AI 中文总结
本文证明了在非正则时间网格上观测的Itô半鞅的非规范化泛函的固定时间稳定收敛定理，给出了极限的跳跃贡献形式，并指出仅靠正性和右连左极正则性不足以替代对倒数采样强度的局部控制。
AI 中文摘要
设一个Itô半鞅在满足$\ au_{i+1}^n-\ au_i^n=(n\ heta_{\ au_i^n})^{-1}$的时间点上被观测，其中$\ heta$是适应的且右连左极的，并且$\ heta$及其倒数在紧区间上有界。在关于半鞅的通常假设(H)下，我们证明了其Hessian在零点处为$o(\ orm{x})$的检验函数的固定时间稳定收敛。极限是跳跃贡献之和，具有采样尺度$\ heta_{T-}^{-1}$以及跳跃前和跳跃后的不同波动率。证明建立了内生网格上局部泊松位置和布朗增量的联合稳定收敛，并通过一致的Itô估计控制了无限多个小跳跃。三个例子确定了左极限采样尺度，并说明了仅靠正性和右连左极正则性不能替代对倒数采样强度的局部控制。
英文摘要
Let an Itô semimartingale be observed at times satisfying $τ_{i+1}^n-τ_i^n=(nθ_{τ_i^n})^{-1}$, where $θ$ is adapted and càdlàg, and $θ$ and its reciprocal are bounded on compact intervals. Under the usual hypothesis (H) on the semimartingale, we prove fixed-time stable convergence for test functions whose Hessian is $o(\norm{x})$ at zero. The limit is a sum of jump contributions with sampling scale $θ_{T-}^{-1}$ and separate pre-jump and post-jump volatilities. The proof establishes joint stable convergence of the local Poisson positions and Brownian increments on the endogenous grid, and controls infinitely many small jumps by a uniform Itô estimate. Three examples identify the left-limit sampling scale and show why positivity and càdlàg regularity alone cannot replace local control of the reciprocal sampling intensity.
