The 2005 Abel Symbosium

Stochastic Analysis and Applications - A Symposium in Honor of Kiyosi Itô

July 29 - August 4, Oslo, Norway

COLLECTIONS OF ABSTRACTS
(alphabetically ordered)

Luigi Accardi, Universita' di Roma "Tor Vergata", Italy

Ito calculus versus white noise calculus
The Wiener and the Poisson processes (together with the deterministic process) are building blocks of all "generic" stochastic processes. From the point of view of classical probability these are quite different objects, not expressible one in terms of the other by means of simple transformations.

Combining white noise techniques with one of the deepest ideas of quantum probability -- the "quantum decomposition of a classical random variable" -- one obtains a decomposition of the standard Poisson noise as the sum of a linear and a quadratic combination of a quantum white noise.

In physics the standard quantum white noise is called the ($1$--dimensional) free Fock boson field and the problem to give a meaning to its powers has been one of the fundamental problems of quantum field theory in the past 50 years. From the mathematical point of view this amounts to the following: give a meaning to "first order differential equations driven by higher powers of white noise", just as Ito calculus gives a meaning to "first order differential equations driven by the first (and some very special second) powers of white noise".

The physicists discovered that the main obstruction to the solution of this problem resides in the so--called" renormalization problem". The most common techniques used to deal with this problem are either "normal order" or "smoothing", which is an analogue of the Stratonovich
approach to the SDE. We will discuss some examples showing that, when applied to nonlinear problems in several important cases, these techniques either lead to wrong
or to trivial results.

The stochastic limit of quantum theory stimulated a new approach to the problem, based on the discovery that: stochastic differential equations (both classical and
quantum) are (causally) normally ordered forms of white noise Hamiltonian equations.

We will discuss why this result allowed the realization of this program for the renormalized square of white noise.

We will also describe some new no--go theorems in dimensions greater or equal than 3, which throw a new light on the renormalization problem, relating it to some old open problems in the theory of classical infinitely divisible processes.

Sergio Albeverio, University of Bonn, Germany

Feynman path integrals: from K. Ito's first papers on them to the present.
We review mathematical work on Feynman path integrals as infinite dimensional oscillatory integrals and their relations with probabilistic path integrals. In particular recent developments and applications will be discussed.

Ole Barndorff-Nielsen, University of Aarhus, Denmark

Volatility and Intermittency
Stochastic Volatility in finance has an analogue in Intermittency, a key concept in turbulence studies. New work in turbulence modelling, connected to Kolmogorov's phenomenological theory and using a semimartingale framework, has partly taken its cue from this analogy and from recent developments in financial modelling and econometrics. A survey of this work is given.

Giuseppe Da Prato, Scuola Normale Superiore di Pisa, Italy

Some results on Kolmogorov equations for stochastic PDEs
See separate file for the abstract

Eugene Dynkin, Cornell University, USA

An application of probability to nonlinear analysis
We apply superdiffusions (a class of branching exit Markov systems) to describe all positive solutions of a semilinear equation $Lu=u^a$ in a smooth domain $E$ in $R^d$. (Here L is a second order elliptic operator and $1<a\le 2}.)

David Elworthy, University of Warwick, UK

Ito maps and Malliavin calculus on path spaces.
Under certain conditions there are versions of Malliavin Calculus for path spaces on compact manifolds with diffusion measures. The standard example being the Brownian motion measures for Riemannian manifolds. Any smooth stochastic differential equation whose solutions have the given measure as law has an Ito map from flat Wiener space to our path space which when evaluated at a given time is smooth in the sense of Malliavin calculus for compact manifold valued functions. Here I shall discuss the intertwining of such Ito maps with the differentiation operators in the flat and curved spaces and describe how it relates to the Markov Uniqueness problem for our diffusion measure. This is taken from: K. D. Elworthy & Xue-Mei Li Itô maps and analysis on path spaces . Warwick University preprint (2005). (Also available from: http://www.xuemei.org/bib.html )

Hans Föllmer, Humboldt University, Germany

Stochastic Analysis of Financial Risk: robust projections and extended martingale measures
We discuss the role of stochastic integrals, martingales, and martingale measures in analyzing the risk of a financial position and in solving some robust optimization problems suggested by recent developments in the theory of convex risk measures.

