# Tagged Questions

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### Independence between the first exit time from an interval and the value of Brownian motion at this first exit time

Suppose you have an arithmetic Brownian motion (or Brownian motion with drift ) called X, started at a level x such that a < x < b, where a and b are two real points . Define tau as the first ...
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### On a condition for a.s. finite stopping times

Assume that $\{\tau_n: n \in \mathbb{N}\}$ is a sequence of stopping times with respect to some filtration such that $P[\tau_n < \infty] = 1.$ Is that true that there must exist a sequence of ...
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### Interlacing stopping times

This question is posed on a measurable space $(\Omega,\mathscr{F}$) equipped with a filtration $\{\mathscr{F}_t\}$. Recall that a random time $\tau\colon\Omega\rightarrow[0,\infty]$ is said to be a ...
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### Prove $S_k$ is a stopping based on $A$ being previsible

Probability with Martingales: It looks like we have a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $A = \{A_n\}_{n \in \mathbb N}$ ...
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### Check that stopping time is a.s. finite

I have the following situation. Let $(X_i)_{i\geq1}$ be a sequence of iid random variables in $\mathbb{Z}$ and consider the random walk $S_n=\sum_{i=1}^n{X_i}$, $S_0=x$. Let $y>x$ and consider ...
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### Is $B_{t\wedge H_a}$ bounded in $L^2$?

Let $a >0$, $(B_t)_{t\geq0}$ be a standard Brownian motion. Define the stopping time $$H_a := \inf\{t \geq 0 : B_t \geq a\}.$$ Then is the martingale $M_t$ where $M_t: = B_{t\wedge H_a}$ bounded ...
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### Brownian Motion maximum process intuition

I am studying the maximum value of a Brownian Motion (BM) on an interval of time (as explained here between boxes 28 and 40) and I am having an issue aligning intuition with the mathematical result. ...
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### Finding optimal strategy in time series game

Let $P_t$ be a time series such that $P_{t+1} = \alpha P_t+S_{t+1}$, where $\forall t\geq0 : S_t \sim N(0,\sigma)$ Consider the following game: In each round $t$, a player sees $P_t$ and decides ...
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### Expected time until pattern (1,0,0,1)

Let $(X_n)_{n\geq 0}$ be i.i.d. with $\mathbb P(X_n = 0 ) = \mathbb P(X_n = 1) = \frac{1}{2}$. Let $\tau_a$ be the stopping times defined as $$\tau_a = \inf\{n: (X_{n-3}, ... , X_n) = (1,0,0,1)\}$$ I ...
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### Where is my mistake in calculating stopping time?

A gambler has $w$ dollars and in each game he loses or wins a dollar with equal $p=1/2$ probability. I want to calculate the average number of games that a gambler can play before he runs out of ...
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### Conditional expectation of a geometric Brownian motion and stopping time

Let $X$ be a geometric Brownian motion, solution of $dX_t = \mu X_t dt + \sigma X_t dW_t, X_0 > 0$ and ${\cal F}$ its natural filtration. Let's consider $t \geq 0$ and $\tau_a$ the first hitting ...
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### Geometric Brownian motion hitting time

Let $X$ be a geometric Brownian motion $dX_t = \mu X_t dt + \sigma X_t dW_t, X_0 > 0$ and ${\cal F}$ its natural filtration. Let $\tau_a$ be the first hitting time of $a$ by $X$. How can we relate ...
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### Recurrence of a state in a finite state space

Suppose $T_A := \inf\{ n \ge 1 : X_n \in A\}$ where $A \subset \mathcal{S}$ is finite. Assume $\mathbb{P}\{T_A < \infty \; | \; X_0 = x\}= 1$ for $\forall x \in \mathcal{S}-A$. I need to show that ...