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 $1/3<\hat{I}\pm \le 1/2$. Given a reference rough path $\mathbf{X}=(1,X,\mathbb{X})$ and an $\mathbf{X}$-controlled driver $\mathbf{Z}=(Z,Z^{\prime })$, 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 $d{Y}_t=F(Y_t)d{\mathbf{Z}}_t.$ 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 $\hat{\mathbf{Z}}$. 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 $(\mathbf{X},\mathbf{Z},Y_0)⟼Y,$ 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.

31 pages