-AOP1066

Primary: 60G44 , 60G51 , 60H05
Secondary: 60G46 , 60G57

Keywords: chaotic representation property , Haar functions , Hermitian polynomials , Lévy processes , Square integrable martingales , Teugels martingales

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 6 • November 2016
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