97 research outputs found
Lower bounds for the density of locally elliptic It\^{o} processes
We give lower bounds for the density of the law of , the
solution of under the following local
ellipticity hypothesis: there exists a deterministic differentiable curve , such that and for all
The lower bound is expressed in terms of a distance
related to the skeleton of the diffusion process. This distance appears when we
optimize over all the curves which verify the above ellipticity assumption. The
arguments which lead to the above result work in a general context which
includes a large class of Wiener functionals, for example, It\^{o} processes.
Our starting point is work of Kohatsu-Higa which presents a general framework
including stochastic PDE's.Comment: Published at http://dx.doi.org/10.1214/009117906000000458 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Riesz transform and integration by parts formulas for random variables
We use integration by parts formulas to give estimates for the norm of
the Riesz transform. This is motivated by the representation formula for
conditional expectations of functionals on the Wiener space already given in
Malliavin and Thalmaier. As a consequence, we obtain regularity and estimates
for the density of non degenerated functionals on the Wiener space. We also
give a semi-distance which characterizes the convergence to the boundary of the
set of the strict positivity points for the density
Regularity of probability laws by using an interpolation method
We study the problem of the existence and regularity of a probability density
in an abstract framework based on a "balancing" with approximating absolutely
continuous laws. Typically, the absolutely continuous property for the
approximating laws can be proved by standard techniques from Malliavin calculus
whereas for the law of interest no Malliavin integration by parts formulas are
available. Our results are strongly based on the use of suitable Hermite
polynomial series expansions and can be merged into the theory of interpolation
spaces. We then apply the results to the solution to a stochastic differential
equation with a local H\"ormander condition or to the solution to the
stochastic heat equation, in both cases under weak conditions on the
coefficients relaxing the standard Lipschitz or H\"older continuity requests
A generic construction for high order approximation schemes of semigroups using random grids
Our aim is to construct high order approximation schemes for general
semigroups of linear operators . In order to do it, we fix a
time horizon and the discretization steps and we suppose that we have at hand some short time approximation
operators such that for some
. Then, we consider random time grids such that for all ,
for some , and
we associate the approximation discrete semigroup Our main result is the following: for any
approximation order , we can construct random grids
and coefficients , with such that % with the expectation concerning the random grids
Besides, and the complexity of the
algorithm is of order , for any order of approximation . The standard
example concerns diffusion processes, using the Euler approximation for~.
In this particular case and under suitable conditions, we are able to gather
the terms in order to produce an estimator of with finite variance.
However, an important feature of our approach is its universality in the sense
that it works for every general semigroup and approximations. Besides,
approximation schemes sharing the same lead to the same random grids
and coefficients . Numerical illustrations are given for
ordinary differential equations, piecewise deterministic Markov processes and
diffusions
Integration by parts formula with respect to jump times for stochastic differential equations
We establish an integration by parts formula based on jumps times in an
abstract framework in order to study the regularity of the law for processes
solution of stochastic differential equations with jumps
An invariance principle for stochastic series I. Gaussian limits
We study invariance principles and convergence to a Gaussian limit for
stochastic series of the form where , , is a sequence of centred independent
random variables of unit variance. In the case when the 's are Gaussian,
is an element of the Wiener chaos and convergence to a Gaussian limit
(so the corresponding nonlinear CLT) has been intensively studied by Nualart,
Peccati, Nourdin and several other authors. The invariance principle consists
in taking with a general law. It has also been considered in the
literature, starting from the seminal papers of Jong, and a variety of
applications including -statistics are of interest. Our main contribution is
to study the convergence in total variation distance and to give estimates of
the error
On the distance between probability density functions
We give estimates of the distance between the densities of the laws of two
functionals and on the Wiener space in terms of the Malliavin-Sobolev
norm of We actually consider a more general framework which allows one
to treat with similar (Malliavin type) methods functionals of a Poisson point
measure (solutions of jump type stochastic equations). We use the above
estimates in order to obtain a criterion which ensures that convergence in
distribution implies convergence in total variation distance; in particular, if
the functionals at hand are absolutely continuous, this implies convergence in
of the densities
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