The Geometric Brownian Motion (GBM) is a standard model in quantitative finance, but the potential function of its stochastic differential equation (SDE) cannot include stable nonzero prices. This article generalises the GBM to an SDE with polynomial drift of order q and shows via model selection that q=2 is most frequently the optimal model to describe the data. Moreover, Markov chain Monte Carlo ensembles of the accompanying potential functions show a clear and pronounced potential well, indicating the existence of a stable price.

翻译：几何布朗运动（GBM）是量化金融中的标准模型，但其随机微分方程（SDE）的势函数无法包含稳定的非零价格。本文将GBM推广为具有q阶多项式漂移的SDE，并通过模型选择表明，q=2通常是描述数据的最优模型。此外，伴随势函数的马尔可夫链蒙特卡罗集成显示出清晰且显著的势阱，表明存在稳定的价格。