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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Strong Markov Property Brownian Motion for Non-Stopping Time

Let $B$ be a Brownian motion and let $\mathcal{F}^B$ be its natural filtration. Define the random variable $$ \tau = \inf\{ t \ge 0 \mid B_t = \sup_{0 \le s \le 1} B_s \}.$$ Now, $\tau$ is not an ...
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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 ...
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Expectation related to Wiener process using strong Markov property

Can you help me to understand a result I found in Krylov's book "Introduction to stochastic calculus". First, I will introduce some notations: $w_t,t\ge 0$ denotes a Wiener process. ...
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27 views

expectation of stopping time in Wiener process

Let $(W_t)$ be a Wiener process and for $a>0$ define stopping time: $$\tau = \inf \left\{t>0: W_t + at = 5\right\}$$ a) show $\tau < \infty$ a.s; b) compute $\mathbb{E}\tau$. I have done ...
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1answer
37 views

Martingale property cannot hold for general random times

Let $\sigma \leq \tau$ be two random times that are no stopping times. I want to create a simple example that shows that for these random times $\mathbb{E}[M_\tau \mid \mathcal{F}_\sigma] = M_\sigma$ ...
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Brownian Motion hitting time is finite yet has infinite expectation?

I've read that a hitting time of a Brownian motion (defined as $T_a = \inf\{t\ge0:W_t=a\}$ where $W_t$ is a standard Brownian Motion, i.e. a Wiener process), has the following two properties, which I ...
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1answer
18 views

identity with Poisson process

I want to show (if it is true) that if $\tau$ is a stopping time and $\mathbb{E}\tau < \infty$ then $\mathbb{E}N_\tau = \mathbb{E}\tau$ for $N_t$ - Poisson process with parameter $1$. I started ...
2
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1answer
34 views

Brownian hitting time of a closed set

I am trying to prove that the first hitting time of a closed set H by a Brownian motion is a stopping time. I have found a proof that states: $$\{\mathcal{T}\leqslant t\} = ...
3
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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} = ...
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1answer
15 views

Distribution of the first exit time of a one-dimensional diffusiom/ Brownian motion

I have a one-dimensional diffusion on $[0,1]$ and I need to calculate the distribution of the first exit time of the interval $(-\epsilon,\epsilon)$ for an $\epsilon > 0$. A good first step would ...
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1answer
56 views

Counterexample for uniform integrability of a stopped process

I want to find an example where $X_n$ is uniformly integrable, $N$ is a stopping time, but $X_n^N = X_{\min\{n,N\}}$ in not uniformly integrable. There is a theorem saying that if $M_n$ is a uniform ...
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1answer
25 views

Exercise on stopping times

Let $(Y_n)_{n \geq 1} $ be a sequence of independent r.v.'s s.t. $$P(Y_n=y) = {n \choose k } \left(\frac1n\right)^y \left(1-\frac1n\right)^{n-y}\quad {\rm if }\;y \in \{0,1,\dots,n\}$$ How to show ...
4
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1answer
44 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 $E[X_N]< \infty$?
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16 views

Stopped process not uniformly integrable

I need to construct a counter example such that the process $\{X_n\}_{n \ge 1}$ is uniformly integrable; however, the stopped process $X_{\tau \wedge n}$ where $\tau$ is a stopping time, is NOT ...
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19 views

Stopped process of maximum stopping times

Suppose $X$ is an adapted process and $\tau_1, \ldots , \tau_k$ are stopping times such that $X^{\tau_1}, \ldots , X^{\tau_k}$ are all martingales. I want to show that then $X^{\tau_1 \vee \ldots \vee ...
4
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1answer
94 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 ...
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1answer
31 views

Question regarding proof of property related to the stopped sigma-algebra.

I have a proof of a property regarding the stopped $\sigma$-algebra, where one part I do not understand, I'll highlight what I do not get, can you please help me? We have a probability space ...
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1answer
22 views

Martingale representation theorem , optimal stopping time and the principal agent problem

I am self-learning some Econ papers. Any suggestion will be appreciated. Even though the questions are from an Econ paper, they are math-related. I provide the economic interpretation as background ...
3
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2answers
27 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 ...
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1answer
44 views

Showing a stopping time is finite

Let $T = \inf\{ n : S_n = a \text{ or } S_n = -b\}$ be a stopping time, where $S_n = X_1 + \dots +X_n$ and each $X_n$ is a martingale. I am looking at a proof which shows that $T < \infty$ almost ...
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1answer
33 views

Stopping Times, the $\inf$ is not a stopping time

I'm having a hard time figuring out why the infimum of a sequence of stopping times is not necessarily a stopping time itself. Indeed, the justification my book gives me is that: Given $(\mathcal ...
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1answer
57 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 ...
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34 views

