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1answer
22 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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0answers
28 views

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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1answer
11 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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0answers
16 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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0answers
16 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 ...
4
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1answer
41 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 ...
0
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1answer
37 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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0answers
23 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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0answers
10 views

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: ...
2
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1answer
29 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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0answers
15 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 ...
2
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1answer
38 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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0answers
30 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 ...
1
vote
1answer
44 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
31 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 ...
2
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1answer
28 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
36 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 ...
3
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1answer
32 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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0answers
25 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 ...
1
vote
1answer
30 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 ...
2
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1answer
39 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 $\newcommand{\ee}[1]{\mathbb{E}[#1]}$ $\newcommand{\pp}[1]{\mathbb{P}[#1]}$ ...
1
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0answers
29 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, ...
4
votes
1answer
73 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 ...
3
votes
1answer
54 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.
4
votes
1answer
39 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 ...
1
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1answer
31 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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2answers
75 views

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 ...
4
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1answer
83 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 ...
0
votes
1answer
32 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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0answers
37 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
31 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 ...
0
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2answers
43 views
0
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1answer
25 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} ...
1
vote
1answer
112 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 ...
0
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1answer
49 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 ...
0
votes
1answer
27 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 ...
0
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1answer
54 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 ...
0
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1answer
39 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 ...
0
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1answer
47 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
17 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
68 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 = ...
1
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1answer
49 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
22 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
42 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
134 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]=?
2
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1answer
28 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?
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0answers
24 views

Proving almost sure finiteness of a stopping time

Let $M_t$ be a local martingale and $S_t = \sup_{0 \leq s \leq t}M_s$ its running supremum. How can I show that the stopping time $T=\inf\{t \geq 0 : S_t - M_t = a\}$ for an arbitrary $a>0$ is ...
1
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1answer
52 views

Almost Surely Finite Stopping Time Inequality

Assume $\tau$ is a $\mathcal{F}_n$- stopping time such that there exists a positive integer $m$ and some $\epsilon>0$ such that for all $n$ $$\mathbb{P}(\tau\leq n+m \,\, \vert \mathcal{F}_n) ...
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1answer
49 views

Prove a thm on stopped processes given fundamental principle 'you can't beat the system'?

How does the principle below imply the thm below? From Williams' Probability w/ Martingales: Principle: Thm: What I tried: $$E[X_{T \wedge n} - X_0 | \mathscr{F_m}] =/ \le X_{T \wedge ...