| This dissertation considers the existence,uniqueness of strong solutions to meanfield stochastic delay differential equations and the error analysis of their numerical schemes.The characteristic of mean-field stochastic delay differential equations is that the coefficients are related to the present state variables,delay state variables,and distribution information of state variables.The studying of numerical schemes for such equations mainly involves two steps:the first step is constructing the interacting particle system associated with the original equation by using stochastic particle method.And as the number of particles tends to infinity,the interacting particle system can converge to the original equation;the second step is constructing suitable numerical scheme for this interacting particle system to achieve the numerical approximation.Combining these two steps yields the convergence error between the mean-field stochastic delay differential equation and the numerical scheme,which depends on both the number of particles and the step size.This dissertation mainly consists of the following four chapters:Chapter 1 introduces the research background,notation explanation,fundamental knowledge,and commonly used inequalities.Chapter 2 discusses the existence,uniqueness of strong solutions to a class of meanfield stochastic delay differential equations with Holder continuous diffusion coefficients and the error analysis of their numerical schemes.When the drift coefficient satisfies a global Lipschitz condition,the existence and uniqueness of strong solution to the equation are proved by using the distribution iteration technique.Subsequently,the propagation of chaos between interacting particle system and non-interacting particle system is shown.Then,the numerical approximation is performed by using the classical Euler-Maruyama scheme,and strong convergence rates in L1 and L2 senses are obtained.Furthermore,when the drift coefficient satisfies a polynomial growth condition with respect to the present state variable,the existence and uniqueness of strong solution to the equation and the corresponding propagation of chaos are analyzed.The tamed Euler-Maruyama scheme is adopted for numerical approximation,and strong convergence rates in L1 and L2 senses are obtained.The Yamada-Watanabe approximation technique plays an important role in the research process.Finally,the numerical simulations are conducted for the examples with linear drift coefficient and superlinear drift coefficient.Chapter 3 analyzes the existence,uniqueness of strong solutions to a class of neutral mean-field stochastic delay differential equations and the error analysis of their numerical schemes.When the neutral term satisfies a polynomial growth condition,the existence and uniqueness of strong solution to the equation are proved by using the distribution iteration technique.Then the propagation of chaos in Lp(p≥2)sense is derived,where the convergence rate between the interacting particle system and non-interacting particle system is slightly lost.Subsequently,the tamed Euler-Maruyama scheme is constructed for the interacting particle system,and strong convergence rate in Lp(p≥2)sense is demonstrated,which reveals the convergence error between neutral mean-field stochastic delay differential equation and tamed Euler-Maruyama scheme in Lp(p≥2)sense.Then the convergence error is provided through simple proof when the neutral term satisfies a contraction condition.Finally,the numerical simulations are conducted for one-dimensional and two-dimensional examples.Chapter 4 considers the existence,uniqueness of strong solutions to a class of meanfield stochastic delay differential equations driven by fractional Brownian motion and the error analysis of their numerical schemes,where the Hurst parameter of fractional Brownian motion is H∈(0,1/2)∪(1/2,1).First,the existence and uniqueness of strong solution to the equation are proved by using the Banach fixed-point theorem.Next,the propagation of chaos in Lp(p≥2)sense between interacting particle system and noninteracting particle system is analyzed.Then the classical Euler-Maruyama scheme is constructed for the interacting particle system,and strong convergence rate is provided for the case of H ∈(0,1/2)∪(1/2,1).It is worth emphasizing that although the delay state variable in the equation are superlinear,the classical Euler-Maruyama scheme can still be used for the numerical approximation without the particle corruption.Finally,the numerical simulations are conducted for a class of stochastic opinion dynamics models which can reflect the external memory and internal memory influences. |
