Integer‐valued Trawl Processes: A Class of Stationary Infinitely Divisible Processes

Abstract

This paper introduces a new continuous-time framework for modelling serially correlated count and integer-valued data. The key component in our new model is the class of integer-valued trawl (IVT) processes, which are serially correlated, stationary, infinitely divisible processes. We analyse the probabilistic properties of such processes in detail and, in addition, study volatility modulation and multivariate extensions within the new modelling framework. Moreover, we describe how the parameters of a trawl process can be estimated and obtain promising estimation results in our simulation study. Finally, we apply our new modelling framework to high frequency financial data.

Indirect inference with time series observed with error

“…As a consequence, the numerical Jacobian of the binding function, necessary for the calculation of the standard errors, contains many values near zero and it is almost singular. This is very informative and may indicate that alternative structural models, involving for example pure-jump Lévy processes as in Barndorff-Nielsen and Shephard (2001) or trawl processes as in Barndorff-Nielsen et al (2014), may be better suited to model stock returns at very high-frequencies and could possibly be estimated by indirect inference.…”

Continuous Time Analysis of Fleeting Discrete Price Moves

“…In the following Proposition, we rephrase the key properties of the stationary process L (A t ) mentioned in Barndorff-Nielsen, Lunde, Shephard, and Veraart (2014) under the squashed trawl variant.…”

“…Barndorff-Nielsen, Pollard, and Shephard (2012) build Lévy processes (continuous time random walks) that are integer-valued. We are also inspired by the stationary integer-valued processes of Barndorff-Nielsen, Lunde, Shephard, and Veraart (2014). Their processes are related to the up-stairs processes of Wolpert and Taqqu (2005) and the random measure processes of Wolpert and Brown (2011).…”

“…Both of these processes are stationary. Barndorff-Nielsen, Lunde, Shephard, and Veraart (2014) also bring out the relationship between their processes and M/G/∞ queues (e.g. Lindley (1956), Reynolds (1968) and Bartlett (1978, Ch.…”

Some Recent Developments in Ambit Stochastics

“…By using integer-valued Lévy bases, integer-valued trawl processes are obtained. These processes are studied in detail in [5] and applied to high frequency stock market data.…”