Fokker-Planck equation - find probability density function

Question

I have problem from my course, that I can't solve. If anyone can do it and explain, would be great.
Find the probability density function $f(x,t)$, of $X_t$ where {$X_t$} is a solution of stochastic differentail equation:
$dX_t = X_t(\frac{\sigma^2}{2}dt+\sigma dW_t), X_0 = 1$
with constant $\sigma > 0$. Show that $f(x,t)$ satisfies corresponding Fokker-Planck equation.

Answer 1

Let $Y_t=ln(X_t)$, by Ito's lemma we derive that the process $Y_t$ follows the SDE

$ dY_t=\sigma dW_t,\quad Y_0=0 $

which has solution $Y_t= \sigma W_t$. Because $W_t$ has distribution $N(0,t)$, $Y_t$ has distribution $N(0,\sigma^2t)$.

Since $X_t=e^{Y_t}$ and $X_t$ has normal distribution, we conclude that $X_t$ has log-normal distribution, so the function $f(x,t)$ is the lognormal probability distribution function $$ f(x,t)=\frac{1}{x\sigma\sqrt{2\pi t}}e^{-\frac{(\ln x)^2}{2\sigma^2t}} $$

The rest of the exercise just consists of showing that this function satisfies the Fokker Planck equation for the process above, i.e., write the Fokker Planck equation for the process $X_t$, replace $f(x,t)$ in it and verify that you obtain an identity.