3. Applications of stochastic analysis in economy and
natural sciences
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Polish supervisor:
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Armen Edigarian
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Cooperating partners:
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Maciej Klimek (Uppsala University)
Regina Y. Liu (Rutgers University)
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Starting from the results of Merton, Scholes and Black in 1973 it
becomes apparent that mathematical methods are quite efficient in
option pricing. The pricing formula obtained by them (so called
Black-Scholes formula) almost immediately was used by
practitioners on stock exchanges.
At the beginning the methods were mainly based on partial
differential equations. However, later it turned out that
stochastic calculus methods (Ito calculus) are more rigorous from
the mathematical point of view.
One of the main notions in the financial mathematics is arbitrage
and martingale measure. It is well-known that on a complete market
without transaction costs the existence of a martingale measure is
equivalent to no arbitrage property (Harrison-Pliska,
Dalang-Morton-Willinger, Rogers, etc). There are also many results
which connect this notion with entropy.
However, the results for markets with transaction costs seem to be
far from sufficient. The main aim of the project will be to
develop the theory of arbitrage pricing, to deepen the
understanding of markets with transaction costs. The student
should develop new stochastic tools to obtain important results of
finance.
From the practitioners point of view, instead of exact formulas,
it is more important to obtain numerical methods which are
sufficiently fast. The idea of the project is also to find methods
based on Monte-Carlo, which could be fast and close to market
prices.
Possible application of the obtained results to mathematical
models in various natural sciences (seismology, weather
forecasting, turbulence, etc.) are a long term aim of the
project.
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