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Generalized stochastic processes revisited

Michael Oberguggenberger

Abstract

The paper addresses the question whether a random functional, a map from a set $E$ into the space of real-valued measurable functions on a probability space, has a measurable version with values in ${\mathbb R}^E$. Similarly, one may ask whether linear random functionals have versions in the algebraic dual. Most importantly, it can be asked which locally convex topological vector spaces $E$ have the ``regularity property'' that any linear random functional on $E$ has a version with values in the dual $E'$, an important issue in the theory of generalized stochastic processes. It has been shown by Itô and Nawata that this is the case when $E$ is nuclear. However, the question of uniqueness has only been partially answered. We build up a framework where these and related questions can be clarified in terms of spaces and mappings. We study classes of spaces $E$ (beyond nuclear spaces) with the said regularity property, prove a seemingly new uniqueness result and exhibit various examples and counterexamples.

Generalized stochastic processes revisited

Abstract

The paper addresses the question whether a random functional, a map from a set into the space of real-valued measurable functions on a probability space, has a measurable version with values in . Similarly, one may ask whether linear random functionals have versions in the algebraic dual. Most importantly, it can be asked which locally convex topological vector spaces have the ``regularity property'' that any linear random functional on has a version with values in the dual , an important issue in the theory of generalized stochastic processes. It has been shown by Itô and Nawata that this is the case when is nuclear. However, the question of uniqueness has only been partially answered. We build up a framework where these and related questions can be clarified in terms of spaces and mappings. We study classes of spaces (beyond nuclear spaces) with the said regularity property, prove a seemingly new uniqueness result and exhibit various examples and counterexamples.
Paper Structure (8 sections, 13 theorems, 45 equations)

This paper contains 8 sections, 13 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Motivation and notations
  3. Random functionals
  4. Linear random functionals
  5. Continuous linear random functionals
  6. Regularity theorems
  7. Counterexamples and mapping properties
  8. Appendix: Regularity and nuclear spaces

Key Result

Lemma 3.1

$X:\Omega\to \mathbb R^E$ is $(\Omega,\Sigma)-(\mathbb R^E,{\mathcal{B}}_K(\mathbb R^E))$-measurable if and only if $\kappa X(e)$ is $(\Omega,\Sigma)-(\mathbb R,{\mathcal{B}}(\mathbb R))$-measurable for all $e\in E$.

Theorems & Definitions (27)

  • Lemma 3.1
  • Lemma 3.2
  • Corollary 3.3
  • Proposition 4.1
  • Proposition 4.2
  • Remark 5.1
  • Definition 5.2
  • Remark 5.3
  • Proposition 5.4
  • Theorem 6.1
  • ...and 17 more