Let $(B_t)_{t\geq 0}$ be a Brownian motion and for $\alpha>0$ consider the drifted Brownian motion \begin{align*} X_t=\alpha B_t+\alpha^2 t,\quad t\geq 0. \end{align*} Is there a simple way to show that for the first passage time \begin{align*} T=\inf\{t\geq 0: X_t\geq h\}, \end{align*} where $h>0$, the probability $\mathbb{P}(T>t)$ decays exponentially as $t\to\infty$?

More generally, can this be extended to Ito diffusions of the form \begin{align} Y_t=\int_0^t\alpha_s dB_s+\int_0^t\alpha_s^2 ds,\quad t\geq 0, \end{align} where we can assume that $\alpha_t\in[a,b]$ with $0<a<b<\infty$. Clearly in this case, $(Y_t)_{t\geq 0}$ is a time-changed drifted Brownian motion: \begin{align*} Y_t = W_{\langle Y\rangle_t} + \langle Y\rangle_t,\quad t\geq 0 \end{align*} where $(W_t)_{t\geq 0}$ is a Brownian motion.

See also 1: Density of first hitting time of Brownian motion with drift


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