The mean reverting Ornstein-Uhlenbeck process is of the equation:

$$dX_t=(a-cX_t) \, dt+\sigma \, dW_t$$

If we are told that both $a$ and $c$ are larger than $0$, what then is the limiting distribution of $X_t$ as $t$ tends to infinity?

Obviously mean-reverting by definition means that the process will tend to drift towards its long-term mean, however I'm not sure how to arrive at the limiting distribution?

Any help would be greatly appreciated...


1 Answer 1


Hint: Set $$ X_t=at+X_0e^{-ct}+\sigma\int_0^te^{-c(t-u)}\,\mathrm dW_u, $$ and check by Itô's formula that this is indeed the unique solution of your stochastic differential equation.

Then, compute the distribution of $X_t$, $t\ge0$ by noting that $$ \int_0^te^{-c(t-u)}\,\mathrm dW_u\sim N\left(0,\int_0^te^{-2c(t-u)}\,\mathrm du\right). $$


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