Consider a mean-reverting equation, generalized in the sense it is driven by a
1-dimensional centered
Gaussian process with Hölder continuous paths on [0,T] (T> 0). Taking that
equation in rough paths sense only gives local existence of the solution because the
non-explosion condition is not satisfied in general. Under natural assumptions, by using
specific methods, we show the global existence and uniqueness of the solution, its
integrability, the continuity and differentiability of the associated Itô map, and we
provide an Lp-converging
approximation with a rate of convergence (p ≫ 1). The regularity of the Itô map ensures a large
deviation 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 a
generalized mean-reverting pharmacokinetic model.