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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Why is stopping time defined as a random variable?

I've been given a crash course in stochastic processes and martingales for the purposes of a semester project on them. The guy I'm working with has been, I feel, a little vague in the definition of ...
0
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1answer
59 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 ...
3
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1answer
269 views

A martingale with bounded increments either converges or diverges to both infinities a.s.

I am reading page 236 "Probability : theory and examples" by R. Durrett. Theorem 31. Let $X_1, X_2,\ldots$ be a martingale with $|X_{n+1}-X_n|\leq M<\infty$. Let $C=\{\lim X_n \;\;\; \text{exists ...
0
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1answer
78 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}}, ...
-1
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1answer
50 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
2answers
821 views

First hitting time for a brownian motion with a exponential boundary

Let $B_t$ be the standard Brownian Motion. Is the distribution/density of the first hitting time of $B_t$ for an exponential decaying boundary known? Trying to be more formal, if ...
1
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1answer
125 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 ...
1
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2answers
96 views

Showing stopping is finite almost surely

Consider a discrete random walk taking values +1 or -1 with probabilities p and q, respectively. Let $S_n = \sum_{k=1}^{n}X_k$. Let $[-A,B]$ be an interval, $A,B \geq 1$. Now define $$\tau =\min(n:n ...
1
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1answer
178 views

Stopping time question $\sigma$

If $S$ and $T$ are stopping time, $S \vee T$ is $\max ({S,T})$, $F_S$ and $F_T$ are stopped sigma algebra, show that $F_{S \vee T} = \sigma(F_S,F_T)$. My thinking : I should take a set $A$ in $F_{S ...
1
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2answers
33 views

Hitting times for Brownian motions

Let $B$ be a standard Brownian motion and let $\alpha, \beta > 0$. Let \begin{align} \tau = \inf\{t \geq 0 : B_t = \alpha \ \ \text{or}\ \ B_t=-\beta\}. \end{align} It can be shown by defining ...
0
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2answers
53 views
0
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1answer
71 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 = ...
7
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2answers
2k views

Density of first hitting time of Brownian motion with drift

I just started learning about Brownian motion and I am struggling with this question: Suppose that $X_t = B_t + ct$, where $B$ is a Brownian motion, $c$ is a constant. Set $H_a = \inf \{ t: X_t =a ...
4
votes
2answers
196 views

Proving Galmarino's Test

Galmarino's Test gives a condition equivalent to being a stopping time. It says: Let $X$ be a continuous stochastic process with index set $\mathbb{R}_+$ (i.e. each sample path is a continuous ...
6
votes
1answer
243 views

How to get closed form solutions to stopped martingale problems?

Way back when, I took a course in stochastic processes in college. I remember being frustrated by the plethora of abstract proofs without much in the way of how to use them to get actual results. It ...
4
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1answer
93 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 ...
4
votes
1answer
96 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 ...
3
votes
1answer
50 views

Exist $\alpha < \infty$, $\beta > 0$ such that $\mathbb{P}\{T_\lambda > t\} \le \alpha e^{-\beta t}?$

Let $B_t$ be a standard one-dimensional Brownian motion. Suppose $\lambda > 0$ and let$$T_\lambda = \min\{t : |B_t| = \lambda\}.$$Do there exist $\alpha < \infty$ and $\beta > 0$ (which may ...
2
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1answer
124 views

Density of running supremum of Brownian motion until a stopping time

I am stuck on an exercise in my book: The question relies on the following fact: Let $M$ be a continuous, non-negative local martingale such that $M_0=1$ and $M_t \rightarrow 0$ almost surely as ...
1
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1answer
115 views

Showing that a hitting time is $\mathbb P-\text{a.e.}$-finite

Let be $\alpha, \beta \in \mathbb R$ such that $\alpha < \beta $ and $x \in [\alpha, \beta ]$. Consider the random time $$T_x = \inf \{ t\geq 0 : x+ B_t \notin [\alpha, \beta]\},$$ where ...
1
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0answers
128 views

Characterization of hitting time's law. (Proof check)

Under the same assumptions of this early question, consider also a the random time $T_a := \inf\{ t > 0: B_t \geq a\}$ which is a stopping time. Since $M^\lambda$ is a continuous martingale, Doob's ...
0
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1answer
59 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 ...
4
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1answer
45 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 ...
3
votes
1answer
57 views