Masatushi Fukushima, Kansai University, Japan

On extensions of Markov processes in duality by point processes of excursions
See separate file for the abstract

Takeyuki Hida, Meijo University, Nagoya, Japan

Some of the recent topics on stochastic analysis
First, we shall quickly explain why and how the space of generalized white noise functionals has been introduced. The space has big advantages to carry on the analysis of nonlinear functionals of white noise (or of Brownian motion) and to apply the theory to various fields. It should be noted that the introduction of generalized functionals was motivated by the Ito formula for Brownian functionals. Using this space we discuss the following two topics.

1. Path integrals.
To formulate Lagrangian path integrals, we have to concretize the expressions of the Lagrangian in terms of paths. We propose that quantum mechanical paths (trajectories) are expressed as a sum of the classical paths and fluctuation which is taken to be a Brownian bridge. It is possible to give a plausible reason why a Brownian bridge is fitting in this case. With this choice of possible trajectories, there arises a difficulty that the kinetic energy becomes a generalized functionals of a Brownian motion. It is now possible to overcome this difficulty to take our favorable space of generalized white noise functionals. Then follows the integration. Our method can be applied to a wider class of dynamics, for instance, to those cases with singular potentials and to some fields over non-euclidean space.

2. Infinite dimensional unitary group.
It is well known that the "infinite dimensional rotation group" has natuarally been introduced in connection with white noise, and the group describes certain invariance of the white noise measure. Hence, we may say that the white noise analysis should have an aspect of an infinite dimensional harmonic analysis. It seems natural, in fact by many reasons, to complexify the rotation group to have "infinite dimensional unitary group". Thus complexified group has various interesting applications to the analysis of nonlinear functionals of complex white noise. In addition, we can find good connections with Lie group theory and theory of quantum dynamics, to which we can give new interpretations.

Yaozhong Hu, University of Kansas, USA

Rough path analysis via fractional calculus
We use fractional calculus to deal with integral with respect to a function of holder continuity of order between 1/3 and 1/2. The differnetial equation driven by such function is also considered. As a consequence, we discuss the almost sure rate of convergence of Wong-Zakai type of approximation
for Brownian motion.

Ioannis Karatzas, Columbia University, USA

Stochastic Analysis of Atlas Models for Large Equity Markets
Atlas-type models are constant-parameter models of uncorrelated stocks for equity markets with a stable capital distribution, in which the growth rates and variances depend on rank. The simplest such model assigns the same, constant variance to all stocks; zero rate of growth to all stocks but the smallest; and positive growth rate to the smallest, the Atlas stock. In this talk we present the basic properties for this class of models, as well as the behavior of various portfolios in their midst. Of particular interest are portfolios that do not contain the Atlas stock.

Key Words and Phrases: Financial markets, portfolios, order statistics, local times, stochastic differential equations, ergodic properties.

(Joint work with Drs. Adrian Banner and Robert Fernholz.)

Claudia Klüppelberg, Technical University of Munich, Germany

Extremal Behaviour of Stochastic Volatility Models
Empirical volatility changes in time and exhibits tails, which are heavier than normal. Moreover, empirical volatility has - sometimes quite substantial - upwards jumps and clusters on high levels. We investigate classical and non-classical stochastic volatility models with respect to their extreme behavior. We show that classical stochastic volatility models driven by Brownian motion can model heavy tails, but obviously they are not able to model volatility jumps. Such phenomena can be modelled by Lèvy driven volatility processes as, for instance, by Lèvy driven Ornstein Uhlenbeck models. They can capture heavy tails and volatility jumps. Also volatility clusters can be found in such models, provided the driving Lèvy process has regularly varying tails. This results then in a volatility model with similarly heavy tails. As the last class of stochastic volatility models, we investigate a continuous time GARCH(1,1) model. Driven by an arbitrary Lèvy process it exhibits regularly varying tails, volatility upwards jumps and clusters on high levels.

Torbjørn Kolsrud, KTH Stockholm, Sweeden

Infinite dimensional Lie algebras related to linear and
non-linear heat equations
See separate file for the abstract

Paul Malliavin, University of Paris VI, France

Itô calculus as providing local charts: applications in
mathematical finance and in mathematical physics
See separate file for the abstract

Henry P. McKean, Courant Institute of Mathematical Science, New York, USA

A new aspect of the invariant distribution for a diffusion
The meaning of the invariant measure for a diffusion is to be clarified. After all, a wide-awake probabilist should ask: OK, it exists, but what is it the distribution of? I will try to answer this question in terms of two auxiliary diffusions allied to the original one.

