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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1answer
74 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
39 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
32 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 $$\mathbb{P}^{X_0}[X_{...
2
votes
1answer
58 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 $\newcommand{\ee}[1]{\mathbb{E}[#1]}$ $\newcommand{\pp}[1]{\mathbb{P}[#1]}$ ...
2
votes
1answer
79 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
129 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
59 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
49 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
38 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 $P(all \ 6) = 1/216$ ...
-1
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2answers
87 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
votes
1answer
99 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
36 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 $\tau^*...
1
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0answers
47 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 ...
1
vote
0answers
40 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
53 views
0
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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} ...
1
vote
1answer
137 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
votes
1answer
69 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
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 ...
0
votes
1answer
71 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 F_n\}...
0
votes
1answer
52 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
votes
1answer
56 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 = ...
2
votes
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 ...
0
votes
1answer
79 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
vote
1answer
68 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} + X_11_{...
0
votes
1answer
28 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
votes
1answer
59 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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votes
2answers
144 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
votes
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
votes
0answers
24 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?
1
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0answers
39 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
vote
1answer
72 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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votes
1answer
53 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 ...
2
votes
0answers
33 views

Optimal stopping time problem

I'm trying to solve a problem: $$ \sup_{0 \leq \tau \leq 1} E W_{\tau - \varepsilon}, $$ where $W$ is a Wiener process and $\varepsilon$ is a fixed real number. I've tried to approximate a Wiener ...
2
votes
0answers
50 views

An intuitive interpretation of stopping time

I have the following definition of exercise time. Let $T\in\mathbb{N}$ with $T>0$, let $(\Omega,\mathcal{F})$ be a probability space with the $\sigma$-algebra $\mathcal{F}=2^{\Omega}$ and let $(\...
0
votes
1answer
44 views

Continuity with respect to hitting time level

Let $\tau(x)$ be the first hitting time of a Lévy process $(X_t)_{t\geq 0}$ to level $x$. Let $f$ be a continuous function and $g(x)=\mathbb{E}[f(\tau(x))]$. Is it obviously true that $g$ is ...
1
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0answers
24 views

Exponential decay of a stopping time for an Ito diffusion process

Let $dX_t=dB_t + a \cot(X_t)dt$, with $X_0=x \in (0,\pi)$, where $a$ is a specific constant so that the lifetime of the process is infinite almost surely. The process has a transition density which ...
2
votes
1answer
139 views

Brownian motion: hitting times for closed sets are stopping times (and more).

Let $(B_t)$ be a $d$-dimensional Brownian motion, and consider the filtrations $(\mathcal{F_t^B}) = \sigma(B_0,...,B_t)$ and $\mathcal{F_t} = \cap_{\epsilon > 0}{\mathcal{F_{t+\epsilon}^B}}$ (the ...
0
votes
1answer
66 views

$X_T = \lim_{n \to \infty} X_{T \wedge n}$ if X is a supermartingale and T is a finite a.s. stopping time?

Given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}, \mathbb{P})$, let $X = (X_n)_{n \geq 0}$ be a $(\{\mathscr{F_n}\}, \mathbb{P})$-supermartingale and $T$ be a finite $\...
-1
votes
1answer
92 views

Prove $X_T$ is integrable if $X$ is a supermartingale, $T$ is stopping time and other conditions

Let $X = ({X_n})_{n \ge 1}$ be a/an $(\{\mathscr{F_n}\}_{n \ge 1}, \mathbb{P})$-supermartingale in the filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}_{n \ge 1}, \mathbb{P})$. ...
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votes
1answer
81 views

Prove $Y_S$ is integrable if $Y$ is a bounded supermartingale and $S$ is an a.s. finite stopping time. [closed]

Let $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}_{n \in \mathbb{N}}, \mathbb{P})$ be a filtered probability space, and let $Y = ({Y_n})_{n \in \mathbb{N}}$ be a/an $(\{\mathscr{F_n}\}_{n \in \mathbb{N}}, \...
0
votes
1answer
54 views

Prove $Y_S$ is integrable if $Y$ is a supermartingale and $S$ is a bounded stopping time.

Given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}_{n \in \mathbb{N}}, \mathbb{P})$, let $Y = ({Y_n})_{n \in \mathbb{N}}$ be a/an $(\{\mathscr{F_n}\}_{n \in \mathbb{N}}, \...
1
vote
0answers
23 views

k-th hitting time is a stopping time

Could you check if my solution is correct? I find the filtrations quite tricky. Here is the problem: Let $\{X_n\}_{n \in \mathbb{N}}$ be a stochastic process and $B$ a borel set in $\mathbb{R}^N$. ...
1
vote
1answer
63 views

Stopping time $\max \{n: S_n \le t\}$

Let $\{X_n\}_{n \in \mathbb{N}}$ be iid and non-negative. Let $$N(t): = \max \{n: \ X_1 + \cdots + X_n \le t \}.$$ Is $N(t)$ a stopping moment with respect to the natural filtration $\{\mathcal{F}_n\...
0
votes
0answers
31 views

Strong Markov property and stopping time

In the book by Jeanblanc, Yor & Cheney, "mathematical methods for financial markets", on page 17 above Prop.1.1.14.3, there is a strange identity of a strong Markov process $X$ that reads $${\bf 1}...
0
votes
1answer
86 views

Questions on Doob's Optional Stopping Theorem (a) and (b)

From Williams' Probability w/ Martingales: What is $X_T$ in red box above? I am fairly certain this was not defined previously in the book. There was this though: I have a feeling $X_T = ...
0
votes
1answer
27 views

Optional Sampling Theorem - Martingales

I have problems with solving the following problem. Can anyone give me a hint how to solve it? Thanks in advance! Consider a contract that at time N will be worth either 100 or 0: Let S(n) be its ...
4
votes
1answer
168 views

Stopping times and hitting times for càdlàg processes

I can't find the proof of the following lemma in any book: LEMMA: If $X=\{X_t\}_{t\in T}$ is adapted and right continuous, then for every closed set $C \subset E $, the variable $\tau_{C}:=\inf\{t\...
2
votes
0answers
53 views

Solving Stochastic Differential Equation

Let $\beta > 0$, $0 < \gamma < 1$, and let $\tau$ be the first hitting time: $$\tau = \inf\{t:t \geq 0, |W_t| = \pi /4\}$$ Solve the SDE in the random interval $0 \leq t \leq \tau$ $$dX_t = -(...
2
votes
1answer
59 views

Augmented filtration martingale proof.

Part a: Consider a Wiener process, $W_t$ and denote by ${\mathscr{F}_t}_{(t \geq 0)}$ the natural filtration generated by W. Let $\mathbb{R}_{+} = \{x : x \geq 0\}$ and $\mathscr{B}$ be a sigma ...