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 $n$, ...

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For the continuous time case, is there any example such that $\tau_1, …, \tau_n$ are stopping times, but $\inf_n \tau_n $ is not a stopping time?

For the continuous time case, is there any example such that $\tau_1, ..., \tau_n$ are stopping times, but $\inf_n \tau_n $ is not a stopping time? We know if the filtration is right continuous, then ...
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42 views

$T$ stopping time, does $\mathcal{F}_T \subseteq \mathcal{F}_\infty$?

I am confused about the definition of $\mathcal{F}_T$, where $T$ is a stopping time. From three different books we find two different definitions: (Karatzas and Shreve; Protter) Events $A \in ...
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Application of Doob's optional stopping theorem to an elementary probability problem

The elementary probability problem is as follows. Let $(X_k)_{k\in\mathbb{N}}$ be a sequence of i.i.d. random variables such that $X_k \sim U(0,1)$ for each $k$. Define $\tau := \inf\{n\geq 0: ...
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1answer
48 views

Expectation of product of iid random variables limited by stopping time

Let $X_1, X_2, \cdots$ be i.i.d. such that $X_i > 0$ and $\mathbb E[X_i]=1$ and consider $\mathbb F = \{\mathcal F_n\}_{n\ge 1}$ to be the discrete filtration. Denote $Y_n = \prod\limits_{i=1}^n ...
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Optimal strategy for a moment to take an exam

In my problem set there was an exercise involving optimal stopping theory. Here is the problem: There is an exam, a list of $n$ questions and $n$ students. Student $A$ knows answers to $k$ of them. He ...
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27 views

Absolute expectation of stopped martingale

Let $M_0,M_1,\dots$ be a martingale with respect to $X_0,X_1,\dots$ and $T$ be a stopping time with respect to $X_0,X_1,\dots$ Define $T_n=\min\{n,T\}$ and let $M_{T_n}$ be the stopped martingale. By ...
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27 views

Linear combination of stopping times

Let $\mathcal F = \{\mathcal F_n\}_{n\ge 1}$ be a discrete filtration. Given $\mathcal F_{\tau} = \{ A \subset \Omega : A \cap \{\tau \le n\} \in \mathcal F_n . \forall n \ge 1\}$ for any stopping ...
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39 views

Expectation of a stopping time on an asymmetric random walk

Let $X_1, X_2, \cdots$ be i.i.d. such that $P(X_i=1)=p , P(X_i=-1)=1-p$. Denote $\tau_a = inf \; \{ n \ge 1 : S_n = a \}$ for any integer $a$, where $\tau_a = \infty$ if $S_n \neq a$ for all $n \ge ...
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67 views

A counterexample for supremum of stopping times

Let $\mathbb{F} = \{ \mathcal{F}_t \}_{t \geq 0}$ be a continuous time filteration. $\tau : \Omega \to [0, \infty]$ is called an $\mathbb{F}$-stopping time if $\{ \tau \leq t \} \in \mathcal{F}_t$ for ...
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51 views

sigma algebra of a stopping time

Let $N$ be a stopping time. i.e $\{N=n\} \in \mathbb{f}_n \forall n$. $\mathbb{f}_n$ is the filtration. $\mathbb{f}_N=\{A\in \mathbb{f}, A\cap \{N=n\}\in \mathbb{f}_n \forall n\}$ is the sigma ...
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28 views

Existence of compensator process under the assumption of local integrability and finite variation

I am reading a proof regarding existence of compensators under the assumption of local integrability in which I don't quite understand: Definition: The compensator of a cadlag adapted process $X$ ...
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Different definitions of local p integrability for local martingales

When talking about cadlag (but not continuous) martingales and local martingales in the context of stochastic integration one can come across different definitions depending on the author. These are: ...
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42 views

$L^p$ integrable local martingale is still $L^p$ integrable when stopped at localizing stopping times.

Assume that $X$ is $L^p$ integrable for $1\leq p\leq \infty$ (i.e., for all $t$, $X_t\in L^p$) and is also a (Cadlag) local martingale. If $T_n$ is a localizing sequence of stopping times for $X$. Is ...
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35 views

How to show a hitting time is finite almost surely?

A one-dimensional symmetric simple random walk starts at $S_0 = 1$. How to show with probability one it passes $x = 0$ (or I guess equivalently, the stopping time of hitting $x = 0$ at the first time ...
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1answer
42 views

Are stopping times the same?

