I am looking for the p-th percentile of a stochastic process $X_t$ that satisfies $dX_t = \mu(X_t) dt + \sigma(X_t) dW_t$ where $W_t$ is a standard Brownian motion.

I believe that the p-th percentile of $dX_t = \mu dt + \sigma dW_t$ is given by $X_0 + \mu t +\sigma \sqrt{t} \Phi^{-1}(p)$ where $\Phi$ is the standard normal cdf, and that the p-th percentile of $dX_t = \mu X_t dt + \sigma X_t dW_t$ is given by $X_0 \exp( (\mu-\frac{1}{2}\sigma^2) t +\sigma \sqrt{t} \Phi^{-1}(p))$.

But I am having trouble generalizing this to more general processes. In particular, I am trying to obtain the p-th percentile of the Ornstein–Uhlenbeck process $dX_t = \lambda(\bar{X}-X_t) dt + \sigma dW_t$.
Thank you for your help.

  • $\begingroup$ You may want to check for en.m.wikipedia.org/wiki/Kolmogorov_equations $\endgroup$ – Yujie Zha May 24 '17 at 16:21
  • $\begingroup$ Thank you for your suggestion! Could you elaborate a little? Do you suggest to describe the evolution of the probability distribution by a Kolmogorov equation? $\endgroup$ – user449277 May 24 '17 at 16:23
  • $\begingroup$ Yep, if you check all the links under that wiki page, you will find different versions of the equations, describing backward/forward, and for different type of diffusions. And basically you could think that SDE relates closely to the instantaneous probability distribution expressed by those partial differential equations. $\endgroup$ – Yujie Zha May 24 '17 at 16:26
  • $\begingroup$ That's a good suggestion, thank you! Indeed, for the Ornstein-Uhlenbeck process the Fokker-Planck equation seems to have a simple closed-form solution according to Wikipedia. But how to generalize this to other cases remains unclear to me, I will see whether I can work it out using the KFE. $\endgroup$ – user449277 May 24 '17 at 16:33
  • $\begingroup$ yes, good you know Fokker-Planck. Sometimes this is used as the synonym for Kolmogorov forward equation. $\endgroup$ – Yujie Zha May 24 '17 at 16:38

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