Finiteness of the hitting time of random walk

Let $X_1,X_2,\ldots$ be an iid sequence such that $P\{X_1 = 1\} = u$, $P\{X_1 = -1\} = d$ and $P\{X_1 = 0\} = 1-(u+d)$. We have that $E[X_1] \neq 0$. Define $S_n = \sum_{i=1}^nX_i$ and $S_0 = 0$ and ...
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2answers
101 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 $\tau \ge \sup_i \tau_i$, ...
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27 views

defining the stopping time sigma algebra

For a stopping time T, define $\mathcal{F}_T$ by $\mathcal{F}_T={A \in \mathcal{F}:A \cap \{T \le t\} \in \mathcal{F}_t, \text{for each t.}}$ Verify that $\mathcal{F}_T$ is a $\sigma$-algebra. ...
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26 views

Hitting times for Brownian Motion (2)

In this post there is shown that for a standard Brownian motion $\mathbb{E}[\tau^p]<\infty$ for all $p \geq 1$, where \begin{align} \tau = \inf\{t \geq 0 : B_t = \alpha \ \ \text{or}\ \ ...
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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 ...
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1answer
22 views

Martingales with bounded increments

It is known that if $T$ is a stopping time such that $E[T] < \infty$ and $(M_n)$ is a martingale with bounded increments, i.e. $\lvert M_n - M_{n-1}\rvert \leq K < \infty$ for every $n$, almost ...
3
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1answer
56 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 ...
0
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1answer
61 views

Is $\tau = \inf \{n :X_1+\cdots+X_{n-1}\leq 1\leq X_1+\cdots+X_{n-1}+X_n \}$ a stopping time?

Let $X_1, X_2, ...$ be a sequence of i.i.d. non-negative random variables. Let $$\tau = \inf \{n :X_1+\cdots+X_{n-1}\leq 1\leq X_1+\cdots+X_{n-1}+X_n \}.$$ Is $\tau$ a stopping time for the ...
7
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1answer
95 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 ...
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Conditional Expectation.

Let $X$ be a random variable in $L^2(\Omega, \Sigma, P)$ and $\mathcal G$ a sub-$\sigma$-algebra of $\Sigma$. Prove that $E[(X-E[X\mid\mathcal G])^2] \le E[(X-E[X])^2]$. As conditional expectation ...
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1answer
42 views

Stopping time $\sigma$-algebras

Let $\tau$ be a stopping time and fix $t \in \mathbb{R}_+$. Let $A \in \mathcal{F}_t$ satsify $A \subseteq \{\tau \leq t\}$. Now, I want to prove that $A \in \mathcal{F}_\tau$. A stopping time is a ...
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1answer
33 views

Stopping time and random variable measurability

Let $\sigma$ be a stopping time and $Z$ a $\mathcal{F}_{\sigma}$-measurable random variable. Now, I want to show that for any $A \in \mathcal{B}_{[0,t]}$, $\mathbb{1}_{\{\sigma \in A\}} Z$ is ...
0
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1answer
30 views

If a stopping time, $T$, satisfies $P(T>k\alpha)\leq (1-\epsilon)^k$ then $E(T)<\infty$

Suppose that $T$ is a stopping time such that for some positive integer $\alpha$ and some $\epsilon>0$ we have for every $n$: $$ P(T\leq n+\alpha|\mathcal{F}_n)>\epsilon \text{ a.s.} $$ I ...
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1answer
66 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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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 ...
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2answers
65 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 = ...
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1answer
56 views

Submartingale and stopping time

Let {$X_1, \dots, X_n$} be a submartingale, and let $T$ be a stopping time for {$X_i, 1\leqslant i \leqslant n$}. Show that $ E(\mid X_T \mid) \leqslant 2E(X_n^+)-E(X_1)$. The corresponding result for ...
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1answer
31 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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40 views

Taylor expansion and Ito when the value function has a non differentiable point

I am trying to solve a multi-period free boundary problem, of an Ornstein–Uhlenbeck process, where each stopping decision at each period adds a different constant (penalty or bonus). Solving with a ...
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21 views

Characterization of Moment Generating Function of Hitting Time

Suppose that $x_t$ follows the stochastic differential equation: \begin{align*} dx_t = (a - b x_t) dt + \sigma x_t dB_t \end{align*} Where $B_t$ is a standard one-dimensional Brownian motion, and ...
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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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1answer
25 views

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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1answer
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 ...
3
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32 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: ...
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1answer
26 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 ...
1
vote
1answer
43 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
35 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 ...
4
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1answer
61 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 ...