580 research outputs found
Limit theorems for nearly unstable Hawkes processes
Because of their tractability and their natural interpretations in term of
market quantities, Hawkes processes are nowadays widely used in high-frequency
finance. However, in practice, the statistical estimation results seem to show
that very often, only nearly unstable Hawkes processes are able to fit the data
properly. By nearly unstable, we mean that the norm of their kernel is
close to unity. We study in this work such processes for which the stability
condition is almost violated. Our main result states that after suitable
rescaling, they asymptotically behave like integrated Cox-Ingersoll-Ross
models. Thus, modeling financial order flows as nearly unstable Hawkes
processes may be a good way to reproduce both their high and low frequency
stylized facts. We then extend this result to the Hawkes-based price model
introduced by Bacry et al. [Quant. Finance 13 (2013) 65-77]. We show that under
a similar criticality condition, this process converges to a Heston model.
Again, we recover well-known stylized facts of prices, both at the
microstructure level and at the macroscopic scale.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1005 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Hierarchy of Temporal Responses of Multivariate Self-Excited Epidemic Processes
We present the first exact analysis of some of the temporal properties of
multivariate self-excited Hawkes conditional Poisson processes, which
constitute powerful representations of a large variety of systems with bursty
events, for which past activity triggers future activity. The term
"multivariate" refers to the property that events come in different types, with
possibly different intra- and inter-triggering abilities. We develop the
general formalism of the multivariate generating moment function for the
cumulative number of first-generation and of all generation events triggered by
a given mother event (the "shock") as a function of the current time . This
corresponds to studying the response function of the process. A variety of
different systems have been analyzed. In particular, for systems in which
triggering between events of different types proceeds through a one-dimension
directed or symmetric chain of influence in type space, we report a novel
hierarchy of intermediate asymptotic power law decays of the rate of triggered events as a function of the
distance of the events to the initial shock in the type space, where for the relevant long-memory processes characterizing many natural
and social systems. The richness of the generated time dynamics comes from the
cascades of intermediate events of possibly different kinds, unfolding via a
kind of inter-breeding genealogy.Comment: 40 pages, 8 figure
Linear processes in high-dimension: phase space and critical properties
In this work we investigate the generic properties of a stochastic linear
model in the regime of high-dimensionality. We consider in particular the
Vector AutoRegressive model (VAR) and the multivariate Hawkes process. We
analyze both deterministic and random versions of these models, showing the
existence of a stable and an unstable phase. We find that along the transition
region separating the two regimes, the correlations of the process decay
slowly, and we characterize the conditions under which these slow correlations
are expected to become power-laws. We check our findings with numerical
simulations showing remarkable agreement with our predictions. We finally argue
that real systems with a strong degree of self-interaction are naturally
characterized by this type of slow relaxation of the correlations.Comment: 40 pages, 5 figure
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