In the context of stochastic integration, we showed how it's possible to define the stochastic integral $\int H dM$ for $H \in L^2(M)$ and $M \in \mathcal M^2_0$ (martingales null at $0$ such that ...
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35 views

Stopping Time Sum of Random Variables

Let $X_1,...,X_t$ be an i.i.d. sequence of random variables with support $\{a,-b\}$, where $a,b>0$, and measure $P(a)=p_1$, $P(-b)=p_2$. Assume $p_1a-p_2b<0$, so that $E[X_t]<0$. Let ...
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1answer
55 views

Random Walk Stopping Time 2

Let $(X_1,X_2,...)$ be i.i.d random variables, with $P(X_t=1)=P(X_t=-1)=1/2$. Then $S_t= \frac{1}{t}\sum_{i=1}^{t}X_i $ is a zero mean random walk. Let $\tau$ be the stopping time corresponding to ...
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2answers
42 views

Random Walk Stopping Time

Let $(X_1,X_2,...)$ be i.i.d random variables, with $P(X_t=1)=P(X_t=-1)=1/2$. Then $S_t= \frac{1}{t}\sum_{i=1}^{t}X_i $ is a zero mean random walk. Let $\tau$ be the stopping time corresponding to ...
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1answer
33 views

Find a Martingale in a game of exchanging hats

$n$ people play a game of exchanging hats, with the following two rules: --They throw their hats in to a pile and everyone chooses one uniformly at random, those who got back their own hat are out of ...
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0answers
48 views

Show that for every $p >0$, $E[\sup \limits_{t \leq L_n} |R_n(t)-t|^p]=O(n^{-p/2})$

I am trying to prove that $E[\sup \limits_{t \leq L_n} |R_n(t)-t|^p]=O(n^{-p/2})$ where $\rho(n)$ is a solution of the following Stochastic differential equation \begin{equation} \rho_n^2(t)=2 ...
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65 views

Filtration of stopping time equal to the natural filtration of the stopped process

Given a probability space $(\Omega,\mathcal{F},P)$ and a process $X_{t}$ defined on it. We consider the natural Filtration generated by the process $\mathcal{F}_{t}=\sigma (X_{s}:s\leq t)$. Let $\tau$ ...
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37 views

Proof finite stopping time and Wiener process bounded

Let $T_{-a,b}=\inf\{t\geq 0: W_{t} \notin [-a,b]\}, a,b>0$. I want to show that this is a finite stopping time ($P(T_{-a,b}<\infty)=1$) and that $|W_{\min(T_{-a,b},t)}|$ is bounded by a ...
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1answer
31 views

Strong Markov property with two stopping times

I have a diffusion $X=(X_t)_{t\ge0}$ and a stopping time $\tau$. From the strong Markov property I know that for any time $t\ge0$ (or a random time independent of $X$) I get that ...
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1answer
53 views

Upper bound for random walk to show stopping time is bounded

I have a simple symmetric random walk (SSRW), and a stopping time: $\tau=\inf\{ n \geq 0 ~:~ |S_n|=N\}$. I am showing that $\newcommand{\ee}[1]{\mathbb{E}[#1]}$ $\newcommand{\pp}[1]{\mathbb{P}[#1]}$ ...
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1answer
70 views

How can a stopping time be independent of its stochastic process?

I was reading about a special case of Wald's equation, which led me to the following question: If $X_t$ is a sequence of iid RV's, and $\tau$ is a stopping time for this discrete stochastic process, ...
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1answer
115 views

Simple Random Walk: Hitting time of 1 is a.s. finite

Let $X_i, i \geq 0$ be i.i.d. random variables with $P[X_i=1]=P[X_i=-1]=1/2$ and consider $S_n = X_1 + \dotsc + X_n$ for $n \geq 1$, $S_0=0$, the symmetric simple random walk on $\mathbb{Z}$. Let ...
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57 views

If $\tau$ is a stopping time, then $E(X_{\tau})=?$

Let $\{X_n \in \mathbb{N}: n \in \mathbb{N}\}$ be a sequence of r.v. and $\tau_k=\min\{n\in \mathbb{N}:X_n=k\}$ Does $E(X_{\tau_k})=E(k)=k$? Any help would be appreciated.
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1answer
47 views

Stopping time in Markov chains

A random variable $T : \Omega \rightarrow ${$1,2,3...$} $\cup$ {$ \infty$} is called a stopping time if the event {$T=n$} depends only on $X_0 , X_1 ,X_2 ,..., X_n$ for $n = 0,1,2,...$ I have trouble ...
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1answer
37 views

Do we need to find an upper bound for the expectation of this stopping time?

From here: It looks like: It is supposed to say 'different from six' rather than 'different from three' $T = \inf\{m: X_{m} = X_{m+1} = X_{m+2} = 6\}$ In every triple $P(all \ 6) = 1/216$ ...
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Show $E[T] < \infty$ by finding an upper bound for $P(T=k)$

Given random variables $X_1, X_2, \ldots \stackrel{iid}{\sim} P(X_i = 1) = p = 1 - q = 1 - P(X_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in ...
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1answer
94 views

Show that $P(T \le n + N \mid \mathscr F_n) > \epsilon$ where T is a stopping time

Given random variables $Y_1, Y_2, \ldots \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in ...
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1answer
35 views

Order statistics for exponential random variables

Let $\tau_1, \tau_2, ..., \tau_K$ be i.i.d. exponential random variables with distribution $P(\tau_k<t) = 1 - e^{-\lambda t}$. Let $\tau^*_i$ be the $i^{th}$ order statistic. The p.d.f. of ...
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45 views

Conditional Probability, Levy Process

Let $(X_t)_{t\in\mathbb{R}}$ be a (double-sided) Levy Process, i.e. $X_0 = 0$ almost surely, the functions $t\mapsto X_t\omega$, $\omega\in\Omega$, are right-continuous with left limits and the ...
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0answers
38 views

Hitting time for Browian motion with upper reflecting boundary

I was wondering if there exist a known distribution function or a nice closed form describing the first hitting time to a given threshold $a$, $T_a$, for a Brownian motion bounded by a upper ...
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2answers
53 views
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1answer
26 views

Random Sampling and Measurability.

