Construction of an Edwards' probability measure on $\mathcal{C}(\mathbb{R}_+,\mathbb{R})$

TL;DR

This paper establishes the convergence of Edwards' probability measures on continuous paths over increasing intervals to a limit measure, providing an explicit martingale representation under Wiener measure.

Contribution

It proves the convergence of Edwards' measures on path space and derives an explicit martingale representation for the limiting measure.

Findings

Convergence of measures $\\mathbb{Q}_T$ to a limit measure $\mathbb{Q}$ as $T \to \infty$.

Existence of a martingale $(D_s)$ such that $\mathbb{Q}$ is absolutely continuous w.r.t. Wiener measure.

Explicit formula for the martingale $D_s$ associated with the limit measure.

Abstract

In this article, we prove that the measures $\mathbb{Q}_T$ associated to the one-dimensional Edwards' model on the interval $[0,T]$ converge to a limit measure $\mathbb{Q}$ when $T$ goes to infinity, in the following sense: for all $s\geq0$ and for all events $\Lambda_s$ depending on the canonical process only up to time $s$, $\mathbb{Q}_T(\Lambda_s)\rightarrow\mathbb{Q}(\Lambda_s)$. Moreover, we prove that, if $\mathbb{P}$ is Wiener measure, there exists a martingale $(D_s)_{s\in\mathbb{R}_+}$ such that $\mathbb{Q}(\Lambda_s) =\mathbb{E}_{\mathbb{P}}(\mathbh{1}_{\Lambda_s}D_s)$, and we give an explicit expression for this martingale.