What is the easiest way to obtain a drift parameter of O-U process given I have $\mu$?

Is it ok to linearize the O-U process like so:

$P_{t} = \mu + \phi(P_{t-1}-\mu)+\xi_t$

Form vectors from historic data:

$$ A = \begin{bmatrix} \ P_0-\mu \\ \ P_1-\mu \\ \ \dots \\ \ P_T-1-\mu \\ \end{bmatrix} $$

$$ b = \begin{bmatrix} \ \mu \\ \ \mu \\ \ \dots \\ \ \dots \\ \end{bmatrix} $$

$$ Y = \begin{bmatrix} \ P_1 \\ \ P_2 \\ \ \dots \\ \ P_T \\ \end{bmatrix} $$

And solve $\phi$ with OLS?



1 Answer 1


yes, using a rolling AR-1 and μ

  • 3
    $\begingroup$ Hi @Anthony Suherli, thanks for your answer. Would you be willing to extend your answer a little bit so that it can be helpful for future readers as well? That'd be great! $\endgroup$ Commented Nov 16, 2020 at 8:48
  • $\begingroup$ dp(t) = α(t) + k (μ- p(t)) dt + σ dW(t), k>0 where lim α(t) -> inf = μ, and dW(t) at time t becomes 0 because we dont use p(t-1) now dp(t) = α(t) + k (μ- p(t)) dt we can estimate using an ar1 using a window (say 60 past observations), and kind of simulate a model with μ- p(t) as input. α(t) should average to μ Now you will get an estimate of k from every observation, you can average them or do anything that best suite your needs $\endgroup$ Commented Nov 17, 2020 at 0:25

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