Jie He, Shuaibin Gao, Weijun Zhan, Qian Guo
In this paper, we establish the theory of chaos propagation and propose an Euler-Maruyama scheme for McKean-Vlasov stochastic differential equations driven by fractional Brownian motion with Hurst exponent $H \in (0,1)$. Meanwhile, upper bounds for errors in the Euler method is obtained. A numerical example is demonstrated to verify the theoretical results.
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In this paper, we present a generic methodology for the efficient numerical approximation of the density function of the McKean-Vlasov SDEs. The weak error analysis for the projected process motivates us to combine the iterative Multilevel Monte Carlo method for McKean-Vlasov SDEs \cite{szpruch2019} with non-interacting kernels and projection estimation of particle densities \cite{belomestny2018projected}. By exploiting smoothness of the coefficients for McKean-Vlasov SDEs, in the best case scenario (i.e $C^{\infty}$ for the coefficients), we obtain the complexity of order $O(ε^{-2}|\logε|^4)$…
In the literatur there exist approximation methods for McKean-Vlasov stochastic differential equations which have a computational effort of order $3$. In this article we introduce full-history recursive multilevel Picard (MLP) approximations for McKean-Vlasov stochastic differential equations. We prove that these MLP approximations have computational effort of order $2+$ which is essentially optimal in high dimensions.
Convergence rate of Euler-Maruyama scheme for McKean-Vlasov SDEs with density-dependent drift
针对漂移项依赖密度、扩散项为常数的McKean-Vlasov随机微分方程，本文借助Euler-Maruyama格式的Hölder稳定性估计与对应Fokker-Planck方程的性质证明其弱适定性，并得到该格式在加权L¹范数下的收敛速率。
Computing the invariant distribution of McKean-Vlasov SDEs by ergodic simulation
针对满足一致汇合性质的遍历McKean-Vlasov随机微分方程，设计可完全实现的方案计算其不变分布，并在自然条件下证明多种收敛结果，包括Wasserstein距离在均方和几乎必然意义下的收敛速率。
In this paper, we introduce adaptive Euler-Maruyama schemes for McKean-Vlasov stochastic differential equations (SDEs) assuming only a standard monotonicity condition on the drift and diffusion coefficients but no global Lipschitz continuity in the state variable for either, while global Lipschitz continuity is required for the measure component only. We prove moment stability of the discretised processes and a strong convergence rate of $1/2$. Several numerical examples, centred around a mean-field model for FitzHugh-Nagumo neurons, illustrate that the standard uniform scheme fails and that t…
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…
On the infinite time horizon approximation for Lévy-driven McKean-Vlasov SDEs with non-globally Lipschitz continuous and super-linearly growth drift and diffusion coefficients
本文研究Lévy过程驱动的McKean-Vlasov随机微分方程的数值逼近，提出了一种驯服自适应Euler-Maruyama格式，并证明了其在非全局Lipschitz连续且超线性增长系数下于有限和无限时间跨度的强收敛性。
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 …
Milstein schemes and antithetic multilevel Monte Carlo sampling for delay McKean-Vlasov equations and interacting particle systems
针对带点时滞、漂移项可能超线性增长的McKean-Vlasov方程对应的相互作用粒子系统，推导了Milstein格式并证明其一阶强收敛与矩稳定，还提出无需模拟Lévy面积的对偶多层Milstein格式以实现解的期望泛函的最优复杂度估计。
