Vector Integration and Stochastic Integration in Banach Spaces
By N. Dinculeanu. 2000
107 citations recorded by the source corpus.
This article deals with vector integration and stochastic integration in Banach spaces. In particular, it considers the theory of integration with respect to vector measures with finite semivariation and its applications. This theory reduces to integration with respect to vector measures with finite variation which, in turn, reduces to the Bochner integral with respect to a positive measure. The article describes the four stages in the development of integration theory. It first provides an overview of the relevant notation for Banach spaces, measurable functions, the integral of step functions, and measurability with respect to a positive measure before discussing the Bochner integral. It then examines integration with respect to measures with finite variation, semivariation of vector measures, integration with respect to a measure with finite semivariation, and stochastic integrals. It also reviews processes with integrable variation or integrable semivariation and concludes with an analysis of martingales.
Subfields: Finance, Mathematical Physics
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In Finance
- Bernhard Schölkopf
- Kenji Fukumizu
- Krikamol Muandet
- Bharath K. Sriperumbudur
- Arthur Gretton
- Bodhisattva Sen
- Promit Ghosal
- Nabarun Deb
In Mathematical Physics
- Kenji Fukumizu
- Bernhard Schölkopf
- Arthur Gretton
- Bodhisattva Sen
- Krikamol Muandet
- Bharath K. Sriperumbudur
Citations in the corpus
- Minimax Estimation of Kernel Mean Embeddings (2016, background): “r X such that X p k(x,x)dP(x) <∞, the kernel mean or the mean element is defined as the Bochner integral µP := Z X k(·,x)dP(x) ∈ Hk. (1) We refer the reader to Diestel and Uhl (1977, Chapter 2) and Dinculeanu (2000, Chapter 1) for the definition of a Bochner integral. The mean element in (1) can be”
- Kernel Mean Shrinkage Estimators (2014, background): “A kernel mean is defined with respect to a probability distribution P over a measurable space X by where µ P is a Bochner integral (see, e.g. , Berlinet and Thomas-Agnan (2004) for a definition of Bochner integral) and H is an RKHS endowed with a reproducing kernel k : X × X → R that is measurable.”
- Measuring Association on Topological Spaces Using Kernels and Geometric Graphs (2020, background): “It is well-defined as a consequence of the Riesz representation theorem, see [93, 2] (equivalently also by Bochner’s theorem, see [32, 33]).”