(Joint work with M. Baldini):

Shige Peng, Shandong University, China

Conditional Nonlinear Expectations, Nonlinear Markov Chains and Dynamic Risk Measures
See separate file for the abstract

Yuri Rozanov, CNR-IMATI, Milano, Italy

Stochastic integrals and derivatives
The stochastics generated by a large number of independent "small" factors (which appear in the corresponding "place" and "time") can be handled through integration and differentiation with respect to stochastic measures with independent values considered over general space-time products. In this framework we are going to discuss some new recent results in the line with the Ito integral and its adjoint derivative (introduced by G. Di Nunno), Skorohod integral and Malliavin derivative (with its application to the Clark-Ocone type formulae).

Paavo Salminen, Åbo Akademy University, Finland

Integral Functionals, Occupation Times and Hitting Times of Diffusions
See separate file for the abstract

Marta Sanz-Solé, Univeristy of Barcelona, Spain

Stochastic calculus of variations and stochastic partial differential equations
See separate file for the abstract

Martin Schweizer, ETH Zürich, Switzerland

Stochastic control, BSDEs and finance
We combine methods from stochastic control and BSDE techniques to study the behaviour of solutions to some problems from mathematical finance. One typical feature is that the resulting BSDEs involve quadratic terms in the driver. Under additional assumptions, one can also study the dependence on a risk parameter and obtain asymptotic results. Examples include utility indifference valuation and utility maximization with uncertain priors.

Michael Sørensen, Univeristy of Copenhagen, Denmark

Stochastic Differential Equations: Recent Statistical Developments
Stochastic differential equation models pose interesting statistical problems that have recently attracted increasing attention. The data can be of several types, for instance direct observations, observations with measurement errors, averages over intervals, sampling at random time-points, partial observation of a multi-dimensional system, and combinations of these types. To study the distribution of estimators and other statistics, the classical large-sample asymptotics can be supplemented by high frequency asymptotics, small diffusion asymptotics, and combinations of the three.

A review will be given of some recent developments in the area of likelihood related parametric statistical inference for stochastic differential equations with emphasis on estimating functions and in particular recent asymptotic results on efficiency of martingale estimating functions. Examples will be presented of applications of to climatological data from ice cores and to a physiological problem.

Esko Valkeila, Helsinki University of Technology, Finnland

Some characterisation results with fractional Brownian motion
We will discuss the following topics:

What is the analogue of the classical Brownian bridge in the case of fractional Brownian motion? [1] What is the extension of Lévy characterisation of standard Brownian motion for fractional Brownian motion? [3] In addition, we will mention some new representation results for fractional Brownian motions. [2]

References:

[1] Gasbarra, D., Sottinen, T., and Valkeila, E. (2004): Gaussian bridges. Report A481, HUT, Insititute of Mathematics.
[2] Jost, C. (2005): A transformation formula for fractional Brownianmotion. Preprint, HU, Department of Mathematics and Statistics.
[3] Mishura, Yu., and Valkeila, E. (2005): Lévy theorem for fractional Brownian motion. Preprint (in preparation).

S. R. Srinivasa Varadhan, Courant Institute of Mathematical Science, New York, USA

Equilibrium fluctuations near infinity of a semi-linear heat
equation
See separate file for the abstract

Shinzo Watanabe, Ritsumeikan University, Japan

Itô calculus and Malliavin calculus - Generalized functionals on Wiener space and applications
By using the Itô calculus, we can realize important probability models on Wiener space as Wiener functionals and important objects in analysis and geometry are obtained as expectation of these Wiener functionals. In analyzing Wiener functionals and their expectations, the Malliavin calculus plays an important role, just as the Schwartz theory of distributions provides us with an effective method in analysis and geometry on Euclidean spaces and manifolds.

Tusheng Zhang, University of Manchester, England

Stochastic differential equations: existence of flows.
In this talk, I will present recent results on the existence of global flow of stochastic differential equations without the global Lipschitz conditions on the coefficients. Existence and uniqueness of solutions will also be discussed.

Xun-Yu Zhou, Chinese University of Hong Kong

Primal and Dual Approaches in Continuous-Time Portfolio Selection
This talk reports the recent research development on solving the continuous-time version of Markowitz's Nobel-prize winning mean-variance portfolio selection problem, using the primal (stochastic control) and dual (hedging of contingent claim) approaches. Complete and explicit solutions to various models, including those with stochastic investment opportunity sets, incomplete markets, regime switching, and mean-semivariance models, will be presented.