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

This paper is concerned with the following Markovian stochastic differential equation of mean-reversion type \[ dR_t= (\theta +\sigma \alpha(R_t, t))R_t dt +\sigma R_t dB_t \] with an initial value $R_0=r_0\in\mathbb{R}$, where $\theta\in\mathbb{R}$ and $\sigma>0$ are constants, and the mean correction function $\alpha:\mathbb{R}\times[0,\infty)\to \alpha(x,t)\in\mathbb{R}$ is twice continuously differentiable in $x$ and continuously differentiable in $t$. We first derive that under the assumption of path independence of the density process of Girsanov transformation for the above stochastic differential equation, the mean correction function $\alpha$ satisfies a non-linear partial differential equation which is known as the viscous Burgers equation. We then develop a Galerkin type approximation scheme for the function $\alpha$ by utilizing truncation of discretised Fourier transformation to the viscous Burgers equation.