2026/08/05 by Guangqian Zhao
Mathematics · #math.PR
75 pages
arxiv created 2026/08/06 · arxiv updated 2026/08/07
Let \mathfrak MΛ0,T be the nondominated family of continuous local-martingale laws on C0([0,T];\mathbb Rd) satisfying \mathrm d[X]Pt\preceqΛId \mathrm dt. We construct a single Borel raw-causal second-order state \mathcal A2=(X,\widehat J,Q) whose realization under every law is the Itô primitive together with quadratic variation; the Itô and Stratonovich step-two lifts are algebraic readouts. The causal exponential resolvent ε Yε=X-Yε=:Dε yields both the finite-scale approximation and the memory required for exact restart. Uniform estimates give sharp O(q√ε) maximal-Lq rates for the tensor and bracket approximations, and rough-path convergence of the smooth signatures at every rate εϑ with ϑ<1/2-α, 1/3<α<1/2. They also produce compact capacity cores with Gaussian tails that, after quadratic reindexing, are stable under stopping, future shifts, and concatenation. Corewise continuity gives quantitative transfer through conditional upper expectations. Analytic conditioning and pasting yield a stopping-time dynamic programming principle, and the maximal bounded-volatility operators extend to regular upper-Lp spaces. Finally, raw Euler schemes complete to one jointly Borel causal cocycle and, outside a single polar set independent of all parameters, agree with the bracket-corrected rough Itô flow generated by \mathcal A2.