Analysing the chinese stock market using the hurst exponent, fractional brownian motion and variants of a stochastic logistic differential equation

Abstract

The Chinese stock market is rapidly developing and is becoming one of the wealth management investment centres. Recent legislation has allowed wealth management products to be invested in the Chinese stock market. By taking data from St. Louis Fed and analysing the Chinese stock market using the Hurst exponent, which was calculated by using two methods, and fractional Brownian motion, it is proved that the Chinese stock market is not efficient. However, further analysis was directed to finding its equilibrium state by using logistic difference and a differential equation. To achieve more precise movement, a stochastic logistic and delayed logistic differential equation have been implemented which are driven by fractional Brownian motion. As there is no explicit solution to a delayed differential equation using the Stratonovich integral, a method of steps has been used. A solution has been obtained that gives the equilibrium state of the Chinese stock market. The following solution is only for a Hurst exponent that is higher than 1/2. If Hurst exponent is close to 1/2, then Brownian motion is an ordinary one and the equilibrium solution is different. Sensitivity analysis in that case has been conducted in order to analyse possible stationary states. The main conclusion is that the Chinese stock market is still not mature enough to be efficient, which represents a hidden risk in wealth management investing. A new approach to valuing the efficiency of the market by using nonlinear dynamics and system analysis has been proposed.