Bounding Ornstein-Uhlenbeck Processes and Alikes

TL;DR

This paper analyzes the long-term behavior of certain Ornstein-Uhlenbeck type stochastic differential equations, establishing almost sure bounds on the process norm growth related to the spectral properties of involved matrices.

Contribution

It provides a precise almost sure asymptotic growth rate for solutions of a class of SDEs with linear drift and noise, extending understanding of their long-term behavior.

Findings

The process norm grows like \\sqrt{2 \\lambda_1 \\log t} almost surely.

The asymptotic growth rate depends on the largest eigenvalue of a specific matrix \\Sigma.

The results hold for SDEs with eigenvalues of A having positive real parts and sublinear growth of F.

Abstract

In this note we consider SDEs of the type $\mathrm{d} X_t=[F (X_t) -A X_t] \mathrm{d} t +D \mathrm{d} W_t$ under the assumptions that $A$'s eigenvalues are all of positive real parts and $F (\cdot)$ has slower-than-linear growth rate. It is proved that $\displaystyle \varlimsup_{t \to \infty} \frac{\|X_t\|}{\sqrt{\log t}} =\sqrt{2 \lambda_1}$ almost surely with $\lambda_1$ being the largest eigenvalue of the matrix $\displaystyle \Sigma :=\int_0^\infty e^{-s A} \cdot (D \cdot D^T) \cdot e^{-s A^T} \mathrm{d} s$; the discarded measure-zero set can be chosen independent of the initial values $X_0=x$.