A Generalized Mean-Reverting Equation and Applications

TL;DR

This paper establishes the global existence, uniqueness, and regularity of solutions to a generalized mean-reverting stochastic differential equation driven by Gaussian processes, with applications to pharmacokinetic modeling.

Contribution

It introduces a framework for solving and analyzing a generalized mean-reverting equation driven by Gaussian processes, including global solutions, regularity, and approximation methods.

Findings

Proved global existence and uniqueness of solutions.

Established regularity and differentiability of the Itô map.

Provided an $L^p$-converging approximation with a specific rate.

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.