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The model for a stock price $$ dS_t=\mu dt + \sigma dB_t $$ where $B_t$ is a Brownian motion on $(\Omega, \mathcal{F},P)$, is commonly attributed to the work that Bachelier has carried out in his PhD thesis. However, it is not until much later that the concept of Brownian motion is formalized by Einstein and/or Lévy, among others.

Without a general definition of Brownian motion, in which way did Bachelier employ such concept and characterize his famous model? In which way did he introduce such non existing construct?

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  • $\begingroup$ I would point out that he was not modeling equity securities. He was modeling rentes, often known in the English-speaking world as consols or perpetuities. I would also argue that his derivation is correct for rentes but not equities. $\endgroup$ Commented May 18, 2022 at 2:57
  • $\begingroup$ He is following Calcul des chances et philosophie de la bourse by Jules Regnault, probably the first quant. $\endgroup$ Commented May 18, 2022 at 3:06
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    $\begingroup$ It is important to realize that axioms of probability did not exist until 1930 and 33 for Kolmogorov's. Bachelier is working in the classical school of probability, these are dice rolls of dice of known parameters. You are imposing later ideas because you are preceding the Hausdorf paradox. $\endgroup$ Commented May 18, 2022 at 3:09
  • $\begingroup$ Probability distributions at that time were seen as the solution to a differential equation. What you are doing is back-fitting newer ideas on older ideas. He isn't using non-existent constructs. He is adapting existing constructs over a novel problem. $\endgroup$ Commented May 18, 2022 at 3:16
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    $\begingroup$ He was academically punished for this. He used perfectly good and respectable math for a pecuniary purpose. He lowered the grand field of mathematics into speculation. He sullied it by taking it from the ivory tower and making something for low use. It is almost six decades before it is rediscovered. By then, the world had discovered important issues that he would not have yet been able to know about. $\endgroup$ Commented May 18, 2022 at 3:32

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