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Let $X_{t}$ be a Wiener process starting at the point $x$. Compute $\mathbb{E}[\tau_{a, b}]$, where $\tau_{a, b}$ is the minimum time $t$ at which the process $X_{t}$ is equal to $a$ or $b$. That is, compute the expected time necessary to hit a boundary.

I've been studying for my stochastic processes exam, and this is one of the questions that I have no idea how to solve. For reference, I've learned about topics like backward Kolmogorov equations, Ito integration, Ito's formula, and so on. I haven't really been able to find many resources online to help me with this problem either.

I would really appreciate some help in solving this problem. I want to gain the insight needed as well, so I would appreciate some sort of explanation along with it.

  • What do you mean by "$X_t$ starts at the point $x+W_t$"...? The starting point shouldn't depend on $t$, right? And what is $W_t$? – saz Apr 29 '19 at 17:42
  • $W_{t}$ is a standard brownian motion. Maybe I copied the problem down and it's just $x$, if that makes more sense? This was given as a practice problem for the exam in class –  Apr 29 '19 at 21:18
  • Could it be that you mean $X_t$ is a Brownian motion started at $x\in\mathbb{R}$? – Mdoc Apr 29 '19 at 23:08
  • Yes that could be it. How would I solve this problem? I've updated my original post. –  Apr 30 '19 at 00:10

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