AI 中文总结
研究有超线性增长系数的反射随机微分方程的强数值逼近，提出耦合驯服欧拉 - 皮亚诺格式，证明约束状态过程和边界调节器强收敛阶为1/2，数值实验验证了格式的约束保持性和理论收敛速率。
AI 中文摘要
我们研究了在可能无界的凸域中具有超线性增长漂移和扩散系数的反射随机微分方程的强数值逼近。在耦合单调性条件和多项式局部利普希茨假设下，我们首先确立了反射随机微分方程的适定性并推导其解的一致矩界。然后引入了一种耦合驯服欧拉 - 皮亚诺格式，其中漂移和平方扩散系数由一个公共因子驯服，所得欧拉 - 皮亚诺路径通过斯科罗霍德问题校正。这个公共驯服因子保持了漂移 - 扩散强制结构并给出数值解的一致矩估计。我们证明了约束状态过程和边界调节器的强二分之一阶收敛性，从而在这种反射设置中恢复了标准欧拉型强阶。针对反射随机金兹堡 - 朗道型系统的数值实验说明了该格式的约束保持性并支持理论收敛速率。
英文摘要
We study strong numerical approximations for reflected stochastic differential equations in possibly unbounded convex domains with super-linearly growing drift and diffusion coefficients. Under a coupled monotonicity condition and polynomial local Lipschitz assumptions, we first establish the well-posedness of the reflected SDE and derive uniform moment bounds for its solution. We then introduce a coupled tamed Euler--Peano scheme, in which the drift and the squared diffusion coefficient are tamed by a common factor and the resulting Euler--Peano path is corrected through the Skorokhod problem. This common taming factor preserves the drift--diffusion coercivity structure and yields uniform moment estimates for the numerical solution. We prove strong convergence of order $1/2$ for both the constrained state process and the boundary regulator, thereby recovering the standard Euler-type strong order in this reflected setting. Numerical experiments for a reflected stochastic Ginzburg--Landau type system illustrate the constraint preservation of the scheme and support the theoretical convergence rate.
Comments36 pages, 4 figures
