Research Group "Stochastic Algorithms and Nonparametric Statistics"
Seminar "Modern Methods in Applied Stochastics and Nonparametric
Statistics" Winter Semester 2012/13
last reviewed: February, 20, 2013, Christine Schneider
M. Tretyakov (University of Nottingham)
Layer methods for Navier-Stokes equations with additive noise
A number of layer methods for stochastic Navier-Stokes equations (SNSE)
with spatial periodic boundary conditions and additive noise are proposed.
The methods are constructed using conditional probabilistic representations
of solutions to SNSE and exploiting ideas of the weak sense numerical integration of stochastic differential equations.
Some convergence results for the proposed methods are proved.
Results of numerical experiments on model problems are presented.
The talk is based on a joint work with Prof. G.N. Milstein (Ural Federal University, Ekaterinburg, Russia).
H. Mai (WIAS Berlin)
Drift estimation for jump diffusions: time-continuous and high-frequency observations
Abstract: The problem of parametric drift estimation for a Lévy-driven jump
diffusion process is considered in two different settings:
time-continuous and high-frequency observations. The goal is to develop
explicit maximum likelihood estimators for both observation schemes that
are efficient in the Hájek-Le Cam sense.
P. Friz (WIAS/TU Berlin)
Solving the KPZ equation
Abstract: "Part one" of a reading group on Hairer's groundbreaking work on the KPZ equation,
a non-linear stochastic PDE which models random surface growth: We introduce a new concept of solution to the KPZ equation which is shown to extend the classical Cole-Hopf solution. This notion provides a factorisation of the Cole-Hopf solution map into a "universal" measurable map from the probability space into an explicitly described auxiliary metric space, composed with a new solution map that has very good continuity properties. The advantage of such a formulation is that it essentially provides a pathwise notion of a solution, together with a very detailed approximation theory. In particular, our construction completely bypasses the Cole-Hopf transform, thus laying the groundwork for proving that the KPZ equation describes the fluctuations of systems in the KPZ universality class.
As a corollary of our construction, we obtain very detailed new regularity results about the solution, as well as its derivative with respect to the initial condition. Other byproducts of the proof include an explicit approximation to the stationary solution of the KPZ equation, a well-posedness result for the Fokker-Planck equation associated to a particle diffusing in a rough space-time dependent potential, and a new periodic homogenisation result for the heat equation with a space-time periodic potential. One ingredient in our construction is an example of a non-Gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge.
Rainer Dahlhaus (Ruprecht Karls Universität Heidelberg)
Cointegration and phase synchronization: Bridging two theories
Abstract:In this talk we present with VEC-state
oscillators a new multivariate time series model
for oscillators with random phases. In particular
the phases may be synchronized. The model is a
nonlinear state space model where the phase
processes follow a vector error correction model
used in econometrics to model cointegration. We
demonstrate the relevance of this model for phase
synchronization. In that way we bridge the
theories of cointegration and phase
synchronization which have been important
theories in econometrics and physics,
respectively. The common ground of both theories
is that they describe the fluctuation of some
multivariate random process around an
equilibrium. We demonstrate how the methods from
cointegration can be applied to phase
synchronization. In particular we consider an
unidirectionally coupled Rössler-Lorenz system
and identify the unidirectional coupling, the
phase synchronization equilibrium and the phase
shifts with cointegration tests.
Solving the KPZ equation (Part II)
Abstract: "Part two" of a reading group on Hairer's groundbreaking work on the KPZ equation,
a non-linear stochastic PDE which models random surface growth.
Christian Bayer (WIAS Berlin)
Asymptotics beats Monte Carlo: The case of correlated local
Abstract: We consider a basket of options with both positive and
negative weights, in the case where each asset has a smile, e.g.
evolves according to its own local volatility and the driving
Brownian motions are correlated. In the case of positive weights,
the model has been considered in a previous work by Avellaneda,
Boyer-Olson, Busca and Friz. We derive highly accurate analytic
formulas for the prices and the implied volatilities of such
baskets. The computational time required to implement these
formulas is under two seconds even in the case of a basket on 100
assets. The combination of accuracy and speed makes these formulas
potentially attractive both for calibration and for pricing. In
comparison, simulation based techniques are prohibitively slow in
achieving a comparable degree of accuracy. Thus the present work
opens up a new paradigm in which asymptotics may arguably be used
for pricing as well as for calibration. (Joint work with Peter
Benjamin Gess (TUB und HU)
Solving the KPZ equation (Part III)
Abstract: "Part three" of a reading group on Hairer's groundbreaking work on the KPZ equation,
a non-linear stochastic PDE which models random surface growth.
M. Zhilova (WIAS Berlin)
Uniform inference for local parametric modeling
Abstract: Local maximum likelihood estimate for generalized regression model is being
considered. We present uniform w.r.t. the local parameters results, i.e.
non-asymptotic versions of classical Wilks and Fisher theorems, uniform
confidence bands and concentration sets. The results are based on the
general finite sample approach by V. Spokoiny (2011). They will be
illustrated by the examples of generalized linear model and local quantile
regression. This is a joint work with Prof. V. Spokoiny.
Andreas Andresen (WIAS Berlin)
Finite sample analysis of maximum likelihood estimators: an approach to show convergence of the alternating procedure?
Abstract: V. Spokoiny's "Parametric estimation. Finite sample theory" (2011) presents a way to analyse the finite sample deviation behaviour of maximum likelihood estimators. The talk explains the main ideas of these results and how the local quadratic bracketing device of that paper allows to extend the results to profile estimators in semi parametric models. Further it is shown how the local quadratic bracketing device and the bounds for parametric MLEs from Spokoiny's (2011) could serve as tools to derive a general convergence result for the alternating procedure to approximate the pMLE.
Hilmar Mai (WIAS Berlin)>
Goncalo dos Reis (TU Berlin)
Root's barrier, viscosity solutions of obstacle problems and
reflected FBSDEs, Part I
Abstract: Following work of Dupire (2005), Carr--Lee (2010) and Cox--Wang (2011)
on connections between Root's solution of the Skorokhod embedding
problem, free boundary PDEs and model-independent bounds on options on
variance we propose an approach with viscosity solutions. Besides
extending the previous results, it gives a link with reflected FBSDEs
as introduced by El Karoui et. al. (1997) and allows for easy
convergence proofs of numerical schemes via the Barles--Souganidis
In part I, we focus on the Skorokhod embedding problem.
Harald Oberhauser (TU Berlin)
An extension of the functional Ito formula
Motivated by questions arising in financial mathematics and the
hedging of exotic options, Dupire introduced a notion of smoothness
for functionals of paths (different from the usual Frechet--Gateaux
derivatives) and arrived at a generalization of Ito's formula
applicable to functionals which have a pathwise continuous dependence
on the trajectories of the underlying process. We revisit this topic
and use old work of Bichteler and Karandikar on pathwise integration
to extend the class of admissible functionals.
Eric Luçon (TU Berlin)
On the fluctuations of the weakly asymmetric exclusion process and
Abstract: In light of the previous talks about M. Hairer's paper "Solving the
KPZ equation", I will make a survey on the seminal paper of Bertini
and Giacomin "Stochastic Burgers and KPZ equations from particle
systems", where they prove the convergence of the fluctuations of the
asymmetric exclusion process (ASEP), under appropriate rescalings, to
the solution of the KPZ equation. The difficulty of ill-posedness of
the KPZ equation is bypassed using an adequate Cole-Hopf transform:
the key result of the paper is to show that the transformed
fluctuation process converges to the solution of the stochastic heat
equation, which is well-posed.