Samuelson; martingales; probability | The Leisure of the Theory Class

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: