alpha-Wiener bridges: singularity of induced measures and sample path properties

TL;DR

This paper studies the singularity of probability measures induced by alpha-Wiener bridges with different parameters and explores their path regularity as time approaches the endpoint.

Contribution

It proves the measures for different alpha parameters are mutually singular and analyzes the sample path properties near the terminal time.

Findings

Measures for different alpha are mutually singular.

Sample paths exhibit specific regularity properties as t approaches T.

The process generalizes the classical Wiener bridge for various alpha values.

Abstract

Let us consider the process $(X_t^{(\alpha)})_{t\in[0,T)}$ given by the SDE $dX_t^{(\alpha)} = -\frac{\alpha}{T-t}X_t^{(\alpha)} dt+ dB_t$, $t\in[0,T)$, where $\alpha\in R$, $T\in(0,\infty)$, and $(B_t)_{t\geq 0}$ is a standard Wiener process. In case of $\alpha>0$ the process $X^{(\alpha)}$ is known as an $\alpha$-Wiener bridge, in case of $\alpha=1$ as the usual Wiener bridge. We prove that for all $\alpha,\beta\in R$, $\alpha\ne\beta$, the probability measures induced by the processes $X^{(\alpha)}$ and $X^{(\beta)}$ are singular on C[0,T). Further, we investigate regularity properties of $X_t^{(\alpha)}$ as $t\uparrow T$.