AI 中文总结
该研究证明加性布朗噪声随机微分方程局部线性化格式离散误差的泛函稳定极限定理，明确误差结构与归一化因子，给出收敛条件及极限过程的驱动项与系数构成。
AI 中文摘要
我们证明了加性布朗噪声随机微分方程的局部线性化格式离散误差过程的泛函稳定极限定理。该格式纳入了漂移项泰勒展开中涉及布朗增量的二阶项的条件均值。其主导误差由布朗增量的中心化二次项构成，精确归一化因子为\textit{n√n}。在漂移项满足\textit{C}³正则性及李雅普诺夫型条件下，经尺度变换的误差过程在\textit{C}([0,1],ℝᵈ)中稳定收敛于极限线性随机微分方程的解。该极限的鞅部分由与原σ域独立的布朗运动驱动，其系数由漂移项的黑塞矩阵与加性噪声的协方差矩阵确定。
英文摘要
We prove a functional stable limit theorem for the discretization error process of a local linearization scheme for stochastic differential equations with additive Brownian noise. The scheme includes the conditional mean of the second-order term involving the Brownian increment in the Taylor expansion of the drift. The leading error is then formed by centered quadratic terms in the Brownian increments, and the sharp normalization is \(n\sqrt n\). Under \(C^3\)-regularity and a Lyapunov-type condition on the drift, the scaled error process converges stably in \(C([0,1],\mathbb R^d)\) to the solution of the limiting linear stochastic differential equation. The martingale part of the limit is driven by a Brownian motion independent of the original \(σ\)-field, and its coefficient is determined by the Hessian of the drift and the covariance matrix of the additive noise.
Comments28 pages
