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Certain problems of dynamic statistical modeling in economics and ecology

2009

The target of the research is the dynamics of options prices and of populations in the World Ocean.The objective of the work is to evaluate the accuracy of existing approaches and algorithms and to suggest improvements.

In the project, we use methods of differential equation theory (ordinary and partial derivatives), theory of random processes and asymptotical methods.

Over the last several years, the role of derivative securities has increased in the world financial markets. This is related to the fact that a rational economic agent usually tries to find investment strategies that would guarantee not only a return, but also risk insurance in case of adverse price change. Thus, when investing the capital, investors are guided not only by the amount of expected profit, but also by potential risks.

Due to the relevance of the problem, new models of price formation for derivative securities, for example options, are being developed. As a rule, new models are based on existing ones. Each one of these models gives an impression that researchers are getting nearer and nearer to a precise description of the derivative price formation. But is this impression justified?

Research was carried out in the three major areas: testing statistical hypotheses that are taken as a basis for the theory, and the stability of results with respect to fluctuations of experimentally determined input parameters; developing asymptotic methods of solving the options price dynamics problems; studying existing databases for the topics mentioned in the research work:

A) Databases for stocks, options, futures etc;

B) Databases of plankton’s quality measurements in different waters;

C) Oceanological models (codes on free access).

As part of the research, we have tried to describe the main problems that need to be solved.

  • We have tested the normality hypothesis in view ofapplying the Wiener process in deriving the Black-Scholes equation.
  • We have obtained asymptotics of the Cauchy problem solutions for large and small values of the time.
  • We have used numerical experiments to evaluate the stability of the solution of ecological models with respect to inaccuracies in parameter setting.

We analyzed various sides of hydro-ecological model creation: starting from measurement methods and measurement result processing to creating a forecast technique. This scheme, based on the integration of a large non-linear system of partial derivatives equations, should describe (along with purely oceanological fields) the density of various fractions of  pelagial. One of the problems that we faced was data assimilation not for one moment in time (the Cauchy problem), but for a period in time of continuous observations. This analysis is based, first and foremost, on a comparison with respective models of atmosphere and ocean dynamics forecasting that are better developed now.

The main conclusion drawn from the research is possibility of creating valid hydro-ecological models that are able to forecast the dynamics of ocean pelagic regions. However their development requires intensive efforts.