Common Causal It\^o Flows under Nondominated Martingale Laws: Capacity Cores and Robust Sensitivities

Abstract

Let \(\Omega=C_0([0,T];\mathbb R)\), and let \(\mathfrak M_\Lambda\) be the laws under which the coordinate process is a continuous square-integrable martingale satisfying \(\mathrm d\langle X\rangle_t\leq \Lambda\,\mathrm d t\). Regularized dyadic square sums define a single Borel causal map \(Q:\Omega\to\mathcal A_\Lambda\) which equals the quadratic variation almost surely under every law in \(\mathfrak M_\Lambda\). The approximants converge uniformly in \(L^r\) at rate \(2^{-n/2}\) and uniformly on common compact sets whose complementary upper capacities tend to zero. For uniformly elliptic scalar coefficients, the Lamperti transform applied to \(Q\) produces a total Borel causal \(C^1\) flow with the cocycle property. Every fixed section coincides almost surely with the classical strong It\^o solution under every model. The flow and its initial-state Jacobian are continuous on the same capacity cores; for differentiable finite-dimensional coefficient families, so is the parameter tangent. A capacity-core \(C^1\) transfer theorem yields class-uniform \(L^p\) Fr\'echet expansions, \(W_p\)-continuity of the first-jet laws, and attainment for continuous field payoffs of polynomial growth. For robust terminal payoffs, the transferred first jet gives a joint Hadamard--Danskin formula in the coefficient parameter and initial state. An explicit constant-volatility family exhibits an active-model switch and a nondifferentiable robust value on an explicit switching surface. The bracket is also invariant under continuous finite-variation translations. The induced response kernel, multiplied by the deterministic volatility, agrees with the Malliavin derivative under Gaussian volatility models.