Intrinsic Tangential Hadamard Differentiability of Rough-BSDE Solution Maps

Abstract

We study first-order sensitivity of a scalar backward stochastic differential equation with a deterministic rough driver. The driver belongs to the nonlinear space of step-two weakly geometric $p$-rough paths, $2<p<3$, so an ordinary Banach-space difference quotient is not available. At a fixed rough path $x$, we use a weakly geometric, finite-$(p,p/2)$-variation tensor realization of the Qian-Tudor tangent structure, represented intrinsically by a first-level direction $h$ and a compatible second-level direction $\kappa$. Admissible rough-path secants are required to converge in a strong levelwise variation topology. Under bounded smooth rough vector fields, a bounded terminal condition, and a globally Lipschitz generator, we construct a continuous linear map $A_x:T_x^p\to S^\infty\times H^2_{\mathrm{BMO}}$. The proof first establishes a uniform four-jet expansion for reset rough flows along bounded realizations of full rough tangents. A Doss-Sussmann transformation transfers this expansion to quadratic generators. Uniform BMO and reverse-Holder estimates then yield a local difference-quotient theorem, which is propagated over a fixed deterministic partition and reconstructed in the original coordinates. The resulting derivative is independent of the joint lift, central decomposition, and radial realization. Consequently, solution difference quotients converge to $A_x(h,\kappa)$ for varying directions and arbitrary admissible secants with strong levelwise variation contact. This is intrinsic tangential Hadamard differentiability on that tensor-coordinate tangent class. Locally bounded ray-homogeneous selections give Frechet-differentiable chart pullbacks at the parameter origin. The latter statement is chartwise; it is not Frechet differentiability of the solution map in the homogeneous rough-path metric.