Bayesian Estimation of the Skew Ornstein-Uhlenbeck Process

Abstract

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.

Appendices

Appendix

Details of Eqs.(6) and (7)

The transition density of $Y_t$ can be constructed by the eigenvalues and eigenfunctions of the Strum–Liouville equation

$$\begin{aligned} ({\mathcal {G}}u)(x)=-\lambda u(x),\quad x\in (-\infty ,+\infty ), \end{aligned}$$

with u satisfying proper conditions at the boundaries. When the spectrum is simple and purely discrete, the spectral expansion of the density reduces to the series Eq.(7). Based on the scale density $\zeta _Y(x)$ of the diffusion $Y_t$, $m_Y(x)$ is the speed density of the diffusion $Y_t$, defined as

$$\begin{aligned} \begin{aligned} \zeta _Y(x)&=\left\{ \begin{array}{ll} \displaystyle \exp (\frac{\kappa (x-(1-p)\theta -pa)^2}{(1-p)^2\sigma ^2}),&{}{{ if}\ a\le x},\\ \displaystyle \exp (\frac{\kappa (x-p\theta -(1-p)a)^2}{p^2\sigma ^2}),&{} {{ if}\ x< a}. \end{array} \right. \\ m_Y(x)&=\left\{ \begin{array}{ll} \displaystyle \exp (\frac{2}{\zeta _Y(x)(1-p)^2\sigma ^2}),&{} {{ if}\ a\le x},\\ \displaystyle \exp (\frac{2}{\zeta _Y(x)p^2\sigma ^2}),&{}{{ if} \ x< a}. \end{array} \right. \\ \end{aligned} \end{aligned}$$

(34)

Let

$$\begin{aligned} \begin{array}{lll} z_1\triangleq \frac{\sqrt{2\kappa }}{(1-p)\sigma }(x-(1-p)\theta -pa),&{}\alpha _1\triangleq -\frac{\sqrt{2\kappa }}{(1-p)\sigma }((1-p)\theta +pa),&{}v\triangleq \frac{\lambda }{\kappa },\\ z_2\triangleq \frac{\sqrt{2\kappa }}{p\sigma }(x-p\theta -(1-p)a),&{}\alpha _2\triangleq -\frac{\sqrt{2\kappa }}{p\sigma }(p\theta +(1-p)a),&{}\varrho \triangleq -\frac{\kappa \theta ^2}{\sigma ^2}, \end{array} \end{aligned}$$

then the eigenvalues $0\le \lambda _1<\lambda _2<\ldots <\lambda _n\rightarrow \infty$ as $n \uparrow \infty$ in Eq.(7) are the simple discrete zeros of the Wronskian

$$\begin{aligned} \omega (\lambda )=e^{\varrho }2^{1-v}v\sqrt{(}\kappa )\left[ \frac{H_v(-\frac{\alpha _1}{\sqrt{2}})H_{v-1}(\frac{\alpha _2}{\sqrt{2}})}{(1-p)\sigma }+\frac{H_{v-1}(-\frac{\alpha _1}{\sqrt{2}})H_v(\frac{\alpha _2}{\sqrt{2}})}{p\sigma }\right] , \end{aligned}$$

(35)

and the normalized eigenfunction $\varphi _n(x)$ follows

$$\begin{aligned} \varphi _n(x)=\left\{ \begin{array}{ll} \displaystyle \text {sign}(\xi (a,\lambda _n)\eta (a,\lambda _n))\sqrt{\frac{\eta (a,\lambda _n)}{\omega ^{\prime }(\lambda _n)\xi (a,\lambda _n)}}\xi (x,\lambda _n),&{}{{ if}\ a\le x},\\ \displaystyle \sqrt{\frac{\eta (a,\lambda _n)}{\omega ^{\prime }(\lambda _n)\xi (a,\lambda _n)}}\xi (x,\lambda _n),&{} {{ if} \ x< a}, \end{array} \right. \end{aligned}$$

(36)

where

$$\begin{aligned} \xi (x,\lambda )=e^{\frac{z_1^2}{4}}D_v(-z_1),\quad \eta (x,\lambda )=e^{\frac{z_2^2}{4}}D_v(z_2), \end{aligned}$$

the special functions $D_v(z)$ and $H_v(z)$ declared above are the parabolic cylinder function and the Hermite function respectively (see, e.g., Buchholz et al. 1970).