Ornstein-Uhlenbeck-Model with irregular time intervals

Question

I have data of the form:

Time t  Price x(t)
0       80
21      82
24      82.3
32      81.5
...     ...

The point is, that the time intervals are highly irregular. I suppose an Ornstein-Uhlenbeck-Process would fit nicely: $$ dx(t)=θ(μ−x(t))dt+σdW(t) $$ The problem in estimating the parameters is the irregularity of the time-intervals. An exact updating formula for discrete time would be: $$ x(t+\Delta t)=x(t)\exp(-θ\Delta t)+\mu (1-exp(-θ\Delta t))+\sigma \sqrt{\frac{1-exp(-2θ\Delta t)} {2θ}} $$ Now, the formula above is autoregressive, so if $\Delta t$ would be constant I could very easily calculate the parameters using OLS-estimation. But I have no idea how to solve the problem of the irregular time intervals. Maybe somebody here has an idea?

Answer 1

Since the conditional transition density of an OU process is known explicitly and is Gaussian, I would suggest to use an ML-estimator. Given the observations $X_0,\dots ,X_n$ at time-points $t_0, \dots, t_n$ the log-MLE is \begin{equation} \operatorname{argmin}_\theta \sum_{i=1}^n \frac{X_{i} - m(X_{i-1}, t_i - t_{i-1})}{2 s^2(t_i-t_{i-1})} + \frac{1}{2}\log s^2(t_i - t_{i-1}), \end{equation} with \begin{equation} m(x, t) = \mathbb E [X_t | X_0 =x] = x\exp(-\theta t) + \mu(1-\exp(-\theta t)) \end{equation} and $s(t) = \sigma \sqrt{\frac{1-exp(-2\theta t)} {2\theta}}$.

Perhaps you find this book useful: S. Iacus, "Simulation and Inference for Stochastic Differential Equations".