Abstracts

This paper introduces the class of integer-valued trawl (IVT) processes for modelling serially dependent and integer-valued data in a continuous-time framework. We show that IVT processes constitute a very flexible and at the same time parsimonious class of stochastic processes which are very well suited for describing the dynamics of high frequency financial.
When defining an IVT process, there are two key components which need to be specified: First, the so-called Levy basis provides the source of randomness determining the marginal distribution of the model. All integer-valued infinitely distributions fall into our modelling framework starting from the Poisson and the negative binomial distribution to more advanced classes obtained as the marginal law from time-changing a Poisson process with a subordinator or by Levy-mixing of an integer-valued infinite divisible distribution.
Second, the choice of the deterministic function specifying the so-called trawl, which is a Borel set, is of key importance and sets the modelling framework apart from many traditional models for count data. We show that the trawl determines the autocorrelation function of the process and, vice-versa, the autocorrelation function determines the integrated trawl function. Here we can allow for very flexible classes of trawl functions e.g. exponential functions and superpositions thereof, which are motivated from Ornstein-Uhlenbeck and supOU processes. In addition, one could also allow for seasonal behaviour in the trawl function, which leads to non-monotonic trawls and autocorrelation functions.
That said, note that the autocorrelation structure and the marginal distribution will be modelled independently of each other, which is a considerable advantage of the trawling framework compared to traditional models for count data, and we can associate a trawl process with any infinitely divisible (integer-valued) distribution.
In addition to establishing the key theoretical properties of the new class of IVT processes, we also derive an efficient simulation scheme for IVT processes and develop an estimation method based on the generalised method of moments, which we prove to be consistent. In a simulation study, we demonstrate that our estimation methods works very well in practice.
Finally, we apply our new class of IVT processes to high frequency financial data from the S&P 600 smallcap index and show that they can describe the stochastic dynamics very well.