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

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.