Stochastic Analysis and its Applications:
The European Union TMR Programme Project ERBFMRXCT960075

Some of the members of the Mathematical Physics Group of the University of Lisbon constitute the portuguese team participating in the European TMR Project Stochastic Analysis and its Applications. This document provides information of a general scientific nature relating to that project. A project outline containing more detailed scientific information is also available.

Similar information is provided by the Project home page, at the project co-ordinator's site.

• Introduction
• Project overview
• Partnership
• Potential applications
• Training aspects: research fellowships
• Participating teams: contact details
• Timescale
• Additional information

Introduction

The European Union programme Training and Mobility in Research has selected a modest number of projects where it provides networking resources to assist teams of scientists to work together towards a common goal. In addition, it enables the employment of young researchers (approximately one for each team) to help push forward the science in the project. By working with key workers in their field, these very able young workers should enhance their own research, and so strengthen the European Science base for the next generation. These young scientists have to be European and must come from outside the EU country where they are appointed.

We provide here information about one of the funded projects---the scientists in this project are mathematicians and they work in an area of mathematics known as Stochastic Analysis.

Project overview

The mathematics developed to study stochastic (or random) systems is difficult but of considerable importance; its impact permeates many aspects of our everyday life.

From sensors monitoring heart beats, through to the management of the risks involved in using stocks to fund pensions, many systems can be profoundly affected by stochastic fluctuations, noise, and randomness. Significant benefits can be achieved if one can be quantitative, and where possible deeply understand, these systems. For example, the ability to price risk, embodied in the famous results of Black and Scholes, has radically changed the financial markets, and is at the present time causing a complete rearrangement of the conventional insurance industry.

The mathematical study of such systems will almost certainly involve tools with names like martingales, conditional expectations, paths spaces, Itô calculus, Malliavin calculus, stochastic integrals, large deviations, log-Sobolev inequalities, measure valued branching processes, stochastic partial differential equations etc. All are tools central to stochastic analysis.

The mathematical analysis of stochastic systems, while perhaps a little inaccessible to outsiders, is undergoing quite rapid scientific development. One could say with some justification that Stochastic Analysis has emerged as a core area of late 20th century of mathematics.

This project is aimed at promoting this fundamental core, and in promoting the flow of expertise and ideas between abstract mathematics and applications.

Partnership

In this project some of the most active European teams in stochastic analysis will join their efforts to further develop a coherent technology for studying random phenomena and dynamical systems.

The central goal will be to unify concepts and develop widely applicable techniques for the analysis of high-dimensional stochastic phenomena.

They have identified six intertwined and vital problem areas where, focusing on specific examples, they hope to fruitfully share expertise and where progress should lead to essential progress in stochastic analysis: Stochastic Differential Equations, Infinite Dimensional Stochastic Geometry, Dynamical Systems, Random Media and Interacting Particle Systems, Super-processes, & Stochastic Analysis of Derivative Securities.

Potential applications

Aside from the applications to finance, Stochastic Analysis has many other potentially important applications. In turn these influence theoretical development. The theoretical progress of the project will undoubtedly increase the range of applications. Random algorithms represent one area of growing importance, they can sometimes be used with considerable effect in high dimensions where classical deterministic algorithms are useless. In one team scientists will research applications of log-Sobolev inequalities to random algorithms, to obtain explicit and tractable bounds on rates of convergence. In another, they will research the application of the measure valued processes to construct efficient algorithms for solving non-linear filtering equations. One goal of the studies in interacting particles is to create a better understanding of stochastic networks such as those arising in communication systems.

Training aspects: research fellowships

The contract will fund approximately 22 man years employment for young visiting researchers. The crucial priority of the teams will be to identity young workers with good training and outstanding research talent, and place them in the very strong research environments of the participant laboratories. In this positive environment it is absolutely reasonable to expect them to assist the principal scientists in an essential way to complete the work of this project, and so simultaneously develop the self confidence, judgement and experience required for their own personal development as outstanding scientists, financial engineers...

Participating teams: contact details

Timescale

The contract commenced on 1 October 1996 and has a duration of 48 months.

Additional information

In addition to this document, you can also find here:

• A project outline containing more detailed scientific information.
• Details on Research Fellowships applications.

URL: http://alf3.cii.fc.ul.pt/gfm/Projects/SA_TMR.en.html
Last modified: Thu Feb 20 17:03:18 WET 1997