On rough differential equations

Abstract

We prove that the Itô map, that is the map that gives the solution of a differential equation controlled by a rough path of finite p-variation with p ∈ [2, 3) is locally Lipschitz continuous in all its arguments and we give some sufficient conditions for global existence for non-bounded vector fields.

Convergence of Multi-Dimensional Quantized SDE’s

“…Several results of local Lipschitz continuity have been established recently, especially in [6], [9], [16], [17], although not completely satisfactory from a practical point of view. So we decided not to reproduce (and take advantage of) them here.…”

“…Following rough paths theory initiated by T. Lyons ([25]) and developed with many co-authors (see e.g. [26,14,16,26,9] …”

“…Furthermore, the Itô map y → x is continuous for the Hölder ρ q distance (and locally Lipschitz in sense described in [16]). …”

Controlled differential equations as Young integrals: A simple approach

“…[7], global existence follows from application of Theorem 6.1 in [7], as well as from the results in [18]. Let us note that here, it is not assume that f is bounded.…”

“…In addition, we are able to deal with the case where f is not bounded, unlike the articles relying on the Picard iteration (however, a global existence result under similar conditions is stated in [7]). Our strategy is the one used in the more complex case for 2 < p 3 for providing bounds and estimation on distances between solutions of rough differential equations [18].…”