Let $X_t$ be an Ornstein-Uhlenbeck process solving $dX_t=−\lambda X_tdt+dW_t$ with $\lambda >0$. Denote $A_t = \frac{1}{t}\int_0^t X_s ds$. Is there a nice way to determine $\limsup_t A_t$?
2026-04-07 00:01:01.1775520061
Limsup of time integral of an Ornstein-Uhlenbeck process
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