Институт вычислительной математики
и математической геофизики

The International Conference on Computational Mathematics


Statistic modeling and Monte Carlo methods

Numerical solution of unstable stochastic differential equations

Averina T.A., Artemiev S.S.

ICM MG SD RAS (Novosibirsk)

Stochastic differential equations (SDEs) with a growing variance arise at modeling price rows of actions and financial futures, and also in the radio engineering under linear and nonlinear oscillatory contours modeling.

Attempts to use numerical methods for statistical simulation of SDEs solution with a growing variance come to failure because of low accuracy of functional estimations from the solution. In this situation use of methods of the high order of convergence in weak sense and simulation of huge quantity of trajectories does not lead to success.

In this work the separation method of initial SDE solution on two components is proposed. The stochastic component is represented by SDEs system with the solution close to stationary process, and nonrandom component is described by unstable ODEs system that can be easily solved by numerical method. The ODEs system includes the equations for the first moments of the initial SDE solution, and these moments can be submitted approximately. We obtained the SDEs system for the stochastic component of initial SDE solution using the Ito formula both in case of linear initial system SDE, and in a nonlinear case.

The numerical experiments were realized that show the efficiency of the suggested method of the unstable SDEs solving. Results of some numerical experiments are presented.

* Supported by scientific program "Russian Universities - basic research", Grant YP.04.01.34; program "Leading scientific schools", Grant HIII - 1271.2003.1, Grant 02-01-01178.

Note. Abstracts are published in author's edition

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