175 research outputs found
An operator approach for Markov chain weak approximations with an application to infinite activity L\'{e}vy driven SDEs
Weak approximations have been developed to calculate the expectation value of
functionals of stochastic differential equations, and various numerical
discretization schemes (Euler, Milshtein) have been studied by many authors. We
present a general framework based on semigroup expansions for the construction
of higher-order discretization schemes and analyze its rate of convergence. We
also apply it to approximate general L\'{e}vy driven stochastic differential
equations.Comment: Published in at http://dx.doi.org/10.1214/08-AAP568 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A probabilistic interpretation of the parametrix method
In this article, we introduce the parametrix technique in order to construct
fundamental solutions as a general method based on semigroups and their
generators. This leads to a probabilistic interpretation of the parametrix
method that is amenable to Monte Carlo simulation. We consider the explicit
examples of continuous diffusions and jump driven stochastic differential
equations with H\"{o}lder continuous coefficients.Comment: Published at http://dx.doi.org/10.1214/14-AAP1068 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Gaussian-type lower bounds for the density of solutions of SDEs driven by fractional Brownian motions
In this paper we obtain Gaussian-type lower bounds for the density of
solutions to stochastic differential equations (SDEs) driven by a fractional
Brownian motion with Hurst parameter . In the one-dimensional case with
additive noise, our study encompasses all parameters , while the
multidimensional case is restricted to the case . We rely on a mix of
pathwise methods for stochastic differential equations and stochastic analysis
tools.Comment: Published at http://dx.doi.org/10.1214/14-AOP977 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Optimal transport bounds between the time-marginals of a multidimensional diffusion and its Euler scheme
In this paper, we prove that the time supremum of the Wasserstein distance
between the time-marginals of a uniformly elliptic multidimensional diffusion
with coefficients bounded together with their derivatives up to the order
in the spatial variables and H{\"o}lder continuous with exponent with
respect to the time variable and its Euler scheme with uniform time-steps
is smaller than . To do so, we use the theory of optimal
transport. More precisely, we investigate how to apply the theory by Ambrosio,
Gigli and Savar{\'e} to compute the time derivative of the Wasserstein distance
between the time-marginals. We deduce a stability inequality for the
Wasserstein distance which finally leads to the desired estimation
A duality approach for the weak approximation of stochastic differential equations
In this article we develop a new methodology to prove weak approximation
results for general stochastic differential equations. Instead of using a
partial differential equation approach as is usually done for diffusions, the
approach considered here uses the properties of the linear equation satisfied
by the error process. This methodology seems to apply to a large class of
processes and we present as an example the weak approximation of stochastic
delay equations.Comment: Published at http://dx.doi.org/10.1214/105051606000000060 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme
In the present paper, we prove that the Wasserstein distance on the space of
continuous sample-paths equipped with the supremum norm between the laws of a
uniformly elliptic one-dimensional diffusion process and its Euler
discretization with steps is smaller than where
is an arbitrary positive constant. This rate is intermediate
between the strong error estimation in obtained when coupling the
stochastic differential equation and the Euler scheme with the same Brownian
motion and the weak error estimation obtained when comparing the
expectations of the same function of the diffusion and of the Euler scheme at
the terminal time . We also check that the supremum over of the
Wasserstein distance on the space of probability measures on the real line
between the laws of the diffusion at time and the Euler scheme at time
behaves like .Comment: Published in at http://dx.doi.org/10.1214/13-AAP941 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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