Several Explicit Numerical Methods For Mean-field Stochastic Differential Equations Driven By Fractional Brownian Motio

This dissertation mainly studies explicit numerical algorithms for mean-field stochastic differential equations driven by fractional Brownian motion and their convergence properties.The numerical approximation of mean-field stochastic differential equations is generally divided into two steps: the first step is to approximate the original equation using the interacting particle system associated with the mean-field stochastic differential equation;the second step is to construct a discrete numerical scheme to approximate the interacting particle system.For mean-field stochastic differential equations driven by fractional Brownian motion,we first prove the propagation of chaos theory of the interacting particle system driven by fractional Brownian motion approximating the original equation.Then,we derive the convergence order of the corresponding numerical scheme based on the properties of fractional Brownian motion.On the other hand,according to the Hurst index,fractional Brownian motion can be divided into three cases: 0 < H < 1/2(the increments of fractional Brownian motion on non-overlapping intervals are negatively correlated),H = 1/2(the increments of fractional Brownian motion on non-overlapping intervals are uncorrelated),and 1/2 < H < 1(the increments of fractional Brownian motion on non-overlapping intervals are positively correlated).When H = 1/2,the increments of fractional Brownian motion on non-overlapping intervals are independent.This dissertation also studies the impact of this correlation on the self-organized dynamics of particle interactions through numerical examples.The main features of this dissertation are as follows:1)When the Hurst index is not equal to 1/2,we propose an explicit numerical method to solve mean-field stochastic differential equations driven by fractional Brownian motion.We give propagation of chaos theories and theoretical analyses for positively and negatively correlated cases,respectively,effectively ensuring the integrity of the convergence analysis of the numerical algorithm.2)When the Hurst index is equal to 1/2,in order to reduce the complexity of the operation of the interacting particle system associated with the mean-field stochastic differential equation,we further develop the random batch method.We prove the boundedness and convergence of the numerical solution for the case where the coefficient of the mean-field stochastic differential equation is linearly or nonlinearly dependent on the distribution,and obtain the relationship between the capacity and convergence order of each partition sample.3)For mean-field stochastic differential equations driven by fractional Brownian motion with a Hurst index of 1/2,and the drift coefficient and diffusion coefficient term of the equation are highly nonlinear with respect to the state variable,we propose the truncated Euler-Maruyama method and adaptive step size method,respectively.This overcomes the problem of classical explicit scheme not converging,and provides theoretical proofs of boundedness and convergence.4)For a class of interacting particle systems with self-organized dynamics,we use explicit numerical methods to explore how their dynamic behaviors change with the size of the Hurst index.Numerical results show that in the positively correlated case,increasing the Hurst index will more significantly break the aggregation behavior of the particle system;however,in the negatively correlated case,decreasing the Hurst index will more help the particle system enter the aggregation state earlier.