Trajectory stratification of stochastic dynamics

TL;DR

This paper introduces a comprehensive mathematical framework for trajectory stratification, enabling efficient simulation of rare events by decomposing trajectories into manageable fragments and combining their averages, applicable to complex stochastic processes.

Contribution

It provides a unified, flexible framework for trajectory stratification that generalizes existing algorithms and allows for defining strata in novel ways to estimate complex averages.

Findings

Framework reveals common structure of rare event sampling algorithms.

Enables defining strata in terms of time points and path-dependent variables.

Demonstrates efficiency in estimating previously intractable averages.

Abstract

We present a general mathematical framework for trajectory stratification for simulating rare events. Trajectory stratification involves decomposing trajectories of the underlying process into fragments limited to restricted regions of state space (strata), computing averages over the distributions of the trajectory fragments within the strata with minimal communication between them, and combining those averages with appropriate weights to yield averages with respect to the original underlying process. Our framework reveals the full generality and flexibility of trajectory stratification, and it illuminates a common mathematical structure shared by existing algorithms for sampling rare events. We demonstrate the power of the framework by defining strata in terms of both points in time and path-dependent variables for efficiently estimating averages that were not previously tractable.