Consider the following 1-dimensional Stochastic differential equation:
$dX_t=b(X_t)dt+\sigma(X_t)dB_t$
With some assumptions, one can have the stationary probability density $p(x)$:
$p(x)=\frac{C}{\sigma^2(x)}e^{\int_{-\infty}^{x}\frac{2b(y)}{\sigma^2(y)}dy}$,
where $C$ is the normalization constant, which can be done from the Fokker-Planck equation. In the simple case where $\sigma(x)=1$ and $b(x)=-ax$ where $a>0$ is a constant,
$dX_t=-aX_tdt+dB_t$
one can find $a$ such that it minimizes the following functional
$F(f;q)=\int_{\bf R}[\sqrt{p(x)}-\sqrt{q(x)}]^2dx$
where $f(x)=-ax$, $p(x)$ is as the probability density defined above, and $q(x)=\frac{1}{\pi}e^{-x^2}$. Without need of calculus of variation, one can simply put $p(x)=q(x)$.
Here is my question:
In the more general cases, say, $b(x)\leq 0$ and $\sigma(x)=1$, can one still be able to find a function $b(x)$ to minimize the functional $F(b;q)$? Here $q(x)=\frac{1}{\pi(1+x^2)}$
Here is what I thought:
One can still "force" $p(x)=q(x)$ to find $b(x)$. However, then one may get $b(x)=\frac{-x}{1+x^2}$, which may cause that $p(x)$ does not exist. Hence calculus of variations may be needed here. But I'm not sure if it will work or not since I am not able to calculate $\frac{d}{d\epsilon}|_{\epsilon=0}F(b+\epsilon h)$ here because of the ugly form of $p(x)$. Is any other technique needed?
