Bayesian Estimation of the Skew Ornstein-Uhlenbeck Process

In this paper, we are particularly interested in the skew Ornstein-Uhlenbeck (OU) process. The skew OU process is a natural Markov process defined by a diffusion process with symmetric local time. Motivated by its widespread applications, we study its parameter estimation. Specifically, we first transform the skew OU process into a tractable piecewise diffusion process to eliminate local time. Then, we discretize the continuous transformed diffusion by using the straightforward Euler scheme and, finally, obtain a more familiar threshold autoregressive model. The developed Bayesian estimation methods in the autoregressive model inspire us to modify a Gibbs sampling algorithm based on properties of the transformed skew OU process. In this way, all parameters including the pair of skew parameters (p, a) can be estimated simultaneously without involving complex integration. Our approach is examined via simulation experiments and empirical analysis of the Hong Kong Interbank Offered Rate (HIBOR) and the CBOE volatility index (VIX), and all of our applications show that our method performs well.