168 research outputs found
The different asymptotic regimes of nearly unstable autoregressive processes
We extend classical results about the convergence of nearly unstable AR(p)
processes to the infinite order case. To do so, we proceed as in recent works
about Hawkes processes by using limit theorems for some well chosen geometric
sums. We prove that when the coefficients sequence has a light tail, infinite
order nearly unstable autoregressive processes behave as Ornstein-Uhlenbeck
models. However, in the heavy tail case, we show that fractional diffusions
arise as limiting laws for such processes.Comment: 16 page
Some explicit formulas for the Brownian bridge, Brownian meander and Bessel process under uniform sampling
We show that simple explicit formulas can be obtained for several relevant
quantities related to the laws of the uniformly sampled Brownian bridge,
Brownian meander and three dimensional Bessel process. To prove such results,
we use the distribution of a triplet of random variables associated to the
pseudo-Brownian bridge together with various relationships between the laws of
these four processes
On the law of a triplet associated with the pseudo-Brownian bridge
We identify the distribution of a natural triplet associated with the
pseudo-Brownian bridge. In particular, for a Brownian motion and its
first hitting time of the level one, this remarkable law allows us to
understand some properties of the process
under uniform random sampling
Asymptotically optimal discretization of hedging strategies with jumps
In this work, we consider the hedging error due to discrete trading in models
with jumps. Extending an approach developed by Fukasawa [In Stochastic Analysis
with Financial Applications (2011) 331-346 Birkh\"{a}user/Springer Basel AG]
for continuous processes, we propose a framework enabling us to
(asymptotically) optimize the discretization times. More precisely, a
discretization rule is said to be optimal if for a given cost function, no
strategy has (asymptotically, for large cost) a lower mean square
discretization error for a smaller cost. We focus on discretization rules based
on hitting times and give explicit expressions for the optimal rules within
this class.Comment: Published in at http://dx.doi.org/10.1214/13-AAP940 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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
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