1,092 research outputs found
Weak approximations. A Malliavin calculus approach
We introduce a variation of the proof for weak approximations that is suitable for studying the densities of stochastic processes which are evaluations of the flow generated by a stochastic differential equation on a random variable that maybe anticipating. Our main assumption is that the process and the initial random variable have to be smooth in the Malliavin sense. Furthermore if the inverse of the Malliavin covariance matrix associated with the process under consideration is sufficiently integrable then approximations for densities and distributions can also be achieved. We apply these ideas to the case of stochastic differential equations with boundary conditions and the composition of two diffusions.Stochastic differential equations, boundary conditions, weak approximation, numerical analysis
On stochastic conservation laws and Malliavin calculus
For stochastic conservation laws driven by a semilinear noise term, we
propose a generalization of the Kru\v{z}kov entropy condition by allowing the
Kru\v{z}kov constants to be Malliavin differentiable random variables.
Existence and uniqueness results are provided. Our approach sheds some new
light on the stochastic entropy conditions put forth by Feng and Nualart [J.
Funct. Anal., 2008] and Bauzet, Vallet, and Wittbold [J. Hyperbolic Differ.
Equ., 2012]
Kolmogorov Equations and Weak Order Analysis for SPDES with Nonlinear Diffusion Coefficient
We provide new regularity results for the solutions of the Kolmogorov
equation associated to a SPDE with nonlinear diffusion coefficients and a
Burgers type nonlinearity. This generalizes previous results in the simpler
cases of additive or affine noise. The basic tool is a discrete version of a
two sided stochastic integral which allows a new formulation for the
derivatives of these solutions. We show that this can be used to generalize the
weak order analysis performed in [16]. The tools we develop are very general
and can be used to study many other examples of applications
Functional It\^{o} calculus and stochastic integral representation of martingales
We develop a nonanticipative calculus for functionals of a continuous
semimartingale, using an extension of the Ito formula to path-dependent
functionals which possess certain directional derivatives. The construction is
based on a pathwise derivative, introduced by Dupire, for functionals on the
space of right-continuous functions with left limits. We show that this
functional derivative admits a suitable extension to the space of
square-integrable martingales. This extension defines a weak derivative which
is shown to be the inverse of the Ito integral and which may be viewed as a
nonanticipative "lifting" of the Malliavin derivative. These results lead to a
constructive martingale representation formula for Ito processes. By contrast
with the Clark-Haussmann-Ocone formula, this representation only involves
nonanticipative quantities which may be computed pathwise.Comment: Published in at http://dx.doi.org/10.1214/11-AOP721 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Dirichlet forms methods, an application to the propagation of the error due to the Euler scheme
We present recent advances on Dirichlet forms methods either to extend
financial models beyond the usual stochastic calculus or to study stochastic
models with less classical tools. In this spirit, we interpret the asymptotic
error on the solution of an sde due to the Euler scheme in terms of a Dirichlet
form on the Wiener space, what allows to propagate this error thanks to
functional calculus.Comment: 15
Stein's method for Brownian approximations
Motivated by a theorem of Barbour, we revisit some of the classical limit
theorems in probability from the viewpoint of the Stein method. We setup the
framework to bound Wasserstein distances between some distributions on infinite
dimensional spaces. We show that the convergence rate for the Poisson
approximation of the Brownian motion is as expected proportional to
where is the intensity of the Poisson process. We
also exhibit the speed of convergence for the Donsker Theorem and for the
linear interpolation of the Brownian motion. By iterating the procedure, we
give Edgeworth expansions with precise error bounds.Comment: Communications on Stochastic Analysis (2013
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