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.
This is the first post in a series on asset price dynamics that are subject to a stochastic clock. The main idea is to not model the volatility as a stochastic process but instead make the rate at which time progresses randomly. In this post, we outline the basic idea and define the terminology.