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TL;DR

A mean-reverting SDE: \( dX_t = \theta(\mu - X_t)dt + \sigma dW_t \). The process is pulled toward \( \mu \) at rate \( \theta \), with solution \( X_t = \mu + (X_0 - \mu)e^{-\theta t} + \sigma \int_0^t e^{-\theta(t-s)}dW_s \).

By Valenke Exam Prep Team·Last updated 2026-06-18

Ornstein-Uhlenbeck Process

A mean-reverting SDE: \( dX_t = \theta(\mu - X_t)dt + \sigma dW_t \). The process is pulled toward \( \mu \) at rate \( \theta \), with solution \( X_t = \mu + (X_0 - \mu)e^{-\theta t} + \sigma \int_0^t e^{-\theta(t-s)}dW_s \).

Why it matters for interviews

The standard mean-reversion model in quantitative finance. Used for interest rate modeling (Vasicek), pairs trading (spread dynamics), and volatility modeling. Understanding its stationary distribution and half-life is interview material.

Definition and Mathematical Foundation

A mean-reverting SDE: \( dX_t = \theta(\mu - X_t)dt + \sigma dW_t \). The process is pulled toward \( \mu \) at rate \( \theta \), with solution \( X_t = \mu + (X_0 - \mu)e^{-\theta t} + \sigma \int_0^t e^{-\theta(t-s)}dW_s \).

Application in Quantitative Finance

The standard mean-reversion model in quantitative finance. Used for interest rate modeling (Vasicek), pairs trading (spread dynamics), and volatility modeling. Understanding its stationary distribution and half-life is interview material.

Related Concepts

Brownian Motion (Wiener Process)

Related Terms

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Frequently Asked Questions

What is the stationary distribution of OU?▼
As \( t \to \infty \), \( X_t \sim N(\mu, \sigma^2/(2\theta)) \). The long-run variance decreases with mean-reversion speed \( \theta \): faster reversion means tighter distribution around the mean.
What is the half-life of mean reversion?▼
The time for the expected deviation from the mean to halve: \( t_{1/2} = \ln(2)/\theta \). For pairs trading, this determines how long you expect to wait for the spread to revert.
How is OU used in pairs trading?▼
The spread between two cointegrated assets is modeled as OU. When the spread deviates significantly from \( \mu \), enter a mean-reversion trade. The half-life determines holding period, and \( \sigma^2/(2\theta) \) determines position sizing.