When Avraham experiences a vision of 'fear, darkness, great, descending upon him', (Gen 15:12), the Midrash says that these four words refer to the 4 exiles. Fear is Babylon, darkness is Medea, great is Greece, and descending is Rome, (Gen rabah 44:17).
Both of the Targumim that expound upon this verse designate Persia as the 4th exile, corresponding to 'descending', (Gen 15:12).
What is the reason for this discrepancy? Did the Targumim not know about the Roman exile?
I suppose it is possible that this passage was composed before the Roman exile, meaning that up until the Roman exile, the persian exile was identified as the fourth exile, as it could have very well been the final exile. Once the second temple was destroyed, however, the Roman exile replaced it, and the Persian exile was clumped together with the Medean, hence the generally used term Persian-Medean exile, (hearsay, source needed). I have no source for this answer other than my own thoughts. Has anyone encountered sources that address this directly?