Rough differential equation ($dY=V(Y)d\mathbf{X}$)

A rough differential equation interprets a differential equation using the higher-order data of its driving rough path. For $2<p<3$, its local expansion includes $V_i(Y_s)X_{s,t}^i+D{V}_j(Y_s)V_i(Y_s){\mathbb{X}}_{s,t}^{ij}$, with a remainder controlled at order greater than one. For a smooth driver it is the usual controlled ordinary differential equation; for enhanced Brownian motion it is the corresponding Stratonovich equation.

A pathlevel solution is the ordinary state path satisfying a local rough Taylor expansion with a remainder of order $\theta >1$ in a control function for rough paths. It specifies the first level of the solution, rather than requiring the full lifted solution as part of the definition. With vector fields of regularity ${\mathrm{Lip}}^{\gamma }$, $\gamma >p$, the Davie lemma bounds discrete Euler errors uniformly; compactness and passage to the limit give such a path.

For a finite time grid, suppose a nonnegative error is zero on neighboring points and obeys $a_{s,t}\le a_{s,u}+a_{u,t}+L\omega (s,t{)}^{1/p}{a}_{s,u}+A\omega (s,t{)}^{\theta }$. For a control function for rough paths and $\theta >1$, sufficiently small control intervals have the displayed bound with constants independent of the grid. Splitting near half the control size makes the higher-order errors contract, and handles a crossing single grid interval by its zero error. Applying this to local Euler composition errors yields compactness of approximate pathlevel solutions.

For sufficiently regular vector fields, for example bounded fields with bounded derivatives through order three when $2<p<3$, the solution map of a rough differential equation exists uniquely and is locally continuous in the rough path metric. Ordinary differential equations driven by smooth approximations therefore converge whenever their enhanced drivers converge. The theorem makes the enhancement, rather than just the original path, the correct input for a continuous solution map.