Differential equations driven by fractional Brownian motion as random dynamical systems: qualitative properties (08frg140)

Arriving in Banff, Alberta Sunday, September 28 and departing Sunday October 5, 2008


David Nualart (University of Kansas)

Björn Schmalfuß (University of Paderborn)

(Purdue University)


Background and relevance: stochastic integration and fractional
Brownian motion

A central mathematical object in Stochastics/Stochastic Processes is the Ito
integral. It plays an important role in many areas of pure and applied
mathematics including: mathematical finance, population dynamics, fluid
dynamics, statistics, signal processing, control, particle systems, to name a
few. The integrator of such an integral is often chosen to be Brownian motion
(the Wiener process) or its semimartingale generalizations. These random
functions are of unbounded total variation, so that their Stieltjes integrals
do not exist. Special properties of the integrators and the integrands are
necessary to generalize the definition of the Stieltjes integral to the Ito
integral, and enable the definition of solutions of differential equations
driven by Brownian motion.

A property of paramount importance to this effect for Brownian motion is the
independence of its increments. To move beyond integrals and processes
constructed using this property is one of the most important tasks in the
theory of Stochastics. We are most interested in using the fractional Brownian
motion (fBm) process $B^{H}$ where $Hin(0,1)$ is fixed. It is a type of
stochastic process which deviates significantly from Brownian motion and
semimartingales, and others classically used in probability theory. As a
centered Gaussian process, it is characterized by the stationarity of its
increments and a medium- or long-memory property which is in sharp contrast
with martingales and Markov processes. Specifically, $B^{H}left( 0right)
=0$ and $Varleft[ B^{H}left( tright) -B^{H}left( sright) right]
=leftvert t-srightvert ^{2H}$. It also exhibits power scaling and path
regularity properties with Holder parameter $H$, which are very distinct from
Brownian motion. In fact, this single textquotedblleft
Hursttextquotedblright parameter $Hin(0,1)$ is also responsible for the low
decorrelation speed of fBm: for increments that are $n$ time units apart, the
correlation is precisely $c_{H}n^{2H-2}$ where the constant $c_{H}=H(2H-1)$.
Note that the standard Brownian motion is included in this family of models:
it is fBm with $H=1/2$, since $c_{H}$ is indeed null in this case. FBm has
become a popular choice of late for applications where classical processes
cannot model these non-trivial properties; for instance long memory, which is
also known as persistence, and corresponds to the case $Hin(1/2,1)$, is of
fundamental importance for financial data and in internet traffic: see
cite{Mandelbrot}, cite{w1}, cite{w2} . Our research group will have these
applications in mind in all discussions.

The mathematical theory of fBm is currently being developed vigorously by a
number of stochastic analysts, in various directions, using complementary and
sometimes competing tools. Ever since the pioneering works of Z{"{a}}hle
cite{Zah98}, Decreusefond and "{U}st"{u}nel cite{DecUst99}, and Lyons
cite{L}, the main thrust has been to understand how to perform stochastic
integration with respect to fBm in a way which is consistent with some
properties of the classical Ito theory for Brownian motion. In the case of
higher regularity ($H>1/2$), simple trajectorial methods, labelled as
emph{pathwise}, can be used which make it easy to translate one integration
theory into another, as emph{fractional derivatives} allow a pathwise
estimate of the integrals in terms of integrand and integrator using special
norms. Pathwise integrals historically gave the first cases where adequate
solutions to stochastic differential equations (sde) were established, see
Nualart and Rascanu cite{NR02}; infinite-dimensional equations have been
treated with the same success as finite-dimensional ones, e.g. Nualart and
Maslowski cite{MN}, Viens et al. cite{TTV}.

Timeliness, motivation, and importance: stochastic differential
equations driven by fractional Brownian motion, from general theory to random
dynamical systems

Solving sde can be considered a benchmark for testing the adequacy of an
integration theory. Among those methods which allow the use of more irregular
fBm ($H<1/2$), the so-called rough path theory, which uses non-probabilistic
constructions even for very irregular signals, is more efficient at defining
solutions of non-linear sde driven by fBm; it also has the advantage of being
applicable to processes that share some regularity properties with fBm, but
are otherwise very different; its main disadvantage is to not allow the same
ease of calculations as for Ito equations for Brownian motion. The Skorohod
(divergence) integral theory, based on stochastic analysis and Malliavin
calculus, is better at exploiting the Gaussian property of fBm, for instance
to generalize the Ito integral and formula with ease, resulting in mean-zero
stochastic integrals, and other convenient constructions: see Cheridito and
Nualart cite{CN05}, Mocioalca and Viens cite{MV05}. But Skorohod integration
has not produced a way to define solutions to fully non-linear equations; this
is one of the longest-standing open problems in stochastic analysis.medskip

Many mathematicians have yet to move decisively beyond the basic "existence-uniqueness" theory of sde driven by fBm. While some problems of this type, such as for Skorohod sde, remain open and are worthy of study, in this focused research group, we will also delve deeper into the qualitative properties of fBm-driven equations. In particular, we will investigate the equations' asymptotics (e.g. in large time). The two most popular theories dealing with the asymptotic qualitative behavior for general sde are: the theory of emph{random dynamical systems} (RDS) and the theory of existence and uniqueness of emph{invariant measures} for the associated Markov semigroup. However, similarly to fBm itself, equations driven by fBm do not generate a Markov process. This precludes the study of invariant measures using classical tools for fBm-driven systems. It motivates our plan to concentrate on the investigation of fBm-driven sde as RDS, the interface between the two being at the heart of our focused research group.

