# Wiki Wiki Web

## Forward Variance Decay of Mean Reverting Processes

we consider the OU process: $$dS = - \lambda S dt + \sigma dW_t$$

We lt $X_t = e^{\lambda t} S_t$, which follows the SDE $dX_t = \sigma e^{\lambda t} dW_t$. Integrating this and using the Ito isometry to determine variance:

\begin{eqnarray} S_T - S_0 e^{-\lambda T} &=& \int_0^T \sigma e^{-\lambda (T-t)} dW_t \\ \frac{1}{T-t} Var(S_T - S_0 e^{-\lambda T}) &=& \sigma^2 \gamma(2 \lambda, t, T)\\ \gamma(x,T-t) &=& \frac{1-\exp(-x(T-t))}{x (T-t)} \end{eqnarray}

Therefore, the volatility of the process observed with return over interval $T$ will be scaled down by a factor $\gamma(2 \lambda, T)^\frac{1}{2}$.