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General alpha-Wiener bridges

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 $α:[0,T)\to R$. We show that if the limit $\lim_{t\uparrow T}α(t)$ 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 $\lim_{t\uparrow T}α(t)\ne 1$ 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 $\lim_{t\uparrow T}α(t)=1$ 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.