Let (Bt)t≥0 be a standard Brownian Motion and let T := inf{t ≥ 0 : Bt = at − b} for some positive constants a, b > 0. Calculate E[T].
How can this be solved? By the optional stopping theorem?
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I am using different symbols than you are, so be careful when reading my answer. Let $$ X_{t}^{\xi}=\xi t+W_{t} $$ where $W$ is a Brownian motion. Let $a$ be a real number and $$ \tau_{a\xi}=\inf\left\{ t\geq0\colon X_{t}^{\xi}=a\right\} $$ be the first hitting time the drifting Brownian motion reaches level $a$. The density of $\tau_{a\xi}$ is $$ f_{a\xi}(t)=\frac{\left|a\right|}{\sqrt{2\pi t^{3}}}\exp\left(-\frac{\left(a-\xi t\right)^{2}}{2t}\right). $$ Then, $$ \mathbb{E}\left[\tau_{a\xi}\right]=\int_{0}^{\infty}tf_{a\xi}(t)dt=\int_{0}^{\infty}\frac{\left|a\right|}{\sqrt{2\pi t}}\exp\left(-\frac{\left(a-\xi t\right)^{2}}{2t}\right)dt=\frac{\left|a\right|}{\left|\xi\right|}e^{a\xi-\left|a\xi\right|} $$ whenever $a\neq0$ or $\xi\neq0$.
parsiad
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Thank you. How can I get the density of $\tau_{\alpha\xi}$ ? – koala Mar 25 '18 at 23:27
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@koala: https://math.stackexchange.com/questions/1053294/density-of-first-hitting-time-of-brownian-motion-with-drift – parsiad Mar 25 '18 at 23:28