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As I say in the subject, How can I put the model $d x_t = \eta\, (\overline{x} - x_t)\,d t + \sigma\, x_t\,d W_t$ into state space form? I mean, which are the observation and transition matrices?

Sonntag
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    What do you observe (measure)? – Juho Kokkala May 13 '14 at 21:20
  • Just found tonight at Schwartz, E. S. (1997). The Stochastic Behavior of Commodity Prices: Implications for Valuation and Hedging. The Journal of Finance, 52(3), 923–973. doi:10.1111/j.1540-6261.1997.tb02721.x – Sonntag May 14 '14 at 05:53
  • This is many years later but, in the article cited, what is put into state space form is the model where the commodity's spot price follows the above SDE (i.e. spot log-price is OU) and is unobservable, and what is observed are the prices of futures contracts on that same commodity. That seems to be missing from the question. There's not really much to say about putting just the log-OU model in state space form so it seems like maybe that was really the question of interest. – Chris Haug Jul 24 '21 at 13:00

2 Answers2

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OU in discrete time $$x_{t+1}-x_t=\eta(\bar x-x_t)+\nu_t$$ is equivalent to AR(1) process $$x_{t+1}=\eta\bar x+(1-\eta)x_t+\nu_t$$

AR(1) model $$x_t=c+\phi_1 x_{t-1}+\nu_t$$ in state space representation is $$x_t=s_t+\nu_t\\ s_t=c+\phi_1s_{t-1}+\phi_1\nu_{t-1}$$

Your model is not OU, strictly speaking, because you have $x_t$ scale of the error term. Yours is something like OU of the logarithm: $$d\ln x_t=\eta(\bar \ln x-\ln x_t)+x_t\nu_t$$

Aksakal
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From my understanding, you may have a situation where there is no observation matrix.

Your transition matrix is $X_t = -\beta(X_{t-1} - \alpha)+\sigma X_{t-1} dW_t$ and your observable is just $Y_T = X_t + \epsilon_t$, assuming that you are slightly uncertain of your observations.

Now, we note here that there are two sources of error, one being the disturbances due to the O-U process in the latent variable, and one observed variable with an observation error (which may be 0 depending on your situation, if you have direct access to the $X_t$ term).

analystic
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