This paper revisits the concept of rough paths of inhomogeneous degree of smoothness (geometric Π-rough paths in our terminology) sketched by Lyons in 1998. Although geometric Π-rough paths can be treated as p-rough paths for a sufficiently large p, and the theory of integration of Lipγ one-forms (γ > p–1) along geometric p-rough paths applies, we prove the existence of integrals of one-forms under weaker conditions. Moreover, we consider differential equations driven by geometric Π-rough paths and give sufficient conditions for existence and uniqueness of solution.

Consider a mean-reverting equation, generalized in the sense it is driven by a1-dimensional centeredGaussian process with Hölder continuous paths on [0,T] (T> 0). Taking thatequation in rough paths sense only gives local existence of the solution because thenon-explosion condition is not satisfied in general. Under natural assumptions, by usingspecific methods, we show the global existence and uniqueness of the solution, itsintegrability, the continuity and differentiability of the associated Itô map, and weprovide an Lp-convergingapproximation with a rate of convergence (p ≫ 1). The regularity of the Itô map ensures a largedeviation principle, and the existence of a density with respect to Lebesgue’s measure,for the solution of that generalized mean-reverting equation. Finally, we study ageneralized mean-reverting pharmacokinetic model.

We present the rough path theory introduced by Lyons, using the swewing lemma of Feyel and de Lapradelle.

We have seen in a previous article how the theory of “rough paths”allows us to construct solutions of differential equations driven by processes generated by divergence form operators. In this article, we study a convergence criterion which implies that one can interchange the integral with the limit of a family of stochastic processes generated by divergence form operators. As a corollary, we identify stochastic integrals constructed with the theory of rough paths with Stratonovich or Itô integrals already constructedfor stochastic processes generated by divergence form operators by using time-reversal techniques.

We show in this article how the theory of “rough paths”allows us to construct solutions of differential equations (SDEs) driven by processes generated by divergence-form operators. For that, we use approximations of the trajectories of the stochastic process by piecewise smooth paths. A result of type Wong-Zakai follows immediately.