178 research outputs found
The snapping out Brownian motion
We give a probabilistic representation of a one-dimensional diffusion
equation where the solution is discontinuous at with a jump proportional to
its flux. This kind of interface condition is usually seen as a semi-permeable
barrier. For this, we use a process called here the snapping out Brownian
motion, whose properties are studied. As this construction is motivated by
applications, for example, in brain imaging or in chemistry, a simulation
scheme is also provided.Comment: Published at http://dx.doi.org/10.1214/15-AAP1131 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Global existence for rough differential equations under linear growth conditions
We prove existence of global solutions for differential equations driven by a
geometric rough path under the condition that the vector fields have linear
growth. We show by an explicit counter-example that the linear growth condition
is not sufficient if the driving rough path is not geometric. This settle a
long-standing open question in the theory of rough paths. So in the geometric
setting we recover the usual sufficient condition for differential equation.
The proof rely on a simple mapping of the differential equation from the
Euclidean space to a manifold to obtain a rough differential equation with
bounded coefficients.Comment: 20 page
Statistical estimation of the Oscillating Brownian Motion
We study the asymptotic behavior of estimators of a two-valued, discontinuous
diffusion coefficient in a Stochastic Differential Equation, called an
Oscillating Brownian Motion. Using the relation of the latter process with the
Skew Brownian Motion, we propose two natural consistent estimators, which are
variants of the integrated volatility estimator and take the occupation times
into account. We show the stable convergence of the renormalized errors'
estimations toward some Gaussian mixture, possibly corrected by a term that
depends on the local time. These limits stem from the lack of ergodicity as
well as the behavior of the local time at zero of the process. We test both
estimators on simulated processes, finding a complete agreement with the
theoretical predictions.Comment: 31 pages, 1 figur
A threshold model for local volatility: evidence of leverage and mean reversion effects on historical data
In financial markets, low prices are generally associated with high
volatilities and vice-versa, this well known stylized fact usually being
referred to as leverage effect. We propose a local volatility model, given by a
stochastic differential equation with piecewise constant coefficients, which
accounts of leverage and mean-reversion effects in the dynamics of the prices.
This model exhibits a regime switch in the dynamics accordingly to a certain
threshold. It can be seen as a continuous-time version of the Self-Exciting
Threshold Autoregressive (SETAR) model. We propose an estimation procedure for
the volatility and drift coefficients as well as for the threshold level.
Parameters estimated on the daily prices of 348 stocks of NYSE and S\&P 500, on
different time windows, show consistent empirical evidence for leverageeffects.
Mean-reversion effects are also detected, most markedly in crisis periods
A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients
The aim of this article is to provide a scheme for simulating diffusion
processes evolving in one-dimensional discontinuous media. This scheme does not
rely on smoothing the coefficients that appear in the infinitesimal generator
of the diffusion processes, but uses instead an exact description of the
behavior of their trajectories when they reach the points of discontinuity.
This description is supplied with the local comparison of the trajectories of
the diffusion processes with those of a skew Brownian motion.Comment: Published at http://dx.doi.org/10.1214/105051605000000656 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
BSDE driven by Dirichlet process and semi-linear parabolic PDE. Application to homogenization
Backward stochastic differential equations (BSDE) also gives the weak solution of a semi-linear system of parabolic PDEs with a second-order divergence-form partial differential operator and possibly discontinuous coefficients. This is proved here by approximation. After that, a homogenization result for such a system of semi-linear PDEs is proved using the weak convergence of the solution of the corresponding BSDEs in the S-topology
Simulation of diffusions by means of importance sampling paradigm
The aim of this paper is to introduce a new Monte Carlo method based on
importance sampling techniques for the simulation of stochastic differential
equations. The main idea is to combine random walk on squares or rectangles
methods with importance sampling techniques. The first interest of this
approach is that the weights can be easily computed from the density of the
one-dimensional Brownian motion. Compared to the Euler scheme this method
allows one to obtain a more accurate approximation of diffusions when one has
to consider complex boundary conditions. The method provides also an
interesting alternative to performing variance reduction techniques and
simulating rare events.Comment: Published in at http://dx.doi.org/10.1214/09-AAP659 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Constructing general rough differential equations through flow approximations
The non-linear sewing lemma constructs flows of rough differential equations
from a braod class of approximations called almost flows. We consider a class
of almost flows that could be approximated by solutions of ordinary
differential equations, in the spirit of the backward error analysis. Mixing
algebra and analysis, a Taylor formula with remainder and a composition formula
are central in the expansion analysis. With a suitable algebraic structure on
the non-smooth vector fields to be integrated, we recover in a single framework
several results regarding high-order expansions for various kind of driving
paths. We also extend the notion of driving rough path. We also introduce as an
example a new family of branched rough paths, called aromatic rough paths
modeled after aromatic Butcher series.Comment: version R0 (august 4, 2020): bibliography updat
Estimation of the Brownian dimension of a continuous It\^{o} process
In this paper, we consider a -dimensional continuous It\^{o} process which
is observed at regularly spaced times on a given time interval .
This process is driven by a multidimensional Wiener process and our aim is to
provide asymptotic statistical procedures which give the minimal dimension of
the driving Wiener process, which is between 0 (a pure drift) and . We
exhibit several different procedures, all similar to asymptotic testing
hypotheses.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ6190 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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