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

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.