Stochastic averaging for SDEs with Hopf Drift and polynomial diffusion coefficients
Subject Areas : Probability theory and stochastic processes
1 - Department of Mathematical Sciences, Isfahan University of Technology,
Isfahan, Iran
Keywords: Stochastic Differential Equation, stochastic averaging, system of complex bilinear equations, stochastic ow, Cholesky decomposition,
Abstract :
It is known that a stochastic differential equation (SDE) induces two probabilisticobjects, namely a difusion process and a stochastic flow. While the diffusion process isdetermined by the infinitesimal mean and variance given by the coefficients of the SDE,this is not the case for the stochastic flow induced by the SDE. In order to characterize thestochastic flow uniquely the infinitesimal covariance given by the coefficients of the SDE isneeded in addition. The SDEs we consider here are obtained by a weak perturbation of a rigidrotation by random fields which are white in time. In order to obtain information about thestochastic flow induced by this kind of multiscale SDEs we use averaging for the infinitesimal covariance. The main result here is an explicit determination of the coefficients of the averagedSDE for the case that the diffusion coefficients of the initial SDE are polynomial. To do thiswe develop a complex version of Cholesky decomposition algorithm.
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