Estimating Stochastic Cusp Model Using Transition Density

This paper focuses on an econometric model which allows discontinuous change in an explained variable for a small continuous change in parameters. This model, given by stochastic differential equation with cubic drift, is called the cusp within standard catastrophe theory. The closed-form solution of density for this process is known only in the stationary case and this density belongs to the class of generalized exponential distributions, which allows for skewness, different tail shapes and multiple equilibria. The transition density has to be numerically approximated. For that purpose, the finite difference method is employed and then parameters are estimated using the maximum likelihood principle. An empirical example deals with the crash known as Black Monday, where parameters of the drift are driven by market fundamentals.