Abstract
Stochastic Cusp model allows discontinuous change in an explained variable for a small continuous change in parameters. 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 approximated and for that purpose, the finite difference method is employed and then parameters are estimated using the maximum likelihood principle. The finite difference method is often used for approximating partial differential equations, however the cubic drift is not handled sufficiently in current software, therefore own computational method is proposed and used.
An empirical example deals with the crash known as Black Monday, where parameters of the drift are driven by market fundamentals.