Is there a stochastic differential equation whose solution follows a stationary Log-normal distribution?
I was thinking in the geometric Brownian motion
$$dx = (\alpha x )dt + (\sigma x )db, \quad \alpha,\beta\in\Re$$
whose probability density function $f_{S_{t}}(s;\mu ,\sigma ,t)$ is a Log-normal distribution
$$f_{x_{t}}(x;\mu ,\sigma ,t)={\frac {1}{\sqrt {2\pi }}}\,{\frac {1}{x\sigma {\sqrt {t}}}}\,\exp \left(-{\frac {\left(\ln x-\ln x_{0}-\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)t\right)^{2}}{2\sigma ^{2}t}}\right)$$.
But it depends on time (it is not stationary). Is there a simple SDE with stationary Log-normal distribution?
Let $F(x)$ be an arbitrary time-independent CDF and $B_t$ be a standard Brownian motion. The CDF of $B_t$ is $\Phi(x/\sqrt{t})$ where $\Phi(x)$ is the standard normal CDF. Then $$ U_t:=\Phi(B_t/\sqrt{t}) $$ is uniform on $[0,1]$ for all $t>0$ and $$ X_t=F^{-1}(U_t)=F^{-1}\circ\Phi(B_t/\sqrt{t}) $$ is a stochastic process that has stationary distribution $F(x)\,.$ If $F^{-1}$ is $C^2$ Ito's formula is applicable and an SDE that is solved by $X_t$ is easily found: $$\tag{1} dX_t=f'(B_t)\,dB_t+\frac{1}{2}f''(B_t)\,dt $$ where $f(x)=F^{-1}\circ \Phi(x/\sqrt{t})\,.$ Writing $$ B_t=\sqrt{t}\,\Phi^{-1}(U_t)=\sqrt{t}\,\Phi^{-1}\circ F(X_t) $$ allows to express (1) as an SDE for $X_t$ driven by $B_t\,.$