I am trying to understand theOrnstein-Uhlenbeck process

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

my question is what is the meaning of the parameters? and assuming that we know those parameters in advance what is the best way to exploit it?


1 Answer 1


$\theta$ is the "mean" for this process. If $X_t > \theta \implies (\theta - X_t) < 0 $, which means that the drift for the process is negative and tends towards $\theta$. The opposite case can be made for $X_t < \theta$ ; the process will have positive drift when $X_t$ is below $\theta$.

Therefore we can consider $\kappa$ to be the "speed" of mean reversion, scaling the distance between $X_t$ and $\theta$ appropriately to match whatever is being modeled.

$\sigma dW_t $ is your standard Wiener process scaled by volatility $\sigma$.

In plain English you can interpret the differentials as a process that reverts to a mean, $\theta$, with speed, $\kappa$, and volatility, $\sigma$.


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