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The Chaotic Representation Property of Compensated-Covariation Stable Families of Martingales
- Source :
- Ann. Probab. 44, no. 6 (2016), 3965-4005
- Publication Year :
- 2015
-
Abstract
- 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.
- Subjects :
- Statistics and Probability
[MATH.MATH-PR] Mathematics [math]/Probability [math.PR]
Monomial
Hermitian polynomials
Haar functions
Gaussian
Chaotic
Mathematics::General Topology
Teugels martingales
chaotic representation property
[MATH] Mathematics [math]
01 natural sciences
Lévy process
Stability (probability)
010104 statistics & probability
symbols.namesake
Mathematics::Probability
60H05
Square integrable martingales
FOS: Mathematics
Filtration (mathematics)
60G46
60G44
[MATH]Mathematics [math]
0101 mathematics
Representation (mathematics)
Mathematics
Discrete mathematics
Mathematics::Functional Analysis
Probability (math.PR)
010102 general mathematics
[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]
Square-integrable function
Lévy processes
symbols
60G57
Statistics, Probability and Uncertainty
60G51
Mathematics - Probability
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Journal :
- Ann. Probab. 44, no. 6 (2016), 3965-4005
- Accession number :
- edsair.doi.dedup.....0c0e905901c82719e348c688c4b1ceb7