Measurability of the zero-crossing time of Brownian motion

I have the following random time $\tau = \inf\{t > 0: W_t = 0\}$ where $(W_t)_{t\geq 0}$ is Brownian motion with almost surely continuous paths and $W_0 = 0$ a.s. I need to prove that $\tau$ is ...
2
votes
1answer
50 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 ...
2
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0answers
29 views

If $(F_t)_t$ is a filtration, $T$ is a stopping time and $Y$ is $F_T$-measurable, then $1_{\left\{T=s\right\}}Y$ is $F_s$-measurable

Let $(\Omega,\mathcal A)$ be a measurable space $I\subseteq[0,\infty)$ $\mathbb F=(\mathcal F_t)_{t\in I}$ be a filtration on $(\Omega,\mathcal A)$ $\tau$ be a $\mathbb F$-stopping time $\mathcal ...
2
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1answer
84 views

Gambling Game martingale

State the optional sampling theorem for martingales and bounded stopping times. You start with a capital of £100 and bet repeatedly on the toss of a coin. On each toss you may bet any whole number of ...
2
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3answers
405 views

Optimal stopping in coin tossing with finite horizon

There's a classic coin toss problem that asks about optimal stopping. The setup is you keep flipping a coin until you decide to stop, and when you stop you get paid $H/n%$ where $H$ is the number of ...
2
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2answers
1k views

Expected stopping time of brownian motion

I've been looking at this for some time now and still have no sensible solutions, can somebody help me out please. Say I define the stopping time of a Brownian motion as followed: $$\tau(a) = \min (t ...
2
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1answer
1k views

proof that a stopped martingale is a martingale?

Defenition. $\mathcal{F}_{\tau}=\{F\subset \Omega: \forall n \in N \cup \{\infty\}, F\cap(\tau\leq n)\in \mathcal{F}_{n}$} is a sigma-algebra. Defenition. $\forall \omega \in \Omega: ...
2
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2answers
610 views

Stopping time proof

Let $\{X_t, t \ge 0\}$ be a continuous stochastic process and adapted to the filtration $\{\mathcal{F}_t,t\ge 0 \}$ and consider $$ \alpha = \inf\{t, |X_t|>1\}, $$ the first time the the process ...
0
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1answer
58 views

Application of CLT to random walks

Let $X_1,X_2,\ldots$ be an iid sequence such that $P\{X_1 = 1\} = p$, $P\{X_1 = -1\} =p$ and $P\{X_1 = 0\} = 1-2p$. We have that $E[X_1] = 0$ and $E[X_1^2] = 2p$. Define $S_n = \sum_{i=1}^nX_i$ and ...
0
votes
1answer
64 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
votes
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 ...
0
votes
1answer
51 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 = ...
0
votes
1answer
50 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}}, ...
0
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1answer
82 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
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1answer
55 views

Expectation of a Wiener process at a Stopping Time

I am working through an answer to the following question and do not understand an expectation which takes place at the end. $\textbf{Question:}$ Define the following stochastic process \begin{align} ...
0
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1answer
550 views

Snowplow Problem

A snowplow can remove snow at a constant rate (in cubic feet per minute). One day, there was no snow on the ground at sunrise, but sometime in the morning it began snowing at a steady rate. At noon, ...
0
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1answer
100 views

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 ...
0
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2answers
241 views

Stopping times, Filtration, Martingales,

I am new here and I have a question. Defenition: Let $ \tau$ be a stopping time, then $\digamma_{\tau}=\{F\subset \Omega: \forall n \in N \cup \{\infty\} , F\cap(\tau\leq n)\in \digamma_{n}$} is a ...
0
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1answer
113 views

How to prove that for Brownian motion in $(a, b)$ $\mathbb{E}^x[\min(H_a, H_b)] = (x-a)(b-x)$?

i'm wondering if anyone can help me with proving the fact that for BM in the interval $(a,b)$ and with $$H_y = \inf\{t>0: X_t = y\},$$ the following is true: $$\mathbb{E}^x[\min(H_a, H_b)] = ...