We present a comprehensive discretization scheme for linear and nonlinear stochastic differential equations (SDEs) driven by either Brownian motions or $\alpha$-stable processes. Our approach utilizes compound Poisson particle approximations, allowing for simultaneous discretization of both the time and space variables in McKean-Vlasov SDEs. Notably, the approximation processes can be represented as a Markov chain with values on a lattice. Importantly, we demonstrate the propagation of chaos under relatively mild assumptions on the coefficients, including those with polynomial growth. This result establishes the convergence of the particle approximations towards the true solutions of the McKean-Vlasov SDEs. By only imposing moment conditions on the intensity measure of compound Poisson processes, our approximation exhibits universality. In the case of ordinary differential equations (ODEs), we investigate scenarios where the drift term satisfies the one-sided Lipschitz assumption. We prove the optimal convergence rate for Filippov solutions in this setting. Additionally, we establish a functional central limit theorem (CLT) for the approximation of ODEs and show the convergence of invariant measures for linear SDEs. As a practical application, we construct a compound Poisson approximation for 2D-Navier Stokes equations on the torus and demonstrate the optimal convergence rate.

翻译：我们提出了一种针对由布朗运动或$\alpha$-稳定过程驱动的线性和非线性随机微分方程（SDEs）的全面离散化格式。该方法采用复合泊松粒子近似，可同时对McKean-Vlasov随机微分方程中的时间和空间变量进行离散化。值得注意的是，该近似过程可表示为取值于格点上的马尔可夫链。重要的是，我们在系数满足相对温和假设（包括多项式增长条件）的情况下证明了混沌传播性质。该结果确立了粒子近似收敛到McKean-Vlasov随机微分方程真实解。仅通过对复合泊松过程强度矩施加矩条件，我们的近似方法即具有普适性。在常微分方程（ODEs）情形下，我们研究了漂移项满足单侧Lipschitz假设的场景，并证明了该条件下Filippov解的最优收敛速率。此外，我们还建立了ODEs近似的泛函中心极限定理（CLT），并展示了线性SDEs不变测度的收敛性。作为实际应用，我们构建了环面上二维Navier-Stokes方程的复合泊松近似，并证明了最优收敛速率。