
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 . We show that if the limit 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 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 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…
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