Two Types Of Parameter Estimation And Empirical Analysis Based On Geometric Mixed Fractional Brownian Motion

This paper proposes two parameter estimation methods based on the geometric mixed fractional Brownian motion model.Geometric mixed fractional Brownian motion is a stochastic process model commonly used in fields such as simulating stock prices and managing financial risks.By modeling and analyzing this kind of stochastic process,market risks can be identified and controlled more effectively,and investment returns can be improved.This article introduces the basic concepts and properties of Brownian motion,fractional Brownian motion,mixed-fractional Brownian motion,and geometric mixed-fractional Brownian motion.By using the property that mixed-fractional Brownian motion is equivalent to a continuous semimartingale when the Hurst index is greater than three-quarters,this article attempts to construct a maximum likelihood estimator based on the observation increment and a "formal" maximum likelihood estimator for the drift coefficient and volatility coefficient in geometric mixed-fractional Brownian motion according to the principle of maximum likelihood estimation,and discusses the asymptotic behavior of the estimators.Next,the effectiveness of the two types of parameter estimation proposed in this paper is examined by Monte Carlo simulation,using numerical simulations with = 0.8.Overall,the result indicates that the estimated values are accurate.Finally,real stock data is selected for empirical analysis,and the proposed model and estimation method are used to estimate the drift and volatility coefficients.The simulated track map is compared with the actual one to verify the characteristics of the estimation proposed in this article.