×

Convolution of functionals of discrete-time normal martingales. (English) Zbl 1267.60079

The paper deals with functionals of stochastic processes in discrete time. At first, the authors recall some basic notions and facts such as discrete-time normal martingale, the full Wiener integral and the chaotic representation property. Then, they define the convolution on square integrable functionals of a discrete-time normal martingale \(M\), which satisfies some mild requirements. After that, the authors investigate its algebraic and analytical properties and show its interesting connection with a certain family of conditional expectation operators associated with \(M\). Finally, the authors present an example of a discrete-time normal martingale to show that the corresponding convolution has an integral representation.

MSC:

60H40 White noise theory
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
60G42 Martingales with discrete parameter
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Kallenberg, Foundations of Modern Probability (1997)
[2] Émery, Séminaire de Probabilités, XXXV pp 123– (2001) · doi:10.1007/978-3-540-44671-2_7
[3] Conway, A Course in Functional Analysis (1990)
[4] DOI: 10.1214/EJP.v15-843 · Zbl 1225.60089 · doi:10.1214/EJP.v15-843
[5] DOI: 10.1063/1.3431028 · Zbl 1310.81107 · doi:10.1063/1.3431028
[6] DOI: 10.1214/08-PS139 · Zbl 1189.60089 · doi:10.1214/08-PS139
[7] DOI: 10.1016/j.jmaa.2010.08.021 · Zbl 1205.60106 · doi:10.1016/j.jmaa.2010.08.021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.