Ergodicity and Scaling Limit of a Constrained Multivariate Hawkes Process

Abstract

We introduce a multivariate Hawkes process with constraints on its conditional density. It is a multivariate point process with conditional intensity similar to that of a multivariate Hawkes process but certain events are forbidden with respect to boundary conditions on a multidimensional constraint variable, whose evolution is driven by the point process. We study this process in the special case where the fertility function is exponential so that the process is entirely described by an underlying Markov chain, which includes the constraint variable. Some conditions on the parameters are established to ensure the ergodicity of the chain. Moreover, scaling limits are derived for the integrated point process. This study is primarily motivated by the stochastic modelling of a limit order book for high frequency financial data analysis. IntroductionSince two decades, the high-frequency price dynamics in a limit order book has received much attention. On the one hand, an extensive statistical study on the limit order book dynamics and limit order book information content can be found in [9,26,39,19,30,10,25,14,41]; on the other hand, many researchers have endeavored to propose different models, including equilibrium model, agent-based model and Markov model, to model the limit order book dynamics, see [36,38,17,15,1,16].More recently point processes have been introduced to financial application. Early works focus on modelling inter-event durations in [23,24] and the effect of duration on price impact and trade sign autocorrelation in [20]. Dufour and Engle [22] extend these models to allow for the censoring of quotes after a trade by intervening trades. Bauwens and Hautsch [8] provide a comprehensive introduction of the application of point processes in financial time series. Hawkes processes belong to the class of selfexciting point processes, where the intensity is driven by a weighted function of the time distance to previous points of the process. Hawkes processes originate from the literature in seismology and are used to model the occurrence of earthquakes, see [28,27,35,18,13]. To the best of our knowledge, the first application of Hawkes processes in financial time series is introduced by [11]. Since then, there is a growing literature on the application of Hawkes processes to the modelling of high frequency financial data, see [7,29,12 [5] and reach to derive and characterize the exact macroscopic diffusion limit of this model in [6].In this contribution we wish to model the dynamics of the limit prices in the limit order book. Nevertheless, we remark that the limits evolve according to specific constraints. For instance the best bid/best ask (buy/sell) price's behavior is constrained by a minimal bid-ask spread. This means that, as soon as the minimal bid-ask spread is attained, the best bid price cannot increase until the best ask price goes up. Therefore, in this work, we propose to add a constraint variable to the classical multivariate Hawkes process to model the dynamics of the limit prices....

General Compound Hawkes Processes in Limit Order Books

“…In this section we give various definitions and some properties of Hawkes processes which can be found in the existing literature (see, e.g., [27], [28], [23] and [48], to name a few). They include in particular one dimensional and non-linear Hawkes processes.…”

Compound Hawkes Processes in Limit Order Books

“…Some properties of Hawkes process can be found in the existing literature. (See, e.g., [Hawkes, 1971] and [Hawkes and Oakes, 1974], [Embrechts et al, 2011], [Zheng et al, 2014], to name a few). However, the notions of compound and regime-switching compound Hawkes processes are new.…”

General Compound Hawkes Processes in Limit Order Books

“…In this section we give various definitions and some properties of Hawkes processes, that can be found in the existing literature (see, e.g., [27,28], [23], [46], to name a few). They include, in particular, one-, D-dimensional and non-linear Hawkes processes.…”

“…Definition 6 (Non-linear Hawkes Process). The non-linear Hawkes process (see, e.g., [46]) is defined by the intensity function in the following form:…”

“…where X k ∈ {1, 2, ..., n} := X is a continuous-time n-state Markov chain, a(x) is continuous and bounded function on X = {1, 2, ..., n}, N (t) is the non-linear Hawkes process (see, e.g., [46]) defined by the intensity function in the following form (see (9), Definition 6):…”