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?