Stationarity, Ergodicity, and Path Dependence
Three assumptions sit underneath most quantitative finance, usually unstated. Each fails in markets, and each failure has a recognisable signature.
Stationarity
Statistical properties do not change over time. This is what allows history to inform the future at all.
Markets violate it constantly: regimes shift, regulation changes, participants enter and leave, and the structure of the market itself has changed beyond recognition in twenty years.
The practical response is to use returns rather than prices, to weight recent data more heavily, to test for structural breaks, and to prefer models with a mechanism you believe over models fitted to a long history you cannot vouch for.
Ergodicity
Time averages equal ensemble averages: one long path tells you what the distribution across many parallel paths would be.
This one is rarely stated and matters more than it appears, because you only get one path. There is one history of markets, and every estimate comes from it.
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