Numerical Solutions And Application Of A Class Of Langevin Equation In Fractional Order Statistics Dynamics

Langevin equation is an important model in fractional order statistics dynamics,and the research on the numerical solution of fractional Langevin equation,a typical fractional stochastic differential equation,has become one of the main topics of abnormal statistical dynamics.This article extends the classical Langevin equation in fractional order,numerically solves a class of fractional Langevin equation,and gives its application in financial market time memory.Concerning a class of fractional Langevin equation,two different methods are used to construct numerical solutions:convolution algorithm and prediction correction algorithm.The convolution algorithm is used to discretize the fractional part,and the numerical solution of the equation is obtained,and its convergence and stability analysis are given.A numerical solution is obtained by using an estimation-correction algorithm,and an estimation value is obtained by using the R0 algorithm,and then the estimation value is substituted into the R1 algorithm to correct the numerical solution,and finally a numerical solution of the estimation correction algorithm is obtained.Error analysis proves that under the condition of 0<?<1,the estimation correction algorithm is(1+?)order convergent.Numerical experiments compare the numerical and true solutions of the two algorithms.Furthermore,the two numerical methods are compared and analyzed in terms of operation time and calculation error.The time-memory application of fractional Langevin equation in financial markets is analyzed,and the autocorrelation function and second-order moment of the fractional Langevin equation are derived,and the expression of standard deviation is solved.The numerical experiments of the convolution algorithm and the estimation and correction algorithm have confirmed that both types of numerical solutions are stable and efficient under different ?,h values.Under the same conditions,the calculation time of the numerical solution of the convolution algorithm is shorter than that of the estimated correction algorithm,but the accuracy of the estimated correction algorithm is better than that of the convolution algorithm.Numerical experiments have also found that,for the convolution algorithm specifically,the loss of accuracy reduces the marginal benefit of reduced operation time.Temporal memory analysis proves that the standard deviation of the fractional Langevin equation is the leading term for early time analysis of measuring market activity.