# Quantiles of a Brownian Motion

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 $dX_t = \lambda(\bar{X}-X_t) dt + \sigma dW_t$.