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.

  • $\begingroup$ What is the law of $\ln X_t$? $\endgroup$ – rlartiga Jan 26 '15 at 20:46
  • $\begingroup$ I have no idea :( $\endgroup$ – Photon Light Jan 26 '15 at 22:52
  • $\begingroup$ @DontHurtMe Expanding on rlartiga's hint, you can solve this much like separation of variables in ODE: divide by $X_t$ and integrate both sides. The right side is very simple while the left side becomes $\ln(X_t)$. $\endgroup$ – Ian Jan 26 '15 at 23:03
  • $\begingroup$ Sorry but it dosen't help much. Maybe I'm dumb, but I need answer, not more questions. My teacher ask only questions in problems and maybe it's great way to develop brilliant mathematicians, but I don't get it. Even if I solve it, how can I find pdf f(x,t) from it and show that it satisfy F-P equation? $\endgroup$ – Photon Light Jan 26 '15 at 23:44
  • $\begingroup$ @DontHurtMe you know Ito rule? $\endgroup$ – rlartiga Jan 27 '15 at 12:38

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.


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