In a probability space $\left(\Omega,(\mathcal{F}_t)_{t=0,..,T},\mathcal{F},\mathbb{P}\right)$ let $\tau$ be a stopping time. Consider the definition of "stopped" filtration as $$ \mathcal{F}_{\tau} ...
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1answer
127 views

Asymmetric Random Walk / Prove that $E[T:= \inf\{n: X_n = b\}] < \infty$

Given random variables $Y_1, Y_2, \ldots \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in ...
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1answer
61 views

Asymmetric Random Walk / Prove that $T:= \inf\{n: X_n = b\}$ is a $\{\mathscr F_n\}_{n \in \mathbb N}$-stopping time

Given random variables $Y_1, Y_2, ... \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in ...
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1answer
32 views

Consider stopping times $S$ and $T$ for a filtration $(\mathcal{F}_n)$. Show that $\mathcal{F}_{\min(S,T)} = \mathcal{F}_{S}\cap \mathcal{F}_{T}$.

I'm trying to solve this question but my argument works for $\mathcal{F}_{\max(S,T)} = \mathcal{F}_{S}\cap \mathcal{F}_{T}$. I'm wondering if anyone can confirm if this question is a typo and should ...
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1answer
66 views

Asymmetric Random Walk / Prove $E[T] = \frac{b}{p-q}$ / How do I use hint?

Given random variables $Y_1, Y_2, \ldots \stackrel{\mathrm{iid}}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr ...
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1answer
46 views

Asymmetric Random Walk / Prove $E[X_{T \wedge n}] = (p-q)E[T \wedge n]$

Given random variables $Y_1, Y_2, ... \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in ...
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1answer
54 views

Symmetric Random Walk / Find $E[X_S]$ and $E[X_T]$

Given a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $\mathscr F_n = \mathscr F_n^Y$, let $Y_1, Y_2, ...$ be iid random variables w/ $P(Y_n = ...
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0answers
18 views

Mean of overcooking time

This question came up this week when I had to put my rice in the microwave for a third time. Suppose the perfect cooking time for a meal is given by a discrete random variable $X$ with values in ...
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1answer
73 views

Symmetric Random Walk / Prove $S = \inf\{n : X_n = 7\}$ and $T = 10^{12} \wedge S$ are $\{\mathscr F_n^Y\}$-stopping times.

Given a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $\mathscr F_n = \mathscr F_n^Y$, let $Y_1, Y_2, ...$ be iid random variables w/ $P(Y_n = ...
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1answer
62 views

Show that $E[X_T | T < \infty] \le E[X_0]$ and $cP(\sup X_n \ge c) \le E[X_0]$

From Probability with Martingales: I'm assuming the semi-colon means condition (Otherwise, why not say $T$ is a finite stopping time?). What I tried: $$X_T1_{T < \infty} = X_01_{T=0} + ...
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1answer
27 views

What segment in 8-bit LED Displays used for Traffic Light timers can be removed causing minimal impact in the readability of the countdown numbers?

I passed by an intersection with traffic lights and noticed that 1 segment of the 8-bit display counter is dimmed (it's not working). When the lowermost segment is dimmed for example, number 4 can ...
2
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1answer
53 views

Is this $X_T$ if the stopping time is $T \le \infty$?

Is this $X_T$ if the stopping time is $T \le \infty$? Let $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}_{n \in \mathbb{N}}, \mathbb{P})$ be a filtered probability space, and let $X = ({X_n})_{n \in ...
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2answers
141 views

Martingales exercise on series [closed]

Given a coin, the probability that a head comes out is p: P(h)=p. We have two series: N1= H,H,T,T,H,T and N2= H,T,H,T,H,T,H How long do I have to wait to see this on average? E[N1]=? E[N2]=?
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1answer
29 views

Future events times and Lévy processes

If I am at time $t$ and I know that in the future, at time $t+h$ a process $X_s$ will jump by a random quantity, can $X_s$ be a Lévy process? ($X_s$ jumps before and after $t+h$ at random times) If ...
0
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0answers
23 views

Is $\tau(\omega )= \infty \forall \omega \in \Omega$ a stopping time?

My guess is yes since $\{\infty \leq t \}=\phi \in \mathcal{F}_t, \forall t \geq 0$ where $\phi$ is the empty set which is always in the sigma algebra. Am I right?