A note about an upper bound for a hitting time of the fractional Ornstein-Uhlenbeck process

In this brief note we give an upper bound for \(P(\tau_u < T)\) with \(T>0\), where \(\tau_u\) is the exit time defined as \(\tau_u:=\inf \{ t\geq 0 \, : \, X_t\geq u \}\) and \((X_t)_{t\geq 0}\) is the fractional Ornstein-Uhlenbeck processes which satisfies the following stochastic differenti...

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Bibliographic Details
Published inarXiv.org
Main Author Wilson, Cabanillas B
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 15.08.2024
Subjects
Differential equations
Ornstein-Uhlenbeck process
Upper bounds
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Summary:In this brief note we give an upper bound for \(P(\tau_u < T)\) with \(T>0\), where \(\tau_u\) is the exit time defined as \(\tau_u:=\inf \{ t\geq 0 \, : \, X_t\geq u \}\) and \((X_t)_{t\geq 0}\) is the fractional Ornstein-Uhlenbeck processes which satisfies the following stochastic differential equation \begin{equation*} dX_t =-\lambda X_t dt+ \epsilon dB_t^H\quad \epsilon>0,\;\lambda>0 \end{equation*} with \((B_t^H)_{t\geq 0}\) as the fractional brownian motion with parameter of Hurst \(H\in ]0,1]\).
ISSN:2331-8422