Parameter Estimation And Application Of Mixed Sub-fractional Geometric Brownian Motion

Brownian motion is a continuous stochastic process with independent increment and normal distribution.It is an important model in financial mathematics.At first,Brownian motion model was often used to describe stock price fluctuation.Later,it was proved that stock price has self-similarity and long-distance dependence,and does not completely obey geometric Brownian motion.Fractional Brownian motion can be obtained by improving Brownian motion,which is the most widely used one.However,fractional Brownian motion does not satisfy the non-stationary increment of stock price returns,so it is modified to obtain a more general type of sub-fractional Brownian motion.The main purpose of this paper is to establish a mixed sub-fractional geometric Brownian motion model to better describe the price change of financial market.In this paper,the model of mixed sub-fractional geometric Brownian motion and its parameter estimation are studied.Firstly,the model is established,and the parameters in the model are estimated.Multiple scalar range method,variation method and maximum likelihood estimation method are selected to estimate Hurst index and unknown parameters respectively;Secondly,the asymptotic property of parameter estimation is proved;Then,the model is used to simulate the sample data.By changing one of the parameters and keeping other parameters unchanged,the influence of Hurst index and parameters on the model is verified;Then,Monte Carlo simulation method is used in MATLAB to verify the accuracy of the estimation method.Set the parameter values,simulate the sample data for many times to estimate the parameter values,compare the set values and the average value of the estimated values,the results prove the effectiveness of the estimation method;Finally,this paper makes an empirical study on the closing price of 601689.sh in Shanghai and Shenzhen stock markets.According to the selected time series,the corresponding model is established,and the unknown parameters are estimated.The fitting trajectory is compared with the actual trajectory,and the results show that the maximum likelihood estimation method is effective for the mixed sub-fractional geometric Brownian motion model.