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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234 views

Does this Condition on Exit Times imply $X_t$ is a Local Supermartingale?

Let $(X_t)_{t\geq 0}$ be a continuous (or càdlàg), real-valued process, and define stopping times $$\tau_{s,a,b}=\inf~ [s,\infty)\cap\{t:X_t\notin (a,b)\}.$$ We can interpret $\tau_{s,a,b}$ as the ...
12
votes
3answers
2k views

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 ...
11
votes
3answers
1k views

What is meant by a stopping time?

TL;DR: is a stopping time some sort of event, or is it a point in discrete time, or something else entirely what is an example of something which is not a stopping time? is my understanding of the ...
7
votes
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 $H_a = \inf \{ t: X_t =a \}$...
7
votes
1answer
96 views

Gambling system theorem given by Doob

Let$\{X_n\}_{n=1}^{\infty}$ be a sequence of i.i.d. random variable. Let $\{\alpha_k\}_{k=1}^{\infty} $be a sequence of strictly increasing finite stopping times. Then $\{X_{\alpha_k+1}\}_{n=1}^{\...
6
votes
1answer
250 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 ...
6
votes
2answers
115 views

Showing that the first hitting time of a closed set is a stopping time.

I found this exercise online: I am stuggling with the last part of the second exercise, that is I am not able to show that $\tau = \sup_i \tau_i$. Obviously we have that $\tau \ge \sup_i \tau_i$, ...
5
votes
1answer
91 views

Bounding an expected hitting time

Consider a stochastic differential equation: $$dX_t = dW_t + \sin(X_t) dt, \, X_0 = x$$ where $W_t$ is a Wiener process. Define $$\tau_1 = \inf \{ t : X_t \in 2 \pi \mathbb{Z} \} \\ \tau_2 = \inf \...
5
votes
1answer
237 views

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 ...
4
votes
1answer
457 views

Stopping time and Brownian motion (specific example)

Let $B$ be a Brownian motion. I want to show that $$ \inf\{t\geq0 \mid B(t)=\max_{x\in [0,1]}B(s)\} $$ is not a stopping time w.r.t. the standard filtration. How can one intuitively see that this ...
4
votes
1answer
53 views

Uniform integrability and stopping times

I want to know whether there is any example where $X_n$ is uniformly integrable, $N$ is a stopping time and $E[X_N] =\infty$? Or uniform integrability of $X_n$ implies that $E[X_N]< \infty$?
4
votes
2answers
203 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 ...
4
votes
2answers
75 views

Stopping time on an asymmetric random walk

Suppose that we are given an asymmetric random walk whose step is defined as $P(\xi_i = 1) = p$ and $P(\xi_i = -1) = 1-p$, where $p >1/2$. The hitting time, $T_x$ is defined as $\inf{\{n : S_n = x\...
4
votes
1answer
48 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 ...
4
votes
1answer
470 views

Conditional Expectation of martingale at stopping time

I am trying to understand the implications of the optimal stopping theorem, which is why I tought of the following problem. Consider the continuous-time Martingale $X = (X_t)_{t \geq 0}$ and the ...
4
votes
1answer
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 ...
4
votes
1answer
72 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 $\tau$ ...
4
votes
1answer
123 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 $...
4
votes
1answer
98 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
158 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\...
4
votes
1answer
157 views

Stopped process of Brownian motion

I am baffled about the following problem: Let $(B_t)$ be a standard Brownian motion. Let $$ \tau:= \inf\{ t \geq 0 :B_t = x \} \wedge \inf\{ t \geq 0 :B_t = -y \}$$ be a stopping time, where $x,y >...
4
votes
1answer
153 views

Uniformly integrable martingale in a finite time horizon

Let $\{ M (t) \mid t \in [0,T] \}$ be a martingale and $\{ \tau_n \mid n = 1, 2, \ldots\}$ be an increasing sequence of stopping times such that $\tau_n \rightarrow \infty$ as $n \rightarrow \infty$. ...
4
votes
0answers
39 views

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: \...
4
votes
2answers
94 views

Localisation in the proof of Ito's formula

I am reading Karatza's and Schreve's book "Stochastic Calculus and Brownian Motion" and I don't understand a strange thing as follows: Let $X=X_0 + A +M $ be a semimartingale, where $A$ is a ...
4
votes
0answers
228 views

Using Girsanov theorem to prove density of stopping time

Let $B$ be a standard Brownian motion and for $a>0$ and $b>0$, and set $$\sigma_{a,b} = \inf\{t\,:\, B_t + bt = a\}.$$ There are at least two ways to solve the following problem (the other one ...
4
votes
0answers
193 views

Dose “optional stopping theorem” imply “optional sampling theorem”?

Suppose $X$ is a martingale,$\tau$ and $\sigma$ are two stopping times which satisfy (a)$\sigma\le\tau$ and (b)the "optional stopping theorem" holds,that is to say: $$\mathbb E[X_\sigma]=\mathbb E[...
4
votes
0answers
713 views

Law of a geometric brownian motion first hitting time (proof checking)

I need to use it in a small step in the middle of a simulation and I think I'm not getting correct results to this probabilities and so for my all subsequent simulation. Could someone ...
4
votes
0answers
79 views

Markov chains and natural filtration

I have the following problem Consider a homogeneous Markov chain $(X_n)$ with countable state set $E$. Suppose that $A$ is a proper subset of $E$ and consider the stopping times $\tau^0=0 $ and $\...
4
votes
0answers
335 views

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. ...
3
votes
2answers
2k views

Sum of two stopping times is a stopping time?

Let $\sigma$ and $\tau$ be two stopping times in $\mathscr{F}_t$ and let this filtration satisfy all the usual conditions. Question: Is $\sigma + \tau$ a stopping time? Attempt at a solution: I ...
3
votes
1answer
61 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 ...
3
votes
2answers
31 views

Simple question regarding stopping times.

