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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
. 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 T.Comment: 26 page
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