AI 中文总结
本文研究2<p<3时具有确定性粗糙驱动项的Rough-BSDE解映射的一阶敏感性，构造了内在切向阿达马可微的连续线性映射，明确了其与弗雷歇可微性的区别。
AI 中文摘要
我们研究具有确定性粗糙驱动项的标量倒向随机微分方程的一阶敏感性。该驱动项属于二阶弱几何p-粗糙路径的非线性空间，其中2<p<3，因此普通巴拿赫空间差商不可用。在固定粗糙路径x处，我们使用Qian-Tudor切结构的弱几何有限-(p,p/2)-变差张量实现，其由一级方向h和兼容的二级方向κ内在表示。容许粗糙路径割线需在强层级变差拓扑中收敛。在有界光滑粗糙向量场、有界终端条件及全局利普希茨生成元的条件下，我们构造连续线性映射Aₓ:Tₓᵖ→S^∞×H²_{BMO}。证明首先建立沿全粗糙切的有界实现的重置粗糙流的一致四阶喷管展开；通过Doss-Sussmann变换将该展开转移至二次生成元；一致BMO和反向赫尔德估计随后给出局部差商定理，该定理在固定确定性划分上传播并在原坐标中重构。所得导数与联合提升、中心分解及径向实现无关，因此对于变化方向及具有强层级变差接触的任意容许割线，解差商收敛至Aₓ(h,κ)，这是该张量坐标切类上的内在切向阿达马可微性。局部有界射线齐次选择在参数原点处给出弗雷歇可微的图表拉回，该结论是逐图表的，并非解映射在齐次粗糙路径度量下的弗雷歇可微性。
英文摘要
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 $κ$. 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 $A_x(h,κ)$ 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.
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