In this pose, we consider an Ornstein-Uhlenbeck stochastic clock whose instantaneous rate of activity process has an inverse Gaussian stationary distribution. We use the previously obtained relationship between the cumulant generation function of the stationary distribution and that of the background driving Lévy process and show that the latter can be represented as the sum of an inverse Gaussian processes and a compound Poisson process with chi-squared distributed increments.
In this post, we consider one particular specification of the background driving Lévy process in the general Ornstein-Uhlenbeck stochastic clock dynamics introduced in a previous post. We show that a compound Poisson process with exponentially distributed increments yields a gamma stationary distribution for the instantaneous rate of activity. We also discuss how the problem could be approached from the other end by imposing the stationary distribution and finding the corresponding background driving Lévy process.
After introducing the general idea of stochastic clocks in the previous post, we now consider the case where the instantaneous rate of activity follows a non-Gaussian Ornstein-Uhlenbeck process. The major reference is the paper by Barndorff-Nielsen and Shephard (2001b), see also Barndorff-Nielsen and Shephard (2001a). In this post, we discuss the general stochastic setup that applies to all such models that are driven by an almost surely non-decreasing Lévy processes. For the moment, we leave the particular dynamics of this subordinator unspecified but we will provide examples in future posts.