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Existence of measurable versions of stochastic processes

Kazimierz Musiał

Abstract

Let $(X, \mfA,P)$, $(Y, \mfB,Q)$ be two arbitrary probability spaces and $¶:=\{(\mfA,P_y):y\in{Y}\}$ be a regular conditional probability on $\mfA$ with respect to $Q$. Denote by $R$ the skew product of $P$ and $Q$ determined by $\{P_y:y\in{Y}\}$ on the product $σ$-algebra $\mfA\otimes\mfB$ and by $\wh{R}$ its completion. I prove that a process $\{ξ_y:y\in{Y}\}$ possesses an equivalent $\wh{R}$-measurable version if and only if it is measurable with respect to a certain particular $σ$-algebra, larger than $\mfA\otimes\mfB$ and uniquely determined by $¶$. It is known that not every process possesses an equivalent measurable version (cf. \cite[§19.5]{St}). My approach is essentially different from earlier trials. It reverts to \cite[Theorem 3]{ta1}, where Talagrand proved existence of an equivalent separable version of a measurable process (in case of $R=P\times{Q}$), provided $Y$ is endowed with a separable pseudometric. The theorem is a strong generalization of \cite[Theorem 6.1]{smm} and \cite[Theorem 5.1]{mms1} where it was proved only that a suitable class of liftings transfer a measurable process into a measurable process.

Existence of measurable versions of stochastic processes

Abstract

Let , be two arbitrary probability spaces and be a regular conditional probability on with respect to . Denote by the skew product of and determined by on the product -algebra and by its completion. I prove that a process possesses an equivalent -measurable version if and only if it is measurable with respect to a certain particular -algebra, larger than and uniquely determined by . It is known that not every process possesses an equivalent measurable version (cf. \cite[§19.5]{St}). My approach is essentially different from earlier trials. It reverts to \cite[Theorem 3]{ta1}, where Talagrand proved existence of an equivalent separable version of a measurable process (in case of ), provided is endowed with a separable pseudometric. The theorem is a strong generalization of \cite[Theorem 6.1]{smm} and \cite[Theorem 5.1]{mms1} where it was proved only that a suitable class of liftings transfer a measurable process into a measurable process.
Paper Structure (3 sections, 3 theorems, 10 equations)

This paper contains 3 sections, 3 theorems, 10 equations.

Table of Contents

  1. Preliminaries
  2. Measurability of processes
  3. Direct product of measures.

Key Result

Theorem 1.2

Assume that ${\mathfrak A}$ contains a countably generated $\sigma$-algebra which is dense in ${\mathfrak A}$ (in the Fréchet-Nikodým pseudo-metric) with respect to $P$. Moreover, let $\{({\mathfrak A},P_y):y\in{Y}\}$ be a rcp on ${\mathfrak A}$ with respect to $Q$. Then for every $y\in Y$ there exi

Theorems & Definitions (10)

  • Definition 1.1
  • Theorem 1.2
  • proof
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Claim 3.1
  • Proposition 3.2
  • proof