The theory of RDS, developed by L. Arnold and coworkers (see cite{Arn98}),
can be used to describe the asymptotic and qualitative behavior of systems of
random and stochastic differential/difference equation in terms of stability,
Lyapunov exponents, invariant manifolds, and attractors. A RDS consists of two
parts. The first part is a model for the noise path $omega$, leading to a
emph{metric dynamical system}. In particular, it is known that the fractional
Brownian motion forms an emph{ergodic} metric dynamical system, see Maslowski
and Schmalfu{ss } cite{MS04}. The second part of a RDS is the dynamics of an
sde: it is given by the solution mapping $phi(t,omega,x)$, which describes a
solution at time $t$ starting at time zero with initial condition $x$ in some
phase space $E$ which is driven be a noise path $omega$. This mapping $phi$
satisfied a generalized (semi)-group property, called emph{cocycle}%
--property, see Arnold cite{Arn98}. In other contexts, some answers have
already been given to basic questions such as stationarity and ergodicity
using fBm: see for instance Hairer cite{Hm05}, Hairer and Ohashi cite{HO07},
where conditions are given so that results familiar to the Brownian case also
hold for fBm. In Maslowski and Schmalfu{ss } cite{MS04},
infinite-dimensional RDS driven by fBm are exhibited as stochastic evolution
systems, and are shown to have unique, exponentially attracting fixed points.

Specific objectives for fractional-Brownian-driven random dynamical
systems: stability, infinite systems, invariant manifolds, Lyapunov

For many infinite dimensional Brownian-driven sde with non-trivial diffusion
coefficients, it is not known whether these equations generate a RDS. The
reason is that typically stochastic differential equations are only defined
$omega$-almost surely because the exceptional set is related to the
definition of an Ito integral as a limit of random variables emph{in
probability}. However, such a family of exceptional sets does not allow to use
the theory of RDS's full power. One advantage of sde driven by an fBm with
$H>1/2$ is that one can consider pathwise integrals, which avoid exceptional
sets. The case of $H>1/2$ is also interesting from the physical standpoint
since it is the case where path memory (persistence) is the longest; in that
sense, it is most different from standard Brownian motion.

On the other hand, when $H<1/2$, as we alluded to above in the relation
between pathwise and Skorohod integration, there does not exist a universally
accepted way of looking at stochastic integration or sde's, even finite
dimensional, and the problem of solving sde's in the Skorohod sense is
entirely open. Our focused research group will address the issue by trying to
understand whether any synergy can exist between the various approaches, which
have thus far been largely disjoint. In particular, it is possible that the
difficulties being experienced by stochastic analysts are related to the
non-existence of a RDS interpretation of sdes driven by irregular fBm. Since
non-linear equations driven by irregular fBm can be solved via the rough path
method (see recent progress by Nualart and Hu cite{NuaHu06}, and also the
original works of Lyons and Qian cite{LioQui98} or cite{LioQia02}), finding
even a single counter-example where such a solution occurs in a semi-explicit
way, but does not allow a RDS to be defined, would help understand the open

A key concept describing the dynamics of RDS generated by fBm-driven sde is
the so-called global attractor, which is an invariant random set attracting
other bounded random sets; it is a hallmark of the concept of emph{stability}%
. The essential dynamics take place in a neighborhood of the attractor. Even
for infinite dynamical systems the attractor often has a finite (fractal)
dimension which allows to describe the dynamics by finitely many parameters.
To show that there is an attractor, the main point is to show the existence of
an absorbing set (see Chueshov and Schmalfu{ss } cite{r-8}). In the
classical theory, it can usually can be constructed via Gronwall's lemma. This
leads to the question of finding a Gronwall lemma with respect to the special
fractional norms used for estimating stochastic integrals of fBm; this forces
one to try and modify the phase space of the RDS accordingly. Some ideas for
this kind of Gronwall lemma can be found for finite dimensional systems in
Garrido, Maslowski and Schmalfu{ss } cite{GarMasSchm08}. Our focused
research group will discuss these findings, their scope, and their extensions
to infinite systems driven by fBm.

