63 research outputs found
Limit Theorems for a Cox-Ingersoll-Ross Process with Hawkes Jumps
In this paper, we propose a stochastic process, which is a Cox-Ingersoll-Ross
process with Hawkes jumps. It can be seen as a generalization of the classical
Cox-Ingersoll-Ross process and the classical Hawkes process with exponential
exciting function. Our model is a special case of the affine point processes.
Laplace transforms and limit theorems have been obtained, including law of
large numbers, central limit theorems and large deviations.Comment: 14 page
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
Alpha-CIR Model with Branching Processes in Sovereign Interest Rate Modelling
We introduce a class of interest rate models, called the -CIR model,
which gives a natural extension of the standard CIR model by adopting the
-stable L{\'e}vy process and preserving the branching property. This
model allows to describe in a unified and parsimonious way several recent
observations on the sovereign bond market such as the persistency of low
interest rate together with the presence of large jumps at local extent. We
emphasize on a general integral representation of the model by using random
fields, with which we establish the link to the CBI processes and the affine
models. Finally we analyze the jump behaviors and in particular the large
jumps, and we provide numerical illustrations
Convergence to equilibrium for time inhomogeneous jump diffusions with state dependent jump intensity
We consider a time inhomogeneous jump Markov process with state
dependent jump intensity, taking values in Its infinitesimal generator
is given by \begin{multline*} L_t f (x) = \sum_{i=1}^d \frac{\partial
f}{\partial x_i } (x) b^i ( t,x) - \sum_{ i =1}^d \frac{\partial f}{\partial
x_i } (x) \int_{E_1} c_1^i ( t, z, x) \gamma_1 ( t, z, x ) \mu_1 (dz ) \\ +
\sum_{l=1}^3 \int_{E_l} [ f ( x + c_l ( t, z, x)) - f(x)] \gamma_l ( t, z, x)
\mu_l (dz ) , \end{multline*} where are sigma-finite measurable spaces describing three different jump
regimes of the process (fast, intermediate, slow).
We give conditions proving that the long time behavior of can be related
to the one of a time homogeneous limit process Moreover, we
introduce a coupling method for the limit process which is entirely based on
certain of its big jumps and which relies on the regeneration method. We state
explicit conditions in terms of the coefficients of the process allowing to
control the speed of convergence to equilibrium both for and for $\bar X.
Limit theorems for nearly unstable Hawkes processes: Version with technical appendix
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 L1 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. 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
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