A Generalized Mean-Reverting Equation and Applications

Abstract

Consider a mean-reverting equation, generalized in the sense it is driven by a 1-dimensional centered Gaussian process with H\"older 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 It\^o map, and we provide an $L^p$-converging approximation with a rate of convergence ($p\geq 1$). The regularity of the It\^o map ensures a large deviation principle, and the existence of a density with respect to the Lebesgue measure, for the solution of that generalized mean-reverting equation. Finally, we study a generalized mean-reverting pharmacokinetic model.