Our plan to discuss stability for fBm-driven sde will continue with the
existence of stable and unstable manifolds and Lyapunov exponents, see Lu and
Schmalfu{ss } cite{r-9}. Such smooth manifolds are invariant under the
dynamics of the systems, and on them, the states are attracted or repelled by
a steady state. Standard methods to obtain these manifolds are the
Lyapunov-Perron method, based on fixed point arguments. We plan to discuss how
to obtain these fixed points with respect to the special fractional norms for
fBm integration, which will again force us to use an appropriate phase

Our focused research groups also intends to deal with the interesting question
of stability expressed in terms of pointwise Lyapunov exponents. The recent
work Viens and Zhang cite{VZ} considers the stochastic heat equation with
multiplicative infinite-dimensional noise which is fBm in time. If the spatial
behavior of the noise is non trivial, it is shown that when $Hleq1/2$, the
main Lyapunov exponent is finite and positive, but when $H>1/2$, it is
infinite, and in fact the solution explodes no slower than $expleft(
ct^{2H}/log tright) $ and no faster than $expleft( ct^{2H}/log
tright) $. Surprisingly, the case of $Hleq1/2$ is the easiest one to
analyze, and leads to familiar results. In the case of long memory ($H>1/2$),
the behavior of the solution is yet unclear. Is the memory length the main
factor in this difficulty for $H>1/2$? Or are there scaling arguments which
would help us show that the magnitude of the noise is the main driving force?
We will discuss these questions which could have important consequences in
statistical physics.

Specific objectives for long memory: memory preservation and
non-linear time series, effect on climate models, delay equations, and
statistical estimation

Further physically intriguing results regarding memory length are provided to
us by engineering and sociological practitioners. In Jumarie cite{Jg02},
arguments are given indicating that the introduction of random decision times
can lead to fractal behavior in dynamical systems, specifically with long
memory. In the opposite direction, numerical analysis on some linear dynamical
systems driven by fBm indicate they may lose the long memory property:
Grigoriu cite{Gm07}. The mathematical issue here is whether fBm's long memory
is preserved when passed through a RDS. As a first step to elucidate the
truths behind the numerical indications, we will discuss the use of
discrete-time RDS driven by long-memory time series. Indeed, it has been known
for a few years that some linear and nonlinear quadratic ARCH($infty$) time
series exhibit precisely fractional Brownian memory: Giraitis et al.
cite{GLRS04}. Here we will focus the discussion on how long memory may effect
products of non-independent matrices; an ARCH($infty$) model may be
considered as a product of infinite-dimensional matrices based on IID noise
terms, but it might be more efficient to model the discrete version of an
fBm-driven system as a product of $2times2$ matrices whose components are
correlated over long ranges. It is possible that for such a problem, using
continuous time and fractional stochastic calculus may provide a distinct
avenue of attack.

FBm is a good candidate to model random long-time influences in climate
systems, see Palmer et al. cite{Palmer2005}, cite{Palmer}. Here, the first
issues to address are existence questions for sde of from fluid dynamics when
they are driven by an fBm. For instance, although several authors appear
interested by the issue of driving a Navier-Stokes equation by fBm, no
definitive publications have yet appeared. The stochastic heat equation
mentioned above can be considered a toy model for this problem, and could
help, via Feynman-Kac representation as in cite{hv} for the corresponding
vorticity equation, understand the full non-linear equation driven by fBm.
After the existence question is better understood, our group will discuss the
dynamics of larger climate systems with long memory.

Stochastic differential delay equations and their asymptotic behavior have
received much attention in recent years (see for instance work of Caraballo
cite{Ca90}, Garrido-Atienza and collaborators cite{CaGaRe03}, cite{GaRe03},
Lisei cite{Lisei01}, Taniguchi et al. cite{TaLiuTru02}): such delayed
problems often appears in applications in physics, biology, engineering,
finance, etc. They offer an alternative to assuming that the driving noise is
fBm, while preserving some of fBm's memory features; but by combining the
effect of delay with the long memory of fBm, more realistic models will
emerge; to our knowledge these have not been considered. During our stay at
BIRS, we plan to discuss whether such stochastic delay equations with long
memory generate RDS, and if so, whether there exist random fixed points.
Judging for instance by the work of Garrido-Atienza et al. cite{CGS}
(stationary solutions for delayed sde driven by Brownian motion), our sde
coefficients will have to satisfy particular conditions.

Stochastic analysis has recently proved useful in tackling statistical
estimation issues for fBm. $H$ can be estimated consistently, and in some
cases with asymptotic normality, using simple power variation statistics, see
Tudor and Viens cite{TV07}. Similar constructs with weights are capable of
giving rise to various Ito integral limits, see for instance Nourdin et al.
cite{NNT07}. The statistics results have also motivated the study of
generalizations of fBm, such as the Rosenblatt process, and higher-order
Hermite processes, which share fBm's covariance structure, but can be
arbitrarily highly non-Gaussian. For simple fBm-driven sde with non-linear
drift scale parameter $a$, Gaussian and Malliavin calculus have yielded the
strong consistency and asymptotic normality of $a$'s maximum likelihood
estimator based on discrete observations: cite{TVstat}. Much remains to be
understood in estimating parameters for RDS driven by fBm; our group will
discuss the future research directions which are most likely to be of interest
to statisticians and other practitioners who need to know how to determine and
test their long-memory models.

Stochastic differential equations with fractional Brownian motion have the potential to provide a wealth of new models in many applied areas; our plan to use ideas from random dynamical systems to study their properties will give them the new perspective they deserve.


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