Graduate Student Seminar: Haipeng Gao
Bayesian Inference for
Stochastic Cusp Catastrophe Model
(Under direction of Prof. Chuanshu Ji)
Continuous-time diffusion processes have been widely used in modern financial economics to model the stochastic behavior of economic variables such as interest rates, exchange rates, and stock prices. The Black Scholes model, the Vasicek model, and the Cox-Ingersoll-Ross model, all assume that the underlying state variables follow diffusion processes. If one believes that the observed values are generated according to some parametric specification, developing rigorous statistical methods to calibrate the underlying model to measured time-series has become a major subject of the field.
The thesis studies the stochastic cusp catastrophe model, often described by the cusp stochastic differential equations. The research problem of this thesis is to develop an accurate and computationally feasible parameter estimation algorithm based on Bayesian principle that can be implemented in the absence of an exact transition distribution for the stochastic cusp catastrophe model using discretely sampled observations.
We take the approach by Ait-Sahalia, using truncated Hermite polynomials expansion to approximate the (actual but intractable) transition density of cusp SDE, and employ Hamiltonian Monte Carlo to obtain posterior distribution to perform Bayesian inference. The problem of sparse sampling with continuous underlying model assumption was tackled by Bayesian imputation. Both accuracy and efficiency of the approach are demonstrated and examined in simulation studies that include a data-generating and a parameter estimation algorithm.