A Markov bridge is a Markov process conditioned to begin at a specified state and arrive at another specified state at a fixed later
time. Let 
denote the transition density of a time-homogeneous Markov
process. The density of the bridge from 
at time 0 to 
at time 
, evaluated at an intermediate
time 
, is
 |
(1)
|
More generally, its transition density between times 
is
 |
(2)
|
These formulas apply where the ratios are defined. They follow from conditional probability and the defining conditional-independence property of a Markov process. They also show that a bridge is generally time-inhomogeneous even when the original process is time-homogeneous.
For a continuous-time Markov chain with transition rates 
, conditioning on arrival at state 
at time 
gives the off-diagonal rates
 |
(3)
|
This formula gives the rates for 
. The diagonal rates are chosen
so that the total rate in each row is zero. This is a space-time Doob
h-transform with 
. A Brownian
bridge is the corresponding construction for Brownian
motion.
