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While studying Brownian motion applied to mathematical finance, I came across these lecture notes by prof Steve Lalley. In the prologue, he gives this explanation for the occurrence of Brownian motion in the theory of asset pricing

In equilibrium, the discounted price process of any tradeable asset, observed at discrete times, is a martingale; therefore, in continuous time, it is also a martingale. Moreover, the prices of traded assets seem to vary continuously with time$^1$ and seem to have finite quadratic variation. Brownian motion now rears its head for the following basic reason, a fundamental theorem of Paul Lévy:

Theorem. Every continuous–time martingale with continuous paths and finite quadratic variation is a time-changed Brownian motion.

$^1$ This, along with the technical assumption that price processes have finite quadratic variation, is somewhat controversial. Discrepancies between theoretical (Black–Scholes) and actual prices of derivative securities may in fact be due to the failure of one or both of these assumptions in real markets.

Summarizing: the discounted price process of any tradeable asset is a martingale, it varies continuosly in time and it has finite variation, so it satisfies all the hypotheses of the Lévy characterization and hence is a Brownian motion.

However, the Lévy characterization of Brownian motion that I studied in my course is a little more specific, in particular, the quadratic variation not only has to be finite but it has to be equal to $t$, moreover the process must start at the origin:

Lévy characterization of Brownian motion. The Wiener process $W$ is an almost surely continuous martingale with $W_0 = 0$ and quadratic variation $[W]_t=t$ (which means that $W_t^2 − t$ is also a martingale).

But how can we prove that the discounted price process of any tradeable has quadratic variation equal to $t$? And what about the other hypotheses? Is it really true that the discounted price process of any tradeable satisfies all the hypotheses of the Lévy characterization and so can be considered a Brownian motion?

Maybe I misunderstood his words and what he actually means is that Brownian motion is just a good candidate to model the price process. However, in these related questions other users explain that the Brownian motion is not that good in modeling prices, so at this point is really hard for me to think that the discounted price process of any tradeable is a Brownian motion.

Why should we expect geometric Brownian motion to model asset prices?

Shortcomings of generalized Brownian motion for asset price modelling

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    $\begingroup$ The key word is time-changed - e.g. a time-changed Brownian motion is not (necessarily) a standard brownian motion. $\endgroup$ Apr 27, 2022 at 18:47
  • $\begingroup$ @rubikscube09 Oh thanks I don't know how I missed that, but what is the definition of time-changed Brownian motion? I never heard about it before. $\endgroup$
    – sound wave
    Apr 27, 2022 at 18:56
  • $\begingroup$ Basically the time becomes a stochastic process on its own, so you take a Brownian motion, sample it at random times (e.g. change your "clock") and get a new process. Here's an econometrically oriented paper on ithttps://www.stats.ox.ac.uk/~winkel/winkel15.pdf $\endgroup$ Apr 27, 2022 at 20:35
  • $\begingroup$ This book, (by Prof. Lalley's colleague Greg Lawler, whose course I took): math.uchicago.edu/~lawler/finbook.pdf discusses that result as well. $\endgroup$ Apr 27, 2022 at 20:36

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