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Bounding Ornstein-Uhlenbeck Processes and Alikes

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 $\displaystyle \varlimsup_{t \to \infty} \frac{\|X_t\|}{\sqrt{\log t}} =\sqrt{2 λ_1}$ almost surely with $λ_1$ being the largest eigenvalue of the matrix $\displaystyle Σ:=\int_0^\infty e^{-s A} \cdot (D \cdot D^T) \cdot e^{-s A^T} \mathrm{d} s$; the discarded measure-zero set can be chosen independent of the initial values $X_0=x$.