How to build a stochastic bridge

Stochastic bridge is a process that is assumed to have known (fixed) values at times 0 and T. I will begin by constructing perhaps the simplest of all bridges called Wiener bridge, which uses the Wiener process for the conditioning. Later, I will show how to systematically build a bridge starting from any stochastic differential equation. Stochastic bridges are of particular interest when the fixed value at T is an atypical event, in which case the problem is to find a typical trajectory leading to such a rare event. Finding such trajectories in general and for various conditionings is beginning to attract a lot of interest in nonequilibrium statistical physics. Particularly surprising case is that of Ornstein-Uhlenbeck bridge condition on a rare event, in which the conditioned process behaves just as the original process until the very end, when it suddenly jumps to the prescribed fixed value.