0
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.