Seminar on Stochastic Analysis, Random Fields and Applications

Frontmatter

On a Semigroup Approach to No-arbitrage Pricing Theory

Abstract

We show that the second order operator characterizing no-arbitrage pricing problems generates an Analytic Semigroup and therefore the Cauchy problem defining the no-arbitrage price of contingent claim contracts admits a solution. The conditions established in this paper are quite general, they encompass the sets of sufficient conditions already established in the literature. With this approach we are also able to give estimates to the derivatives of the no-arbitrage price.

Emilio Barucci, Fausto Gozzi, Vincenzo Vespri

Generalized Random Vector Fields and Euclidean Quantum Vector Fields

Abstract

We study generalized random vector fields in the framework of Euclidean quantum field theory. A recent no-go theorem about the non-existence of covariant reflection-positive random vector fields with locally integrable covariance is discussed and some of its implications are pointed out. We study Euclidean quantum fields obtained as pullbacks by translation invariant, covariant and weakly local operators and present examples of Gaussian random vector fields that fulfil all axioms for Euclidean random fields and that have a covariance which is not locally integrable. Finally we point out that in space-time dimension 2 there exist interacting Euclidean quantum vector fields obtained as pullbacks of P(Φ)₂-theories. However, the different components of these fields do not couple.

Claas Becker, Hanno Gottschalk, Jiang-Lun Wu

Central Limit Theorem for the Local Time of a Gaussian Process

Abstract

We prove a central limit theorem for the local time of real stationary Gaussian process via its expansion in terms of Hermite polynomials. The limiting process is Gaussian, and we give conditions ensuring its sample paths continuity. Other new asymptotics are also proved for such a local time.

Samir Ben Hariz, Paul Doukhan, José Rafael León

Explicit Solutions of Some Fourth Order Partial Differential Equations via Iterated Brownian Motion

Abstract

We give, via the iterated Brownian motion, an explicit formula for the fundamental solution of \( \frac{{\partial u}}{{\partial t}} = - \frac{1}{8}\frac{{{\partial^4}u}}{{\partial {x^4}}}\,in\,\left] {0, + \infty } \right[ \times \mathbb{R} \). We are also able to solve more general linear differential equations than the previous one.

S. Benachour, B. Roynette, P. Vallois

A Microscopic Model of Phase Field Type

Abstract

We introduce a microscopic and stochastic model of phase field type and discuss its macroscopic limits.

Lorenzo Bertini, Paolo Buttà, Barbara Rüdiger

Ergodic Backward SDE and Associated PDE

Abstract

By replacing the final condition for backward stochastic differential equations (in short: BSDEs) by a stationarity condition on the solution process we introduce a new class of BSDEs. In a natural manner we associate to such BSDEs the solution of ergodic second order partial differential equations.

Rainer Buckdahn, Shige Peng

Statistical Manifolds, Self-Parallel Curves and Learning Processes

Abstract

Introduced is some formalism of information geometry, new domain relating differential geometry to probability theory. Analysis and examinations of structure of some special spaces called statistical manifolds have been done. For the new geometry, covariant properties have been introduced. For that the new formalism the so-called α-geometry provides natural interpretation for the learning process, the learning rules have been discussed. The Boltzmann machine is studied using the previously described analysis of the manifold.

G. Burdet, Ph. Combe, H. Nencka

Law of Iterated Logarithm for Parabolic SPDEs

Abstract

We prove a version of Strassen’s functional law of iterated logarithm for a family of parabolic SPDEs. The lack of scaling due to the Green function makes it impossible to reduce the proof to the comparison of one single process at several times.

Fabien Chenal, Annie Millet

Random Production Flows. An Exactly Solvable Fluid Model

Abstract

The fluctuations of the buffered flow of parts delivered by failure prone production processes are analyzed by using a fluid model. The presence of storage regions between the production centers introduces memory effects into the dynamics of the flow of parts. As a consequence, the production output delivered by the factory can be approximately described by stochastic differential equations with noises being non-markovian alternating renewal processes. The relevant probabilistic properties of the solutions of such stochastic differential equations are discussed. Using results available in the context of dam’s theory, we derive, as an illustration, an exact characterization of the output process delivered by a production dipole composed of two machines separated by a single storage zone.

Philippe Ciprut, Max-Olivier Hongler, Yves Salama

A Compactness Principle for Bounded Sequences of Martingales with Applications

Abstract

For H ¹-bounded sequences of martingales, we introduce a technique, related to the Kadeč-Pełczynski-decomposition for L ¹ sequences, that allows us to prove compactness theorems. Roughly speaking, a bounded sequence in H ¹ can be split into two sequences, one of which is weakly compact, the other forms the singular part. If the martingales are continuous then the singular part tends to zero in the semi-martingale topology. In the general case the singular parts give rise to a process of bounded variation. The technique allows to give a new proof of the Optional Decomposition Theorem in Mathematical Finance.

