# Tagged Questions

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### Hitting times for Brownian motions

Let $B$ be a standard Brownian motion and let $\alpha, \beta > 0$. Let \begin{align} \tau = \inf\{t \geq 0 : B_t = \alpha \ \ \text{or}\ \ B_t=-\beta\}. \end{align} It can be shown by defining ...
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### Prove previsibility and $E[X_{T \wedge n}] \le E[X_{S \wedge n}]$

From Probability with Martingales:
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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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### Stopping times, Filtration, Martingales,

I am new here and I have a question. Defenition: Let $\tau$ be a stopping time, then $\digamma_{\tau}=\{F\subset \Omega: \forall n \in N \cup \{\infty\} , F\cap(\tau\leq n)\in \digamma_{n}$} is a ...
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### Expectation of a Wiener process at a Stopping Time

I am working through an answer to the following question and do not understand an expectation which takes place at the end. $\textbf{Question:}$ Define the following stochastic process \begin{align} ...
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### How to prove that for Brownian motion in $(a, b)$ $\mathbb{E}^x[\min(H_a, H_b)] = (x-a)(b-x)$?

i'm wondering if anyone can help me with proving the fact that for BM in the interval $(a,b)$ and with $$H_y = \inf\{t>0: X_t = y\},$$ the following is true: \mathbb{E}^x[\min(H_a, H_b)] = (x-a)(...
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### equality of value implies equality of stopping time

Question: Let X be a stochastic process and T a stopping time of ${\mathcal{F}^{X}_{t}}$. Suppose that for some pair $\omega$, $\omega$' $\in$ $\Omega$, we have $X_{t}(\omega)=X_{t}(\omega')$ for all ...
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### Application of CLT to random walks

Let $X_1,X_2,\ldots$ be an iid sequence such that $P\{X_1 = 1\} = p$, $P\{X_1 = -1\} =p$ and $P\{X_1 = 0\} = 1-2p$. We have that $E[X_1] = 0$ and $E[X_1^2] = 2p$. Define $S_n = \sum_{i=1}^nX_i$ and \$...