What are the transition distribution (or density) functions of processes defined by
$dX_t=\mu dt +\sigma dW_t$
and
$dX_t= \theta(\mu-X_t) dt +\sigma dW_t,$
where $\theta>0$, $\mu$ is a real number, $\sigma>0$, and $W_t$ is a standard Brownian motion.
I know it is a solved problem, but I cannot find a reference that presents the detailed steps of the derivations. Could you please provide some good references? Or, could you please come with the derivations?
We consider the first one, that is, $X_t = X_s + \mu (t-s) + \sigma (W_t-W_s)$, for $t>s$. Then, \begin{align*} P(X_t \le y \mid X_s) &= P(X_t-\mu(t-s)-X_s \le y-\mu(t-s)-X_s \mid X_s)\\ &=P(\sigma(W_t-W_s) \le y-\mu(t-s)-X_s\mid X_s)\\ &=\Phi\left(\frac{y-\mu (t-s) -X_s}{\sigma\sqrt{t-s}}\right). \end{align*} That is, \begin{align*} P(X_t \le y \mid X_s=x) &=\Phi\left(\frac{y-\mu (t-s) -x}{\sigma\sqrt{t-s}}\right). \end{align*} Here, $\Phi$ is the cumulative distribution function of a standard normal random variable. The transition density function can be obtained subsequently by taking the derivative with respect to $y$.
For the second one, note that, for $t>s$, \begin{align*} X_t = e^{-\theta(t-s)}X_s + \mu\left(1-e^{-\theta(t-s)} \right)+\sigma\int_s^te^{-\theta(t-v)}dW_v. \end{align*} Then, \begin{align*} &\ P(X_t \le y \mid X_s)\\ =&\ P\left(X_t-e^{-\theta(t-s)}X_s - \mu\big(1-e^{-\theta(t-s)} \big) \le y-e^{-\theta(t-s)}X_s - \mu\big(1-e^{-\theta(t-s)} \big) \mid X_s\right)\\ =&\ P\left(\sigma\int_s^te^{-\theta(t-v)}dW_v \le y-e^{-\theta(t-s)}X_s - \mu\big(1-e^{-\theta(t-s)} \big) \mid X_s\right)\\ =&\ \Phi\left(\frac{y-e^{-\theta(t-s)}X_s - \mu\big(1-e^{-\theta(t-s)} \big)}{\sigma\sqrt{\frac{1}{2\theta}\big(1-e^{-2\theta(t-s)} \big)}} \right). \end{align*} That is, \begin{align*} P(X_t \le y \mid X_s=x) &=\Phi\left(\frac{y-e^{-\theta(t-s)}x - \mu\big(1-e^{-\theta(t-s)} \big)}{\sigma\sqrt{\frac{1}{2\theta}\big(1-e^{-2\theta(t-s)} \big)}} \right). \end{align*}
