This paper revitalizes the investigation of the classical cusp catastrophe model in catastrophe theory and tackles the unsolved statistical inference problem concerning stochastic cusp differential equation. This model is challenging because its associated transition density hence the likelihood function is analytically intractable. We propose a novel Bayesian approach combining Hamiltonian Monte Carlo with two likelihood approximation methods, namely, Euler approximation and Hermite expansion. We validate this novel approach through a series of simulation studies. We further demonstrate potential application of this novel approach using the real USD/EUR exchange rate.

本文重振了突变理论中经典尖点突变模型的研究，解决了随机尖点微分方程尚未解决的统计推断问题。该模型具有挑战性，因为其相关的过渡密度因此似然函数在分析上是难以处理的。我们提出了一种新的贝叶斯方法，将哈密顿蒙特卡罗与两种似然近似方法（即欧拉近似和 Hermite 展开）相结合。我们通过一系列模拟研究验证了这种新颖的方法。我们使用实际美元/欧元汇率进一步展示了这种新方法的潜在应用。