I have this exercise regarding stopping times, but I am not able to solve it. You have a probability space $(\Omega, \mathcal{F},P)$, with a filtration $\{\mathcal{F}_t\}$}. You have two stopping ...
3
votes
1answer
98 views

how to prove $(X_{n})_{n\in \mathbb N}$ and $(Y_{n})_{n\in \mathbb N}$ are supermartingale.and $(Y_{n})_{n\in \mathbb N}$ is convergence to -7

Let $p \in [0 , \frac{1}{2}] $ and $\eta_{i}$ be i.i.d random variables and $P(\eta_{i}=1)=p$ and $P(\eta_{i}=-1)=1-p$ and $\mathcal F_{n}=\sigma(\eta_{1},\cdots,\eta_{n})$ and $X_{n}=\sum_{i=1}^{n}\...
3
votes
1answer
133 views

Find the distribution of $T_a=\inf\{n\ge 0: R_{n}\gt a\}$ for fixed number $a\gt 0$

Suppose $R_{n}=\sum_{i=1}^{n} X_{i}$ for $n\ge 1$ and $R_{0}=0$ , where $X_{i}\gt 0$ are independent and identically distributed. Find the probability law of the stopping time $T_a=\inf\{n\ge 0: R_{n}...
3
votes
2answers
267 views

coupon collector problem for different number of copies of each coupon type

I would like to pose a question on a variation on the classical coupon collector's problem: coupon type $i$ is to be collected $k_i$ times. What is the expected stopping time or the expected number of ...
3
votes
1answer
39 views

A martingale characterization

I saw the following characterization of martingales (without proof) in some lecture notes I found on the web and I haven't been able to produce a proof it. Let $X$ be an adapted process. If $E[X_{\...
3
votes
1answer
49 views

Determine $E\sum_0^\infty X_n1_{(T=n)}$

$X_T = \sum_0^\infty X_n 1_{(T=n)}$ where $T$ is a stopping time and $(X_n)$ is a martingale. Show that if $T$ is bounded then $EX_T = EX_0$: $T \leq N$, and then consider $X_T = X_{T\wedge N} = \...
3
votes
1answer
65 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 ...
3
votes
1answer
52 views

On the proof of lemma 1.2.4 of Stroock and Varadhan A question concerning stopping times

In the book Multidimensional diffusion processes, of Stroock and Varadhan one reads (page 23): This is the proof of $(i)$. Here the authors say Define $f_t$ on $(\{\tau \leq t\}, \mathcal{F}_t ...
3
votes
1answer
108 views

Markov and strong Markov properties

In my study of strong Markov property of an RCLL canonical Markov process I encounter the following definition: Suppose $Y_t:\omega\rightarrow \omega(t)$ is canonical Markov process with respect to ...
3
votes
1answer
118 views

Brownian motion proof of Dirichlet problem

I am reading the proof of the Dirichlet theorem stated in the following form: Theorem: Let $D$ be a bounded domain in $\mathbb{R}^d$ such that every boundary point satisfies the Poincare cone ...
3
votes
1answer
92 views

Tower Property for Expectations and Stopping Times

Let $(\Omega,(\mathcal{F_t})_{t\geq0},P)$ be a filtered proability space with $X\in L^1(P)$ and two stopping times $S$ and $T$. Show that \begin{equation*} \mathbb{E}(\mathbb{E}(X|\mathcal{F}_T)|\...
3
votes
1answer
238 views

The expected time until reaching a specified set in a Markov chain

I am reading an article in which they discuss a specific Markov chain in an example, and it turns out I need to sharpen up my Markov knowledge. First the setup. I have a continuous time Markov chain ...
3
votes
1answer
68 views

Determining if some random variable is a stopping time

I am stuck on this issue: Let $(B_t)$ be a Brownian motion. We know that since $\{0\}$ is a closed set in $\mathbb{R}$ and that $(B_t)$ is a continuous adapted process, $$ \tau:= \inf \{ t\geq 0 : ...
3
votes
1answer
272 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 ...
3
votes
1answer
526 views

Expectation of brownian motion at hitting time

Am i correct in my derivation? I want to calculate $\mathbb{E}B_{\tau_a}$. From the definition of the hitting time i get $B_{\tau_a}=a$, so $$\mathbb{E}B_{\tau_a}=\mathbb{E}a=a$$ I am new to the ...
3
votes
2answers
111 views

Stochastic variables independent given Tau

Say we have a filtration $(\mathbb{F}_s)$, and a stopping time $\tau$ w.r.t. to that filtration.Let $X_t$ be a continuous stochastic process (not required to be adapted to the mentioned filtration), ...
3
votes
1answer
115 views

Proof of stopping theorem for bounded stopping times

Let $\tau$ be a bounded stopping time and $X=X_n$ a martingale. Then $X_\tau$ is integrable and $E(X_\tau)=E(X_0)$. I need help with the proof at discrete time, at one step I am not sure I understood,...
3
votes
1answer
113 views

Hitting time process of Brownian motion [closed]

I am stuck with this problem: Let $(B_t)$ be a standard Brownian motion in $\mathbb{R}$. For $t \geq 0$, let $$ H_t = \inf \{ s \geq 0 : B_s = t \}, \quad S_t = \inf \{ s \geq 0 : B_s > t \}. $$ ...
3
votes
1answer
102 views

Extended (or augmented) stopping times

I am trying to prove that $\tau$, defined as: $$ \tau = inf\{t > 0 \mbox{ }|\mbox{ } B_t < t-1 \} $$ is a stopping time with respect to the filtration $(\mathscr{F}_{t+}^B)_{t\geq 0}$ where $\...