# Ornstein-Uhlenbeck-Model with irregular time intervals

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?

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 $s(t) = \sigma \sqrt{\frac{1-exp(-2\theta t)} {2\theta}}$.