General alpha-Wiener bridges

TL;DR

This paper extends the concept of alpha-Wiener bridges to variable alpha functions, analyzing their convergence properties and relationship to Ornstein-Uhlenbeck processes, revealing new insights into their probabilistic structure.

Contribution

It generalizes alpha-Wiener bridges to arbitrary continuous alpha functions and characterizes their convergence and relation to Ornstein-Uhlenbeck processes.

Findings

If the limit of alpha(t) as t approaches T is positive, the process converges to zero at T.

For alpha(t) approaching a value not equal to 1, the law differs from any non time-homogeneous Ornstein-Uhlenbeck bridge.

When alpha(t) approaches 1, the paper identifies all Ornstein-Uhlenbeck processes that generate the alpha-Wiener bridge.

Abstract

An alpha-Wiener bridge is a one-parameter generalization of the usual Wiener bridge, where the parameter alpha>0 represents a mean reversion force to zero. We generalize the notion of alpha-Wiener bridges to continuous functions $\alpha:[0,T)\to R$. We show that if the limit $\lim_{t\uparrow T}\alpha(t)$ exists and is positive, then a general alpha-Wiener bridge is in fact a bridge in the sense that it converges to 0 at time T with probability one. Further, under the condition $\lim_{t\uparrow T}\alpha(t)\ne 1$ we show that the law of the general alpha-Wiener bridge can not coincide with the law of any non time-homogeneous Ornstein-Uhlenbeck type bridge. In case $\lim_{t\uparrow T}\alpha(t)=1$ we determine all the Ornstein-Uhlenbeck type processes from which one can derive the general alpha-Wiener bridge by conditioning the original Ornstein-Uhlenbeck type process to be in zero at time T.