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Following the death of Paul Samuelson, I looked at his seminal paper `Proof that properly anticipated prices fluctuate randomly’. I found it intriguingly related to my beloved puzzlement about the meaning of probability, and I have something to say about it. Since I didn’t look at the forty-five years of literature that followed this paper I am probably going to flaunt my ignorance in public, but this is just a blog so what harm can it do.

The purpose of Samuelson’s paper is to formalize the intuition that competitive prices must, in some sense, look like random walk: Let {\dots,X_{t-1},X_t,X_{t+1},\dots} be a stochastic process that represents the spot price of some product, say wheat: {X_t} is the price in which you can buy wheat at day {t}. Fix a day {T^\ast} and let {Z(t)}, for {t\leq T^\ast}, be the price at day {t} of a contract that requires the delivery of wheat at day {T^\ast}. Samuelson proves the following theorem:

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