Journal Article10.1017/S0305004100070420
On homogeneous chaos
Nigel J. Cutland,Siu-Ah Ng +1 more
- 01 Sep 1991
- Vol. 110, Iss: 2, pp 353-363
TL;DR: In this paper, the authors discuss the Wiener-Ito chaos decomposition of an L 2 function <p over Wiener space, and are concerned in particular with the identification of the integrands /
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Abstract: This paper discusses the Wiener-Ito chaos decomposition of an L 2 function <p over Wiener space, and is concerned in particular with the identification of the integrands /„ in the chaos decomposition = &+ S [fndb™. n- l J First these are identified as Radon-Nikodym derivatives. Two elementary non-standard proofs of the Wiener-Ito chaos decomposition are given, based on Anderson's construction of Brownian motion and Ito integration. The non-standard framework then gives rigorous sense to Wiener's intuitive recipe i. db t where b, = —-, at which is meaningless in standard terms, but conveys the right idea. Finally, a Loebspace chaos decomposition result is used to give a generalization of a result of Nualart, Ustunel and Zakai.
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Citations
Robust linear estimation by a first order polynomial chaos expansion
Burak Seymen,Mubeccel Demirekler +1 more
- 20 Apr 2011
TL;DR: A mathematical model for stochastic uncertain systems where the system uncertainty is handled by polynomial chaos method is developed and the performance of the proposed robust estimation algorithm is shown.
Coherent chaos interest-rate models
Dorje C. Brody,Stala Hadjipetri +1 more
TL;DR: In this paper, the authors derive general expressions for the pricing kernel and the associated bond price and short rate processes in the case of a generic nth order chaos model, for each n ∈ ℕ.
Peer Review
Galaxy bias in the era of LSST: perturbative bias expansions
Andrina Nicola,Boryana Hadzhiyska,Nathanial B Findlay,David Alonso,Anze Slosar,Zhi-Hui Guo,N. Kokron,Raul E. Angulo,Alejandro Aviles,Jonathan Blazek,Jo Dunkley,Bhuvnesh Jain,James M. Sullivan,C. W. Walter,Matteo Zennaro +14 more
- 06 Jul 2023
TL;DR: The results of the LSST bias challenge as mentioned in this paper show that the systematic uncertainties associated with current nonlinear bias models are likely to be subdominant compared to other sources of error for tomographic analyses of upcoming photometric surveys.
Uncertainty quantification for integrated circuits: stochastic spectral methods
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TL;DR: The recent advances of stochastic spectral circuit simulators based on generalized polynomial chaos (gPC) can handle both Gaussian and non-Gaussian random parameters, showing remarkable speedup over Monte Carlo for circuits with a small or medium number of parameters.
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References
•Book
Nonstandard Analysis and its Applications
Nigel J. Cutland
- 30 Sep 1988
TL;DR: Non-standard analysis (NSA) is a subject of great research interest both in its own right and as a tool for answering questions in subjects such as functional analysis, probability, mathematical physics and topology.
202
•Book
The Malliavin calculus
Denis Bell
- 01 Jan 1987
TL;DR: The Malliavin calculus of variations as discussed by the authors is an infinite dimensional differential calculus on the Wiener space, which was initiated by Malli-avin in 1974 to give a probabilistic proof of the Hörmander's theorem and its importance was immediately recognized.
188
Nonstandard Analysis and its Applications: AN INVITATION TO NONSTANDARD ANALYSIS
Ton Lindstrøm
- 01 Sep 1988
TL;DR: Nonstandard analysis has been used to obtain new results in such diverse fields as Banach spaces, differential equations, probability theory, algebraic number theory, economics, and mathematical physics just to mention a few as mentioned in this paper.
179