Bounding Ornstein-Uhlenbeck Processes and Alikes

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 $\underset{t\to \infty }{\overline{\lim }}\frac{\Vert X_t\Vert }{\sqrt{\log t}}=\sqrt{2{\lambda }_1}$ almost surely with ${\lambda }_1$ being the largest eigenvalue of the matrix $\Sigma :={\int }_0^{\infty }{e}^{-sA}\cdot (D\cdot D^T)\cdot e^{-s{A}^T}\mathrm{d}s$; the discarded measure-zero set can be chosen independent of the initial values $X_0=x$.

6 pages

References in corpus (1)

• Bounding the Solutions to Some SDEs via Ergodic Theory

Cited by in corpus (1)

• Bounding the Solutions to Some SDEs via Ergodic Theory