504 research outputs found
Multiscale Analysis for SPDEs with Quadratic Nonlinearities
In this article we derive rigorously amplitude equations for stochastic PDEs
with quadratic nonlinearities, under the assumption that the noise acts only on
the stable modes and for an appropriate scaling between the distance from
bifurcation and the strength of the noise. We show that, due to the presence of
two distinct timescales in our system, the noise (which acts only on the fast
modes) gets transmitted to the slow modes and, as a result, the amplitude
equation contains both additive and multiplicative noise.
As an application we study the case of the one dimensional Burgers equation
forced by additive noise in the orthogonal subspace to its dominant modes. The
theory developed in the present article thus allows to explain theoretically
some recent numerical observations from [Rob03]
Isotropic Ornstein-Uhlenbeck flows
Isotropic Brownian flows (IBFs) are a fairly natural class of stochastic
flows which has been studied extensively by various authors. Their rich
structure allows for explicit calculations in several situations and makes them
a natural object to start with if one wants to study more general stochastic
flows. Often the intuition gained by understanding the problem in the context
of IBFs transfers to more general situations. However, the obvious link between
stochastic flows, random dynamical systems and ergodic theory cannot be
exploited in its full strength as the IBF does not have an invariant
probability measure but rather an infinite one. Isotropic Ornstein-Uhlenbeck
flows are in a sense localized IBFs and do have an invariant probability
measure. The imposed linear drift destroys the translation invariance of the
IBF, but many other important structure properties like the Markov property of
the distance process remain valid and allow for explicit calculations in
certain situations. The fact that isotropic Ornstein-Uhlenbeck flows have
invariant probability measures allows one to apply techniques from random
dynamical systems theory. We demonstrate this by applying the results of
Ledrappier and Young to calculate the Hausdorff dimension of the statistical
equilibrium of an isotropic Ornstein-Uhlenbeck flow
Multi-dimensional parameter estimation of heavy-tailed moving averages
In this paper we present a parametric estimation method for certain
multi-parameter heavy-tailed L\'evy-driven moving averages. The theory relies
on recent multivariate central limit theorems obtained in [3] via Malliavin
calculus on Poisson spaces. Our minimal contrast approach is related to the
papers [14, 15], which propose to use the marginal empirical characteristic
function to estimate the one-dimensional parameter of the kernel function and
the stability index of the driving L\'evy motion. We extend their work to allow
for a multi-parametric framework that in particular includes the important
examples of the linear fractional stable motion, the stable Ornstein-Uhlenbeck
process, certain CARMA(2, 1) models and Ornstein-Uhlenbeck processes with a
periodic component among other models. We present both the consistency and the
associated central limit theorem of the minimal contrast estimator.
Furthermore, we demonstrate numerical analysis to uncover the finite sample
performance of our method
Regularization by noise and stochastic Burgers equations
We study a generalized 1d periodic SPDE of Burgers type: where , is
the 1d Laplacian, is a space-time white noise and the initial condition
is taken to be (space) white noise. We introduce a notion of weak
solution for this equation in the stationary setting. For these solutions we
point out how the noise provide a regularizing effect allowing to prove
existence and suitable estimates when . When we obtain
pathwise uniqueness. We discuss the use of the same method to study different
approximations of the same equation and for a model of stationary 2d stochastic
Navier-Stokes evolution.Comment: clarifications and small correction
Quasi maximum likelihood estimation for strongly mixing state space models and multivariate L\'evy-driven CARMA processes
We consider quasi maximum likelihood (QML) estimation for general
non-Gaussian discrete-ime linear state space models and equidistantly observed
multivariate L\'evy-driven continuoustime autoregressive moving average
(MCARMA) processes. In the discrete-time setting, we prove strong consistency
and asymptotic normality of the QML estimator under standard moment assumptions
and a strong-mixing condition on the output process of the state space model.
In the second part of the paper, we investigate probabilistic and analytical
properties of equidistantly sampled continuous-time state space models and
apply our results from the discrete-time setting to derive the asymptotic
properties of the QML estimator of discretely recorded MCARMA processes. Under
natural identifiability conditions, the estimators are again consistent and
asymptotically normally distributed for any sampling frequency. We also
demonstrate the practical applicability of our method through a simulation
study and a data example from econometrics
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