A Galerkin approximation scheme for the mean correction in a mean-reversion stochastic differential equation

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Wu, JiangLun and Yang, Wei (2013) A Galerkin approximation scheme for the mean correction in a mean-reversion stochastic differential equation. Preprint / Working Paper. UNSPECIFIED.

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Abstract

This paper is concerned with the following Markovian stochastic dierential equation of mean-reversion type dRt = ( + (Rt; t))Rtdt + RtdBt with an initial value R0 = r0 2 R, where 2 R and > 0 are constants, and the mean correction function : R [0; 1) 7! (x; t) 2 R is twice continuously dierentiable in x and continuously dierentiable in t. We rst derive that under the assumption of path indepen- dence of the density process of Girsanov transformation for the above stochastic dierential equation, the mean correction function sat- ises a non-linear partial dierential equation which is known as the viscous Burgers equation. We then develop a Galerkin type approxi- mation scheme for the function by utilizing truncation of discretised Fourier transformation to the viscous Burgers equation.

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