Preliminaries on Stochastic Calculus in Infinite Dimension

Fabbri, Giorgio; Gozzi, Fausto; Święch, Andrzej

doi:10.1007/978-3-319-53067-3_1

Giorgio Fabbri¹⁷,
Fausto Gozzi¹⁸ &
Andrzej Święch¹⁹

Part of the book series: Probability Theory and Stochastic Modelling ((PTSM,volume 82))

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Abstract

We recall some basic notions of measure theory and give a short introduction to random variables and the theory of the Bochner integral.

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Notes

1.
This example has been suggested to us by Mauro Rosestolato.
2.
Note that \(\underline{f}\) is not always Borel measurable, see [61] Volume 2, Exercise 6.10.42(ii), p. 59.
3.
In measure theory it is more often called the push-forward of \(\mathbb {P}\) and denoted by \(X_\#\mathbb {P}\).
4.
Note that if a process X is progressively measurable and satisfies (1.17) and Y is \(dt \otimes \mathbb P\)-equivalent to X, then Y must also satisfy (1.17) since for \(\mathbb P\)-a.s. \(\omega \), \(X(\cdot ,\omega )=Y(\cdot ,\omega )\), a.e. on [t, T].
5.
Without assuming (1.65) such continuity of trajectories may fail to hold, see e.g. [357].

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Authors and Affiliations

Aix-Marseille School of Economics, CNRS, Aix-Marseille University, EHESS, Centrale Marseille, Marseille, France
Giorgio Fabbri
Dipartimento di Economia e Finanza, Università LUISS - Guido Carli, Rome, Italy
Fausto Gozzi
School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA
Andrzej Święch

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Giorgio Fabbri
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Fausto Gozzi
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Andrzej Święch
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Correspondence to Fausto Gozzi .

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Fabbri, G., Gozzi, F., Święch, A. (2017). Preliminaries on Stochastic Calculus in Infinite Dimension. In: Stochastic Optimal Control in Infinite Dimension. Probability Theory and Stochastic Modelling, vol 82. Springer, Cham. https://doi.org/10.1007/978-3-319-53067-3_1

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DOI: https://doi.org/10.1007/978-3-319-53067-3_1
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-53066-6
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