The adaptive EM schemes for McKean-Vlasov SDEs with common noise in finite and infinite horizons
针对带公共噪声且系数满足超线性增长条件的 McKean-Vlasov SDE，提出自适应 Euler-Maruyama 格式，建立有限与无限时域下依赖粒子数和步长的 L^p 收敛速率，并以数值算例验证理论。
Multilevel Picard approximations for McKean-Vlasov stochastic differential equations with jumps
该研究针对带跳的McKean-Vlasov SDE提出MLP近似方法，证明其无维数灾难，计算成本呈多项式增长，且在维数达200的数值实验中验证了实用性。
Rectified deep neural networks overcome the curse of dimensionality when approximating solutions of McKean--Vlasov stochastic differential equations
该研究证明校正深度神经网络逼近 McKean–Vlasov SDE 解时参数数量仅随维数与精度倒数多项式增长，从而克服维数灾难。
Multilevel Picard approximations for McKean-Vlasov stochastic differential equations with nonconstant diffusion
本文提出了一种多级皮卡近似法用于求解具有非常数扩散系数的麦肯-瓦尔萨夫随机微分方程，在标准Lipschitz条件下，证明该方法在L²意义下近似解，且计算成本随维度和误差容忍度的倒数呈多项式增长。
An Euler scheme for McKean SDEs with Besov drift: convergence rate and implementation
本文提出了一种适用于具有Besov漂移的麦肯SDE的欧拉方案，通过平滑分布漂移与PDE近似，证明了强收敛性并推导了收敛速率。
Particle Method for the McKean-Vlasov equation with common noise
本文研究带共同噪声的McKean-Vlasov方程的数值模拟，通过Euler格式和粒子方法进行时空离散，在系数满足特定连续性条件下证明了收敛速度，并推广了无共同噪声情形的结论。
Long time strong convergence analysis of one-step methods for McKean-Vlasov SDEs with superlinear growth coefficients
本文分析超线性增长McKean-Vlasov SDE单步法在无限时间上的强收敛速率，推导混沌传播与均方收敛率，并应用于投影Euler和后退Euler格式，数值实验验证理论。
Uniform-in-time error estimates for McKean-Vlasov SDEs with common noise and stochastic algorithms
通过构造反射渐近耦合，对带共同噪声的McKean-Vlasov SDE与相互作用粒子系统诱导的测度值过程，在漂移项仅长程耗散下得到概率距离的均匀时间估计，并应用于数值方法的误差估计和条件混沌传播的量化。
We present an implicit Split-Step explicit Euler type Method (dubbed SSM) for the simulation of McKean-Vlasov Stochastic Differential Equations (MV-SDEs) with drifts of superlinear growth in space, Lipschitz in measure and non-constant Lipschitz diffusion coefficient. The scheme is designed to leverage the structure induced by the interacting particle approximation system, including parallel implementation and the solvability of the implicit equation. The scheme attains the classical $1/2$ root mean square error (rMSE) convergence rate in stepsize and closes the gap left by [18, "Simulation of…
Optimal stopping of McKean-Vlasov diffusions via regression on particle systems
本文通过粒子系统回归算法求解非线性马尔可夫过程的最优停止问题，证明了算法收敛性，并通过数值例子展示其性能。
Convergence rate of nonlinear delayed neutral McKean-Vlasov SDEs driven by fractional Brownian motions
本文研究了由分数布朗运动驱动的超线性延迟中立型McKean-Vlasov随机微分方程的强收敛性，证明了解的唯一存在性、矩有界性与混沌传播，并给出了Euler-Maruyama格式的强收敛结果及数值验证。
On modified Euler methods for McKean-Vlasov stochastic differential equations with super-linear coefficients
本文针对具有超线性系数的McKean-Vlasov随机微分方程提出一类修正Euler数值方法，在非全局Lipschitz条件下证明其半阶强收敛性，并借助混沌传播结果建立完整收敛速率，通过数值实验验证理论结论。
Convergence and stability of truncated Euler-Maruyama algorithm for stochastic proportional delay Mckean-Vlasov models with jump process
研究具有跳跃过程的随机比例延迟Mckean-Vlasov模型，应用截断Euler-Maruyama算法，探讨其收敛速度和几乎必然指数稳定性，并通过数值例子支持理论分析。
Well-posedness and approximation of reflected McKean-Vlasov SDEs with applications
本文研究了反射麦肯-沃拉斯随机微分方程的适定性及其粒子近似，并应用于受限域中的采样和优化问题。
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