Traditional probability theory and Bayesian decision criteria developed on this basis focus on the frequency of events, provided that events can occur repeatedly. It is not applicable to the probability estimation of one-off events. In reality, people really often need to make their own comments on different propositions. The research shows that people can't get the best result because of their own conditions and limited knowledge and ability, but can only get a certain degree of satisfactory solution, which enables people to solve complex problems by intuition and personal experience. The essential characteristics of the actual decision-making process urge scientists to study the decision-making process from the perspective of behavioral cognition, among which amos tversky and Daniel Kahneman are the best. They tried to use heuristic method instead of Bayesian analysis. This research is called subjective probability research, and expectation theory is actually a kind of subjective probability theory.

Second, from linear to nonlinear-the application of nonlinear science

In human understanding, first of all, a relatively simple linear relationship (linear model) is used to describe the quantitative relationship of linear problems. For those cases where nonlinear factors cannot be ignored, linear approximation or linear iteration is often used to deal with them, which can sometimes get better results, but this kind of situation usually only appears in relatively simple nonlinear problems, or only studies some "conventional" behavior characteristics of the system. With the deepening of people's understanding of society and nature, people are increasingly afraid to "underestimate" nonlinear problems. First of all, in essence, nature is nonlinear. Secondly, the strong nonlinear action and long-time scale system behavior in many problems cannot be described by linear methods (including linear approximation). Thirdly, even some seemingly simple systems may show amazing complexity (such as deterministic randomness), so people pay more and more attention to the study of nonlinear phenomena widely existing in society and nature, and nonlinear science is born.

Stuzer, an American economist, first applied nonlinear science to economic research. In his paper "Chaotic Dynamical Systems and Bifurcation Theory in Macroscopic Models" published in 1980, Li-York Theorem and Bifurcation Technology are applied to Havelmo growth model, and the conditions for chaos in this model are found. After that, more and more scholars began to use nonlinear scientific methods to study economic and financial systems.

Benoit B.Mandelbrot( 1997), the founder and famous mathematician of fractal, applied his research results to the study of financial market price changes, which can be explained by the model derived from the research results in fractal geometry. The purpose of fractals (multifractals) is not to accurately predict the future, but they do give a more realistic description of market risks. Fractal is a geometric shape, which is characterized by being divided into several parts, and each part is a replica of the original whole on a smaller scale. In finance, this concept is not an unfounded abstraction, but a restatement of a simple and clear market common sense from a theoretical height.

Edgar E. Peters's research (1996) provides a lot of evidence for the fact that the securities market does have fractal and chaotic characteristics. It is believed that the stock price is not random, but influenced by a certain trend and highly sensitive to initial fluctuations. In other words, the stock price movement is chaotic. On this basis, Peters (1994) put forward the fractal market hypothesis, which holds that: (1) the market is composed of many investors with different investment expectations; (2) The information set related to each investment expectation is different. As long as the market maintains this fractal structure and has no characteristic time scale, the market will remain stable. When the investment expectations of the market become consistent, the market will become unstable, because everyone trades on the same information set (Peters, 1994).

References: stock and investment, finance and market research