General alpha-Wiener bridges

Abstract

An α-Wiener bridge is a one-parameter generalization of the usual Wiener bridge, where the parameter α > 0 represents a mean reversion force to zero. We generalize the notion of α-Wiener bridges to continuous functions α : [0, T ) → R. We show that if the limit limt↑T α(t) exists and is positive, then a general α-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 limt↑T α(t) 6= 1 we show that the law of the general α-Wiener bridge cannot coincide with the law of any non time-homogeneous Ornstein-Uhlenbeck type bridge. In case limt↑T α(t) = 1 we determine all the Ornstein-Uhlenbeck type processes from which one can derive the general α-Wiener bridge by conditioning the original Ornstein-Uhlenbeck type process to be in 0 at time T .