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

Abstract

In this article, we prove that the measures ℚ_T associated to the one-dimensional Edwards’ model on the interval [0, T] converge to a limit measure ℚ when T goes to infinity, in the following sense: for all s≥0 and for all events Λ_s depending on the canonical process only up to time s, ℚ_T(Λ_s) → ℚ(Λ_s).

Moreover, we prove that, if ℙ is Wiener measure, there exists a martingale (D_s)_s∈ℝ+ such that $\mathbb{Q}(\Lambda_{s})=\mathbb{E}_{\mathbb{P}}(𝟙 _{\Lambda_{s}}D_{s})$, and we give an explicit expression for this martingale.