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The European Union TMR
|Stochastic Analysis and its applications is a project which aims to develop an improved mathematical understanding of random systems and at the same time provide training and research opportunities for young researchers. It draws on the expertise of scientists in 6 European countries and has employed at least 20 young researchers.|
Feature: Women in the Stochastic Analysis Network.
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|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.
This Internet site
provides 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.
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|Project overview||The mathematics
developed to study stochastic (or random) systems is difficult but of
considerable importance; it's impact permeates many aspects of our
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.
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|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.
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|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.
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|Training Aspects||The contract will fund
approximately 12 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,
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|Timescale||The contract commenced on 1
October 1996 and has a duration of 48 months
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|Contact Details||Scientific coordinator:
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|Other Participant Teams||The scientists responsible
for the other teams are:
Mathematicians from a number of
other institutions play an active part in the contract as members of the
above teams. In particular:
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1.University of Oxford