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SOME FUNDAMENTAL THEOREMS IN MATHEMATICAL FINANCE:
A STOCHASTIC PROGRAMMING DUALITY PERSPECTIVE
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Lisa A. Korf
Department of Mathematics
University of Washington, Seattle
Date. Tuesday, 6th of June 2000
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Time. 16.30
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Location. Seminarraum 107, 6th floor, green area, TU WIEN - Freihaus
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Abstract.
Stochastic programming concerns the theory of making optimal decisions
under uncertainty, in which a probability distribution has been assigned
to the uncertain parameters of a problem. These are generally
optimization problems in infinite-dimensional spaces. Much of the theory
revolves around how to approximate such problems in finite dimensions so
that they might be solved in a practical mathematical programming setting.
In addition, a nice duality theory has been developed in the infinite-
dimensional setting.
It was only natural that financial applications took their place as one of
the primary application areas in this field. Mathematical programming
provides a flexible framework in which to model all kinds of stochastic
price processes, as well as the unavoidable complications of constraints,
costs, etc. which arise in practice. While much attention has been
focused on the practical aspects of solving these problems, little
attention has been paid to deriving some of the rich (finite and
infinite-dimensional) theory of mathematical finance in a stochastic
programming duality setting, where extensions to problems with transaction
costs, etc. would be considered very natural.
This lecture introduces stochastic programming duality, and delves into
some of the fascinating issues involved in trying to derive the
``fundamental theorems of asset pricing'' (equating no arbitrage
conditions with the existence of an equivalent martingale measure for the
underlying asset price process) in a stochastic programming framework.
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Further information about the schedule of seminars at the Department of
Financial and Actuarial Mathematics is available at
http://www.fam.tuwien.ac.at/schedule !
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