F. Delbaen, W. Schachermayer

Risk Minimizing Hedging Strategies Under Partial Observation

Abstract

We use stochastic filtering techniques to solve the hedging problem in a financial market, where the price of the underlying risky asset may not be accessible to full observation and evolves according to one of two possible models. The coefficients of the models are furthermore driven by a hidden finite-state Markov process, of which one can observe the instants of jump and which can be interpreted as describing the state of the “economy”.

Paul Fischer, Eckhard Platen, Wolfgang J. Runggaldier

Multiparameter Markov Processes and Capacity

Abstract

This survey is essentially based on our papers [13] and [14]. In the first section, we shall present our definition of symmetric multiparameter Markov processes and introduce the capacities associated with them. This definition is the starting point of our study, but it is not sufficient to develop the whole theory. Rather than give explicitly the other assumptions which are used in the proofs, we shall present in the second section the fundamental examples for which the main results that we shall give in the third section, are valid.

Francis Hirsch, Shiqi Song

Iterated Brownian Motion and its Intrinsic Skeletal Structure

Abstract

This is an overview of some recent results on the stochastic analysis of iterated Brownian motion. In particular, we make explicit an intrinsic skeletal structure for the iterated Brownian motion which can be thought of as the analogue of the strong Markov property. As a particular application, we derive a change of variables (i.e., Itô’s) formula for iterated Brownian motion.

Davar Khoshnevisan, Thomas M. Lewis

Heavy Traffic and Optimal Control Methods for a Communications System

Abstract

There has been much work on the heavy traffic analysis of queueing systems, but very little on controlled problems. The paper deals with the heavy traffic approach to the control problem for multiplexing-type systems, a basic component of communications systems. There are many mutually independent sources which feed into a single channel. Due to the random and widely varying data rates, control over admission or bandwidth is needed for acceptable performance. Optimal control and heavy traffic analysis has been shown to yield systems with greatly improved performance, and the general techniques provide a useful and practical approach to problems of analysis and control arising in modern telecommunications.

Due to the presence of the control and to the ergodic cost function, the problem cannot be treated by classical methods of heavy traffic analysis for queueing networks, and requires methods from stochastic control. A basic result is that the optimal average costs per unit time for the physical problem converge to the optimal cost per unit time for the limit stationary process as the number of sources and the time interval goes to infinity. Furthermore, a “nice” nearly optimal control for the limit system provides nearly optimal values for the physical system, under heavy traffic.

Harold J. Kushner

Stochastic Wess-Zumino-Witten Model for the Measure of Kontsevitch

Abstract

We define a measure over C ¹ loops and give integration by parts formulas. We give different Hilbert structures over the tangent space of a loop such that the energy functional belongs to all the Sobolev spaces. We modify a little bit the measure in order that this measure is invariant under time reversal. We introduce a connection which is invariant under time reversal. This allows us to construct the Wess-Zumino-Witten supercharge and the Wess-Zumino-Witten laplacian which are associated to the energy functional.

Rémi Léandre

Independence of a Class of Multiple Stochastic Integrals

Abstract

We show that two multiple stochastic integrals I _n (f _n ), I _m (g _m ) with respect to the solution (M _t )t∈R₊ of a deterministic structure equation are independent if and only if two contractions of f _n and g _m , denoted as f _n o ₁ ⁰ g _m , fn o ₁ ¹ g _m vanish almost everywhere.

Nicolas Privault

Existence of Invariant Measures for Diffusion Processes on Banach Spaces

Abstract

In this paper, we obtain existence of invariant measures for diffusion processes with unbounded drifts on Banach spaces.

Michael Röckner, Tusheng S. Zhang

On Some New Type of Infinite Dimensional Laplacians

Abstract

A new class of infinite dimensional Laplacians is presented and its main properties are highlighted. We also discuss some specific example and refer to [11] for the general theory.

Sergio Scarlatti

Stochastic PDE’s of Schrödinger Type and Stochastic Mehler Kernels — a Path Integral Approach

Abstract

We present a rigorous path integral derivation of stochastic Mehler kernel formulae with applications to stochastic partial differential equations used in the theory of quantum measurement and filtering.

Aubrey Truman, Tomasz Zastawniak

Probability and Quantum Symmetries in a Riemannian Manifold

Abstract

We show how to use the theory of diffusion processes in order to discover new features of quantum symmetries on a Riemannian manifold.

Jean-Claude Zambrini