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### Density of first hitting time of Brownian motion with drift

I just started learning about Brownian motion and I am struggling with this question: Suppose that $X_t = B_t + ct$, where $B$ is a Brownian motion, $c$ is a constant. Set $H_a = \inf \{ t: X_t =a \}$...
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### Dominated convergence problems with Wald's identity for the Brownian Motion

In the course of proving Wald's second identity $E(B^2_T)=E(T)$, where $(B_t)_{t\geq0}$ is the Brownian motion and $T$ is a stopping time with $E(T)<\infty$, I got stuck with the following problem. ...
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### Hitting time of an open set is not a stopping time for Brownian Motion

Let $(B_t)$ be a standard Brownian motion and $\mathcal F_t$ the associated canonical filtration. It's a standard result that the hitting time for a closed set is a stopping time for $\mathcal F_t$ ...
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### Stopping time and Brownian motion (specific example)

Let $B$ be a Brownian motion. I want to show that $$\inf\{t\geq0 \mid B(t)=\max_{x\in [0,1]}B(s)\}$$ is not a stopping time w.r.t. the standard filtration. How can one intuitively see that this ...
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