It depends a bit on what you mean. History, as in written texts that we today can interpret and which is the very definition of history, started to be written down as earliest in Egypt around 3100 BCE. It was in the form of hieroglyphs on the Narmer palette (in the British Museum I believe, unless Egypt has reclaimed it).

If I recall correctly the earliest written material from Sumeria that has been found is slightly older, but is in the form of financial records - which is still historical material, but doesn't tell a story.

If you on the other hand mean actual written stories then Homeros is the earliest in the late 8th/early 7th century BCE. Most of his work is of course pure fiction, but some parts of it can still be interpreted as facts.

Furthermore, if you speak about written down history as done by historians, then you may be looking for the Greek historian Herodotus in the 5th century BCE. He's widely regarded as unreliable as a source, but is proclaimed as the father of history writing.

I can't speak for East Asia though. Considering how the culture looks like, I would guess that there may have been historians like Herodotus even before the 5th century BCE there, even if the actual sources don't remain. There are a couple of people with such flaires on the subreddit though so you might want to PM them if you want answers for that part of the world as well.

With a fairly generous definition and slightly anachronistic definition of “historiography” (writing down events that are considered by modern historians to be a reasonably faithful rendition of actual events), you could count Near Eastern king lists as the oldest form of historic writings (even though they generally include in their lists mythical rulers, and are fairly dry, since they only record the length of each reign, changes of dynasties, and a handful of events). The best-known artifact of this tradition, the Sumerian one, was redacted after 1753 BC, but it drew on earlier recensions, some of them probbly as early as the 3rd millenium BC—which would make them more or less contemporaneous with fragments of the Egyptian Royal Annals, whose most famous fragment is the Palermo Stone. While these Egyptian records are called “annals,” they are in fact closer in many ways to the Sumerian king list—though they give us more anecdotal information on the deeds of rulers. As such, they are predecessors of later royal annals, which tracked information year by year, focusing on battles and victories. The genre was certainly well-developed by the 2nd millenium (e.g. in the Hittite kingdom), but I don't know when the innovation appeared.