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Questions tagged [stopping-times]

This tag is for questions about stopping times. Let $X = {X_n : n \geq 0}$ be a stochastic process. A stopping time $\tau$ with respect to $X$ is a random time such that for each $n \geq 0$, the event ${\tau = n}$ is completely determined by (at most) the total information known up to time $n$, ${X_0, . . . , X_n}$.

This tag is for questions about stopping times. Let $X = \{X_n : n \geq 0\}$ be a stochastic process. A stopping time $\tau$ with respect to $X$ is a random time such that for each $n \geq 0$, the event $\{\tau = n\}$ is completely determined by (at most) the total information known up to time $n$, $\{X_0, . . . , X_n\}$.

1527 questions
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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 stopping time if $\{\tau \leq t\}\in\mathscr{F}_t$…
parsiad
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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 t $\in [0,T(\omega)]\cap[0,\infty]$. Show that…
swang999
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stopping time expectation for gambler's ruin

2 players A and B start with x & y dollars respectively, and they bet against each other 1 dollar each time by tossing a fair coin. I let $X_n = x + \sum_{i=1}^{n}\xi_i$ where $\xi_i$ are i.i.d. with $P(\xi_i=\pm 1)=\frac{1}{2}$. Let $\tau_0 =…
meta_warrior
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IID sequence and stopping time

Let $X_0=0, X_1, X_2,\dots, X_N$ be i.i.d. random variables, with Gaussian distribution $\cal N (0,1)$. For $k=0,\dots, N, S_k=\sum_{i=1}^k X_i$ and $\tau=\min\{k:S_k^2\geq N-k\}$. So $\tau$ is a stopping time, while $\tau-1$ may not be one. My…
Richard
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If $T$ is a stopping time, prove that $X_T$ is $\mathcal F_T$ measurable.

Let $(X_t)$ a progressively measurable process, i.e. $[0,t]\times \Omega \ni (t,\omega )\mapsto X_t(\omega )\in (\mathbb R, \mathcal B(\mathbb R))$ is $\mathcal B([0,t])\otimes \mathcal F$ measurable. Prove that $X_T$ is $\mathcal F_T$ measurable on…
user657324
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Question about stopping time : what would be $M_{\tau}$ if $\{\tau\leq t\}\notin \mathcal F_t$?

I really have difficulties by really understand in why stopping times are interesting. Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space, $(M_t)_{t\geq 0}$ a stochastic process and let $(\mathcal F_t)_{t\geq 0}$ an adapted filtration. Let…
user657324
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How to compute $P(S_\tau=10)$

Let $X_1,X_2,X_3,\cdots$ variables i.i.d., such that $P(X_1=2)=1/2=P(X_1=-1)$ Let $S_n=X_1+\cdot+X_n$ consider stopping time $\tau=\{n\geq 1:S_n=10, \ or\ S_n=-10\}$. If $E(\tau )=8$. Compute $P(S_\tau = 10)$. How can I approach this problem, as…
apa
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show $\max$ and $\min$ are stopping times.

I need your opinion about my proof. Let $\tau$ and $\sigma$ two stopping time on $( \Omega , \mathscr{F} , (\mathscr{F}_n)_n , \mathbb{P} )$. Let $\mu_1: \Omega \rightarrow \mathbb{N} \cup \left\{ + \infty \right\} $ defined by $$ \forall \omega \in…
Tohiea
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Very simple gambler's ruin, Martingale and optional stopping theorem

I've been trying very hard to understand the "simple Martingale + stopping theorem" solution to Gambler's Ruin. Sorry if this is a repeat question, but I really don't understand some of the (seemingly trivial) solutions other people have…
Hinton
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If $T$ is a stopping time, what represent $\mathcal F_T$?

Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space and let $T:\Omega \to \mathbb N$ a stoping time. Let $(X_n)_{n\geq 1}$ a stochastic process and $\mathcal F_n=\sigma (X_1,...,X_n)$, and $\mathcal F_\infty =\sigma (X_1,X_2,...)$. I don't…
user657324
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What is the largest optimal stopping time, and what are the "in-between" times?

Given an adapted process $X_t$ and it's Snell envelope $S_t$, I know that the smallest optimal stopping time is to stop as soon as the Snell envelope equals $X$. This is very intuitive and makes perfect sense. You could explain it to a kid. However,…
nemesis
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A relationship between stopping times and their $\sigma$-algebras

I'm studying Stochast integration and stochastic differential equation by Protter and got confused by the solution of the very first exercise provided here. So Exercise 1. Let $S,T$ be stopping times, $S \leq T$ a.s. Show $\mathcal{F}_S \subset…
zer0hedge
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Is $\sigma_1 I_A + \sigma_2 I_{A^c}$ a stopping time for stopping times $\sigma_1,\sigma_2$ and $A\in\mathscr{F}_{n+1}$, $n\geq 0$?

Let $\sigma_1$ and $\sigma_2$ be stopping times of a filtration $(\mathscr{F}_n)^\infty_{n=0}$, and let $A\in\mathscr{F}_{n+1}$ for some $n\geq 0$. Question: Is $\sigma_3 = \sigma_1 I_A + \sigma_2 I_{A^c}$ a stopping time? Here $A^c$ is the…
Bart
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Limit of monotone sequence of stopping times.

I am wondering if the next statement is true: Let $\{\tau_n\}$ a non-increasing sequence of stopping times, then if we define $\tau_0=\lim_{n\rightarrow\infty}\tau_n$ is also a stopping time. I think is true but I am not sure. Thanks for any help.
Don P.
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identifying a time period spent in a place that is longer than expected by chance

I have an object that moves randomly back and forth across a line (L) with frequency of movement (F) and speed (S) dependent on temperature (T), and temperature raising at a rate(r). I would like to know whether the object is staying within a tract…
Agus camacho
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