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In this question, I asked about validating the assumption of geometric Brownian motion in a analytic model using ARIMA. Here, I want to generalise this idea.

If I'm building a decision model that requires some assumption about how prices, say, change in time, suppose I begin by looking at a time-series of said prices and discover that it is fit well by some $ARIMA(p, d, q)$ model. How can I use this information in an analytic, i.e.~continuous-time model? For instance, if I see that my data is fit well by an $ARIMA(0, 1, 0)$ model, I can with some degree of confidence model my prices by a Wiener process and compute integrals, expected values, variances and so on.

Analogously, is there a continuous-time, non-recursive equation associated to each $ARIMA(p, d, q)$ model?

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    $\begingroup$ The middle paragraph sounds like something really difficult. I would google for what you are looking for. maybe something will come up ? Note that Ornstein Uhlenbeck process is continuous and has a discrete time AR(1) analog but I doubt that this result holds in general. Bergstrom and A.Q Phillips are top people in continuous time econometrics. Not sure of the people in statistics. $\endgroup$
    – mlofton
    Nov 20 '20 at 14:23

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