Universal limit theorem for rough differential equations driven by controlled rough paths

Abstract

We study rough differential equations driven by controlled rough paths in the level-$2$ regime $1/3<\alpha\le 1/2$. Given a reference rough path $\mathbf X=(1,X,\mathbb X)$ and an $\mathbf X$-controlled driver $\mathbf Z=(Z,Z')$, we first give a point-removal construction of the controlled rough integral $ \int_s^t Y_r\,d\mathbf Z_r $ and prove the corresponding remainder estimates. We then establish local and global well-posedness for the controlled-driven rough differential equation $ dY_t=F(Y_t)\,d\mathbf Z_t. $ A key structural result is the canonical lift of the controlled driver: from the controlled data $(\mathbf X,\mathbf Z)$ we construct a level-$2$ rough path \[ \widehat{\mathbf Z}=(1,Z,\mathbb Z), \qquad \mathbb Z_{s,t}:=\int_s^t Z_{s,u}\otimes dZ_u, \] and show that the controlled-driven equation is equivalent to the classical rough differential equation driven by $\widehat{\mathbf Z}$. This equivalence shows compatibility with classical rough path theory, while the controlled formulation keeps track of the dependence of the effective driver $Z$ on the reference rough path $\X$. Finally, we prove a universal limit theorem for the solution map $ (\mathbf X,\mathbf Z,Y_0)\longmapsto Y, $ which gives stability with respect to perturbations of the initial condition, the reference rough path, and the controlled driver. These results provide a natural framework for layered rough systems and equations driven by transformed or previously evolved rough signals.