含时滞的McKean-Vlasov方程与相互作用粒子系统的Milstein格式及对偶多层蒙特卡洛采样

一种用于非交换噪声随机偏微分方程的对偶多层蒙特卡洛–Milstein格式

An Antithetic Multilevel Monte Carlo-Milstein Scheme for Stochastic Partial Differential Equations with non-commutative noise

本文将有限维对偶Milstein格式推广到Hilbert空间值SPDE，提出截断Milstein时间步进与对偶耦合结合的多层蒙特卡洛方法，在不引入额外偏差的前提下显著降低计算复杂度。

First order convergence of Milstein schemes for McKean-Vlasov equations and interacting particle systems

In this paper, we derive fully implementable first order time-stepping schemes for McKean--Vlasov stochastic differential equations (McKean--Vlasov SDEs), allowing for a drift term with super-linear growth in the state component. We propose Milstein schemes for a time-discretised interacting particle system associated with the McKean--Vlasov equation and prove strong convergence of order 1 and moment stability, taming the drift if only a one-sided Lipschitz condition holds. To derive our main results on strong convergence rates, we make use of calculus on the space of probability measures with…

弱抗性多级蒙特卡罗估计SDEs的Milstein方案用于低维维纳过程

Weak Antithetic MLMC Estimation of SDEs with the Milstein scheme for Low-Dimensional Wiener Processes

本文提出了一种弱Milstein方案用于模拟低维随机微分方程，结合抗性多级蒙特卡罗估计和弱SDE近似方法，实现了在估计期望值时具有二次计算复杂度的根均方误差的倒数，无需模拟莱维区域或要求底层SDE近似方法的强收敛性。

On Explicit Milstein-type Scheme for Mckean-Vlasov Stochastic Differential Equations with Super-linear Drift Coefficient

We develop an explicit Milstein-type scheme for McKean-Vlasov stochastic differential equations using the notion of derivative with respect to measure introduced by Lions and discussed in \cite{cardaliaguet2013}. The drift coefficient is allowed to grow super-linearly in the space variable. Further, both drift and diffusion coefficients are assumed to be only once differentiable in variables corresponding to space and measure. The rate of strong convergence is shown to be equal to $1.0$ without using Itô's formula for functions depending on measure. The challenges arising due to the dependence…

针对含共同噪声与不可微漂移系数的相互作用粒子系统及McKean--Vlasov随机微分方程的显式Milstein型格式

An explicit Milstein-type scheme for interacting particle systems and McKean--Vlasov SDEs with common noise and non-differentiable drift coefficients

针对含共同噪声与不可微漂移的McKean--Vlasov SDE及相互作用粒子系统，提出显式漂移随机化Milstein格式，在低正则假设下证得1阶强收敛，引入Spijker型范数处理粒子相互作用并讨论了格式复杂度。

Well-posedness and tamed schemes for McKean-Vlasov Equations with Common Noise

In this paper, we first establish well-posedness of McKean-Vlasov stochastic differential equations (McKean-Vlasov SDEs) with common noise, possibly with coefficients having super-linear growth in the state variable. Second, we present stable time-stepping schemes for this class of McKean-Vlasov SDEs. Specifically, we propose an explicit tamed Euler and tamed Milstein scheme for an interacting particle system associated with the McKean-Vlasov equation. We prove stability and strong convergence of order $1/2$ and $1$, respectively. To obtain our main results, we employ techniques from calculus …