An Approximate Solution Of The Fokker-Planck Equation In The Stock Market Dynamics

Since the establishment of statistical physics in nineteenth century,probability and statistics methods have been used to to describe and analyze the macroscopic physical phenomena and physical system. By the bridge of statistical mechanics connecting macro and micro,it moved a key step towards the development of physical science.With the deepening of research,random force has been introduced in the statistical system.Non equilibrium statistical physics problems gradually has been widely explored, and then stochastic process and the nonlinear equation method is more and more application in statistical physics.By the end of the twentieth century, quantitative statistical methods is applied to from the humanities to a wide range of subjects,such as science, industry and trade.Individual in the society had also been regarded as statistical individual element, to construct the model of social, cultural, linguistic and economic. This type of model system increased along with produce a new subject:complex systems.Some physicists were trying to use statistical physics models to study the specific economic and financial system and achieved some good results.Econophysics came into being, and became an active branch of physics.Random walk with Brown motion model is used to describe a lot of problems in econophysics. Brown motion theory always describe the crowd random motion particle theory, and random walk model is a stochastic process with similar be-havior to Brownian motion in Wiener process.From the aspect of dynamics,Brown motion can be described by Langevin equation in mathematics.However, in many-body system, Langevin equation is too complex to solve.At this time, in order to get a good result,The Fokker-Planck equation should be constructed with stochas-tic process method by calculated the drift and diffusion coefficients from Langevin equation.Then the distribution of Langevin equation’s solution can be get to study movement of the system by the relation of the two equations.In order to establish a dynamic system similar to the Brown motion in the stock market, we must take effective market hypothesis at first, so that the stock market can satisfy the Markov process.By price fluctuations, the log return of stock and in-stantaneous gain can be get.To construct a suitable transient force and random force function by instantaneous gain,a Langevin equation can be wrote down to describe the behavior of the stock market under a certain condition.Then calculate the drift and diffusion coefficients, the corresponding Fokker Planck equation is obtained to fit the stock system dynamics.The Fokker Planck equation in the stock market system is easy to solve than the Langevin equation,but it also very hard.So we consider the stationary solution at first to find the distribution of Langevin equation’s stationary solution.Then try to find the perturbation solutions of the Fokker Planck equation.By improving the degree of approximation,the solutions will be more close to the actual distribution in the stock market.