Bayesian Inference in Reducible Stochastic Differential Equations

The linear Ornstein-Ulenbeck diffusion model is too simple to describe the movement of short term interest rates. However diffusions with a non-linear drift and volatility function have no closed form likelihood function which make inference either classical or Bayesian very problematic. A vast range of approximation were proposed in the literature. In this paper, we develop the idea of a non-linear diffusion model, which after transformation can be reduced to an Ornstein-Uhlenbeck. At the price of a constrained drift function, we get a model equipped with a closed form likelihood function. We test this class of models on the the US Federal fund rate data. We propose a Bayesian approach to compare the performance of various specification of the volatility function.