AI 中文总结
该研究针对非支配鞅律构造因果预解粗糙提升，证明相关张量、括号及流的收敛性，建立紧路径集的稳定性，给出Wong–Zakai转移等结果，为相关随机分析问题提供理论支撑。
AI 中文摘要
设$\boldsymbol{\frak{M}}_{\boldsymbol{\frak{\u039b}}}$为$C_0([0,T];\boldsymbol{\frak{R}}^d)$上满足$\boldsymbol{\frak{d}}[X]^P_t\boldsymbol{\frak{\u2270}} \boldsymbol{\frak{\u039b}} I_d \boldsymbol{\frak{d}}t$的连续局部鞅律类。从因果预解$\boldsymbol{\frak{\u03b5}}\boldsymbol{\frak{\u1e26}} Y^\boldsymbol{\frak{\u03b5}}=X-Y^\boldsymbol{\frak{\u03b5}}=:D^\boldsymbol{\frak{\u03b5}}$出发，构造单个Borel因果Itô张量与二次变差选择子。有限尺度括号满足路径恒等式$Q_t^\boldsymbol{\frak{\u03b5}}=(D_t^\boldsymbol{\frak{\u03b5}})^{\boldsymbol{\frak{\u2297}}2}+\frac{2}{\boldsymbol{\frak{\u03b5}}}\boldsymbol{\frak{\u222b}}_0^t (D_r^\boldsymbol{\frak{\u03b5}})^{\boldsymbol{\frak{\u2297}}2}\boldsymbol{\frak{d}}r$，且张量与括号逼近在$\boldsymbol{\frak{M}}_{\boldsymbol{\frak{\u039b}}} $上以$L^q$阶$O(q\boldsymbol{\frak{\u221a{\frak{\u03b5}}})$一致收敛。对$1/3<\boldsymbol{\frak{\u03b1}}<1/2$，$Y^\boldsymbol{\frak{\u03b5}}$的典范特征在$\boldsymbol{\frak{\u03b1}}$-Hölder粗糙路径拓扑下收敛到公共Stratonovich提升，收敛率为$O(q\boldsymbol{\frak{\u03b5}}^\boldsymbol{\frak{\u03b8}})$（对任意$\boldsymbol{\frak{\u03b8}}<1/2-\boldsymbol{\frak{\u03b1}}$）。构造递增紧路径集族，其互补上容量具高斯尾；经显式重索引后，这些集合对停时、时移、时间拼接稳定。在每个紧集上，公共提升与括号连续，经典预解驱动方程一致收敛到公共Stratonovich与Itô流。该紧结构对有界波动率对应给出停时条件Wong–Zakai转移；对最大无历史依赖对应，还给出Bellman连续性、正则上-$L^p$条件算子、有限网格非线性Snell包络及首次接触策略的定量稳定性。最后，上-$L^p$欧拉构造生成全Borel因果Itô流场，几乎必然与粗糙流一致。
英文摘要
We study the least stable state determined by a causal response program and its quantitative calibration. For nondominated families of continuous Hilbert-valued semimartingale laws with uniform drift and trace-clock bounds, the calibrated quadratic Itô program selects a state $\mathcal A_2$ with canonical coordinates $(X,A,Q)$, where $A$ is Hilbert--Schmidt skew area and $Q$ is trace-class covariance. Brownian experiments identify the Hilbert--Schmidt/operator coefficient gauges and hence the dual $\mathcal S_2/\mathcal S_1$ state geometry. A lossless encoder--decoder pair shows that every joint realization factors through $\mathcal A_2$. A single sequence of total Borel causal approximants constructs the common state, while the laws verify its classical semantics. A dimension-free defect--energy estimate gives independence from the finite-variation regularization, and spatial trace tightness yields restart-stable compact capacity cores in infinite dimensions. Each fixed higher signature level is a continuous readout of $\mathcal A_2$. Rough flows inherit corewise continuity, while Euler schemes construct a jointly Borel raw-causal Itô field outside one parameter-independent polar set. On compact covariance-envelope subclasses, covariance responses select a backward first jet which, under the stated completion and continuation hypotheses, has modelwise Galtchouk--Kunita--Watanabe semantics and agrees with the solver's martingale coordinate.
Comments96 pages
