Brownian Bridge Approach
The Brownian bridge algorithm has been implemented for stress testing within the Risk Management framework. It is used for generation of multidimensional random paths whose initial and ending points are predetermined and fixed.
In the context of stress testing this algorithm is used for efficient generation of specific scenarios subject to certain extreme and generally unlikely conditions. If paths were generated by a conventional Monte-Carlo method only a very small portion of all the paths would satisfy such conditions.
The Brownian Bridge algorithm belongs to the family of Monte Carlo or Quasi-Monte Carlo methods with reduced variance. It generates sample paths which all start at the same initial point and end, at the same moment of time, at the same final point.
The conditional probability is a product of the two factors: the first is the probability of getting to point xf at time tf starting from point xi at time ti, and the second is the probability of passing through point x at time t given those initial and final points.
It is the second factor that gives the distribution of points on the paths connecting fixed initial and final points, i.e. those generated by the Brownian bridge algorithm. In particular, for any given time t between tf and ti the distribution of points x should be normal with the mean.
The Brownian bridge algorithm generates points on a path sequentially, starting from the initial point and progressing toward the end. However, the symmetry of the formula for the intermediate point with respect to the initial and final points and the recursive nature of the algorithm suggest that the generator may work better
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