AI 中文总结
本文研究耦合常微分方程与依赖历史的随机变分不等式的微分随机变分不等式，证明其相关性质，构造样本平均近似并推导迁移稳定性界，将其应用于老年健康监测系统实现高效迁移学习。
AI 中文摘要
本文提出并研究一类微分随机变分不等式（DSVI），其中常微分方程（ODE）与依赖历史的随机变分不等式（SVI）耦合。该框架对具有时变随机均衡的闭环随机系统进行建模，且包含带优化约束的ODE作为特例。在适当技术条件下，本文建立了第二阶段响应关于状态的唯一性、可测性和Lipschitz连续性，进而得到诱导状态轨迹的存在性与唯一性。此外，本文基于独立样本路径构造了样本平均近似（SAA），并证明了近似轨迹的一致收敛性。对于相关随机环境间的迁移，本文推导了具有移动可行集的参数变分不等式的局部1/2-Hölder估计，以及关于初始状态差和外生路径律Wasserstein距离的定量轨迹稳定性界。数值实验验证了SAA收敛性和迁移稳定性结果，进一步将该框架应用于老年健康监测系统：预计算响应的相似性加权复用实现了接近完全重计算基准的0.97准确率，同时将在线批处理运行时间从86秒缩短至1秒以内；扰动实验和延迟更新实验还刻画了对传感器噪声的鲁棒性，以及响应新鲜度、预测准确率与计算成本间的权衡。这些结果为依赖历史的DSVI系统中的高效迁移学习提供了理论和计算支撑。
英文摘要
In this paper, we propose and study a class of differential stochastic variational inequalities (DSVIs), in which an ordinary differential equation (ODE) is coupled with history-dependent stochastic variational inequalities (SVI). This framework models closed-loop stochastic systems with time-varying random equilibria and includes optimization-constrained ODEs as special cases. Under appropriate technical conditions, we establish uniqueness, measurability, and Lipschitz continuity with respect to the state of the second-stage response, and consequently the existence and uniqueness of the induced state trajectory. Moreover, we construct a sample average approximation (SAA) based on independent sample paths and prove uniform convergence of the approximate trajectories. For transfer between related stochastic environments, we derive a local $1/2$-Hölder estimate for parametric variational inequalities with moving feasible sets and a quantitative trajectory-stability bound in terms of the initial-state difference and the Wasserstein distance between exogenous path laws. Numerical experiments illustrate the SAA convergence and transfer-stability results. We further apply the framework to an elderly-health monitoring system. Similarity-weighted reuse of precomputed responses achieves an accuracy close to the full-recomputation benchmark of 0.97, while reducing the online batch runtime from 86 seconds to less than one second. Perturbation and delayed-update experiments additionally characterize robustness to sensor noise and the trade-off between response freshness, predictive accuracy, and computational cost. These results provide theoretical and computational support for efficient transfer learning in history-dependent DSVI systems.
Comments38pages,8 figures
