The Brownian Bridge Process
The Brownian Bridge is a classical brownian motion on the interval [0,1] and it is useful for modelling a system that starts at some given level and it is expected to return to that same level at some specified future time.
It is used for the publicly traded prices of bonds having a specified redemption value on a fixed expiration date.
It is modelled by conditioning on the event that the brownian motion W is zero both at the start and at the end of the process. Written formally:
SDE of the Brownian Bridge Process
The Parameters
X0 = a is the lower bound of the process
XT = b is the upper bound of the process
W is a brownian motion
Solving The SDE:
1. Solve the Ordinary Differential Equation (ODE) of the deterministic part of the SDE
We have to get this ODE in the following form and find a factor, called the Integrating Factor, that would make the left-hand side an exact differential:
Rearranging the terms we get:
The Integrating Factor (IF) is always of the form:
