Identifying the Median Path of a Stochastic Processes

This paper presents a method to characterize the typical path of a stochastic process, which I refer to as the Median Path. The paper describes how to estimate the Median Path in simulation and compares it to a different estimate of the typical path that is defined as the median value at each time step, like an ensemble average. The Median Path is an actual path from the stochastic process, whereas the path of median values is not. Therefore, the two paths may have very different properties. The Median Path is a single path from a set of simulated paths and is identified using a Ranking Algorithm that calculates the rank of each path at each time and then averages the ranks over time, similar to a time average. The Median Path is potentially useful in simulation applications where it is important to characterize the actual behaviour of a path generated by the stochastic process rather than the behaviour of statistics of the process over time.