Modelling market returns as independent random variables/martingales is the same as modelling the solar system as a geocentric system with the planets and Sun circling around Earth in epicycles. Predictions of the future are often vastly incorrect in both models. Quite surprisingly, this solar system model survived for thousands of years, despite it being totally incorrect. Then came Tycho Brahe who introduced a modified version of this Ptolemaic system. In Brahe’s model the planets orbit the Sun which orbits the Earth. While this model improved the accuracy of planetary motions, it failed to model reality. Perhaps it could be said that stochastic jump processes are equivalent to Brahe’s model of the solar system. While these jump process do a better job at modelling the returns than simple stochastic processes, they fail to grasp the underlying true model of returns.
Poor Market Forecasting

And as we now know, the true model (for now) of the solar system was introduced by Aristarchus (Copernicus and Kepler helped bring forward this model) and predicts planetary motions with near perfection and represents the actual state of the solar system. I believe that the analogous model for stock returns has been introduced by Didier Sornette, Anders Johanson and others.

These scientists have expounded the idea that market returns are a function of individual agents in a complex system. Just as the human body is a collection of individual cells which make “decisions” based on communications with neighbours through chemical processes, traders make their own decisions based on communication with neighbours. With this perspective the market is a complex system of interacting agents. Thus, returns should be a function of these interactions. Under this complex system model, bubbles and crashes which dot the history of finance (which are not explained fully by independent returns/martingales) are straightforward results. In addition, these models still explain why returns are close to normal “most” of the time.
So, it seems that we need to modify or throw away the old models in favour of these new complex system models. These complex models offer better prediction of the overall market and more fully represent reality.

There is the interesting possibility of this: the stochastic volatility model referred to as the Ornstein-Uhlenbeck process represents the physical process of a “noisy relaxation process.” The Wiener Process represents Brownian motion or motion of a particle through a gas or liquid. So, if we consider the movement of a stock through a virtual container of many stocks (these stocks are the atoms in the Brownian motion) then we need to ask ourselves: What does the price, interest rate, returns, etc. mimic? It is NOT the equations! BUT the physical processes themselves.  Why is an interest rate in a state of disequilibrium in the first place… that it must try to relax? Who put the stock in swarm of human hands all independently moving… It more correctly seems that the traders are following its movement at every second, waiting to grab it when the time if right (thus not independent)?