Eschatology has to do more with the "end of humankind" specifically than the "end of things". So, if Eschatology is concerned with the final events of history, or the ultimate destiny of humanity, then Anthropogeny is concerned with study of human origins.
Anthropogeny comes from Greek anthropo- (human) + Greek -geny (birth, origin).
But, if you want to take it in more general sense i.e. "final events of history", as history1 can be of humans, earth, or cosmos, then it is Cosmogony.
The word comes from the Koine Greek κοσμογονία (from κόσμος "cosmos, the world") and the root of γί(γ)νομαι / γέγονα ("come into a new state of being"). In astronomy, cosmogony refers to the study of the origin of particular astrophysical objects or systems, and is most commonly used in reference to the origin of the Universe, the Solar System, or the Earth–Moon system.
And on the theological side, you can consider Theogony: "the genealogy or birth of the gods."
1. 2nd definition of history according to Oxford Dictionary: The whole series of past events connected with a particular person or thing.