Transition Distributions for Brownian Motion and the Ornstein–Uhlenbeck Process
Summary
The document derives conditional transition distributions for two diffusion processes: Brownian motion with constant drift and an Ornstein–Uhlenbeck process that reverts toward a long-run level. For the first process, independent Brownian increments make the future value, conditional on the current state, normally distributed with a mean shifted by drift and a variance growing with elapsed time.
For the mean-reverting process, solving the stochastic differential equation gives an exponentially weighted current state plus a Gaussian stochastic integral. Its conditional distribution is therefore normal, with a mean that moves toward the long-run level and a variance determined by the mean-reversion speed and time interval. The cumulative distribution formulas are provided, and differentiating with respect to the endpoint yields the transition densities. These results assume constant parameters and Brownian noise; the treatment does not address estimation, boundary constraints, or model fit to data.
Key ideas
- A drifted Brownian process has a normal transition distribution conditional on its current value.
- The Brownian transition mean shifts linearly with elapsed time, while its variance increases with the interval.
- The Ornstein–Uhlenbeck solution combines the decayed current state with a Gaussian stochastic integral.
- Mean reversion makes the conditional mean approach the long-run level, with variance depending on time and reversion speed.
- The transition density follows by differentiating the conditional cumulative distribution.
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# How to find the transition distribution functions of these two processes?
# How to find the transition distribution functions of these two processes?
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?
## Answer by Gordon (score 4, accepted)
https://quant.stackexchange.com/a/32397
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*}Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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