194 research outputs found
An extension to the Wiener space of the arbitrary functions principle
The arbitrary functions principle says that the fractional part of
converges stably to an independent random variable uniformly distributed on the
unit interval, as soon as the random variable possesses a density or a
characteristic function vanishing at infinity. We prove a similar property for
random variables defined on the Wiener space when the stochastic measure
is crumpled on itself
Differential calculus for Dirichlet forms : the measure-valued gradient preserved by image
In order to develop a differential calculus for error propagation we study
local Dirichlet forms on probability spaces with square field operator
-- i.e. error structures -- and we are looking for an object related to
which is linear and with a good behaviour by images. For this we
introduce a new notion called the measure valued gradient which is a randomized
square root of . The exposition begins with inspecting some natural
notions candidate to solve the problem before proposing the measure-valued
gradient and proving its satisfactory properties
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
Some Historical Aspects of Error Calculus by Dirichlet Forms
We discuss the main stages of development of the error calculation since the
beginning of XIX-th century by insisting on what prefigures the use of
Dirichlet forms and emphasizing the mathematical properties that make the use
of Dirichlet forms more relevant and efficient. The purpose of the paper is
mainly to clarify the concepts. We also indicate some possible future research.Comment: 18 page
The Lent Particle Method, Application to Multiple Poisson Integrals
We give a extensive account of a recent new way of applying the Dirichlet
form theory to random Poisson measures. The main application is to obtain
existence of density for thelaws of random functionals of L\'evy processes or
solutions of stochastic differential equations with jumps. As in the Wiener
case the Dirichlet form approach weakens significantly theregularity
assumptions. The main novelty is an explicit formula for the gradient or for
the "carr\'e du champ' on the Poisson space called the lent particle formula
because based on adding a new particle to the system, computing the derivative
of the functional with respect to this new argument and taking back this
particle before applying the Poisson measure. The article is expository in its
first part and based on Bouleau-Denis [12] with several new examples,
applications to multiple Poisson integrals are gathered in the last part which
concerns the relation with the Fock space and some aspects of the second
quantization
On some errors related to the graduation of measuring instruments
The error on a real quantity Y due to the graduation of the measuring
instrument may be represented, when the graduation is regular and fines down,
by a Dirichlet form on R whose square field operator do not depend on the
probability law of Y as soon as this law possesses a continuous density. This
feature is related to the "arbitrary functions principle" (Poincar\'{e}, Hopf).
We give extensions of this property to multivariate case and infinite
dimensional case for approximations of the Brownian motion. We use a Girsanov
theorem for Dirichlet forms which has its own interest. Connections are given
with discretization of stochastic differential equations.Comment: 23
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