Instantons in Monte Carlo simulation of geometric Brownian motion in a confining quartic potential

Abstract

This study examines the behavior of the geometric Brownian motion (GBM) in a static confining quartic potential. The potential adopts the form of a double-well, indicating two distinct stable states separated by a barrier of finite height. We used Markov chain Monte Carlo (MCMC) simulation to sample paths of the time series and investigate the resulting statistics of paths. This sampling follows a path-integral-like representation of the probability weights that permits us to constrain the paths to pass through desired values at particular times if needed. For the case when we only constrain the initial state, most paths are confined in a single stable state, with rare instances of paths crossing the barrier into the other stable state. These tunnelling events which are often suggested to model rare catastrophic events in various contexts (e.g. crashes in finance and economics) can be viewed as non-perturbative instanton solutions in the presence of a double-well quartic potential. To efficiently sample these rare tunnelling trajectories, we constrain both initial and final states of the paths to the different stable states. The resulting statistics of paths display the asymmetric effect of multiplicative noise in the local variances around the two stable states. It is also notable that some segments of paths spend a considerable amount of time within the barrier separating the two stable states.