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

Arriving Sunday, September 28 and departing Sunday, October 5, 2008

Organizers: David Nualart (University of Kansas), Björn Schmalfuß (University of Paderborn), Frederi Viens (Purdue University).

Confirmed Participants

Information for Participants

Final Report (PDF file)

Objectives

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 exponents

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 problem.

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 space.

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.

Conclusion 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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