Mean Reversion vs. Stationarity

What is the difference between mean reversion and stationarity?

Answer

Stationarity is a more formally defined statistical concept, while mean reversion is often used more informally.

In finance, the two terms are often used interchangeably.

However, theoretically they are not the same.

Example 1: Stationarity does not imply mean-reverting.

Let $$Z = -1$$ with probability $\frac{1}{2}$ , and $$Z = 1$$ with probability $\frac{1}{2}$ . Define

X_t = Z \quad \text{for all } t.

Then $$X_t$$ is clearly stationar. On the other hand, it is not mean-reverting, because if the process starts at $$-1$$ , it stays at $$-1$$ , and if it starts at $$1$$ , it stays at $$1$$ . There is no drift back toward the mean $$0$$ .

Example 2: Mean-reversion does not imply stationarity.

Consider the Ornstein-Uhlenbeck process:

dX_t = \kappa(\theta - X_t)dt + \sigma dW_t.

This process is mean-reverting because when $X_t < \theta$ , the drift is positive, and when $X_t > \theta$ , the drift is negative. However, if the process starts from a fixed point $$X_0 = x_0$$ , which is not drawn from the stationary distribution, then the distribution of $$X_t$$ changes over time, so the process is not stationary.

If the starting point is drawn from the stationary distribution, then the whole Ornstein-Uhlenbeck process is both stationary and mean-reverting.