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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Is $B_{t\wedge H_a}$ bounded in $L^2$?

Let $a >0$, $(B_t)_{t\geq0}$ be a standard Brownian motion. Define the stopping time $$H_a := \inf\{t \geq 0 : B_t \geq a\}.$$ Then is the martingale $M_t$ where $M_t: = B_{t\wedge H_a}$ bounded ...
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21 views

Brownian Motion maximum process intuition

I am studying the maximum value of a Brownian Motion (BM) on an interval of time (as explained here between boxes 28 and 40) and I am having an issue aligning intuition with the mathematical result. ...
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1answer
36 views

The Second Hearts Problem

According to the last part of these lecture notes, if we have a standard deck of playing cards and turn cards until the first heart appears, the probability that the next card is a heart is ...
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16 views

Finding optimal strategy in time series game

Let $P_t$ be a time series such that $P_{t+1} = \alpha P_t+S_{t+1}$, where $\forall t\geq0 : S_t \sim N(0,\sigma)$ Consider the following game: In each round $t$, a player sees $P_t$ and decides ...
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1answer
33 views

Expected time until pattern (1,0,0,1)

Let $(X_n)_{n\geq 0}$ be i.i.d. with $\mathbb P(X_n = 0 ) = \mathbb P(X_n = 1) = \frac{1}{2}$. Let $\tau_a$ be the stopping times defined as $$\tau_a = \inf\{n: (X_{n-3}, ... , X_n) = (1,0,0,1)\}$$ I ...
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1answer
43 views

Conditional independence of stopping times from i.i.d. stochastic processes

My question is somewhat arbitrary but I was thinking about independence of processes and stopping times. Say that we define two processes $X,Y$ on different probability spaces ...
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33 views

Sigma algebra generated by the stopped process.

Let $(X_n)_{n \geq 0}$ be a sequence of random variables. Let $\mathcal{F}_n = \sigma (X_0, \dots, X_n)$ be a filtration and $T$ is a $(\mathcal{F}_n)_{n\geq 0}$-stopping time. I want to understand ...
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16 views

Understanding of Second Arcsine law for Brownian motion

Ok I'm trying to understand the second arcsine law which states: Let $g_t:=\sup\{s\leq t:W_s=0\}$, then $$\mathbb{P}(g_t\leq s)=\frac{2}{\pi}\arcsin \left(\sqrt{\frac{s}{t}}\right )$$ This won't be ...
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13 views

Conditional expectation of a geometric Brownian motion and stopping time

Let $X$ be a geometric Brownian motion, solution of $dX_t = \mu X_t dt + \sigma X_t dW_t, X_0 > 0$ and ${\cal F}$ its natural filtration. Let's consider $t \geq 0$ and $\tau_a$ the first hitting ...
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2answers
52 views

Where is my mistake in calculating stopping time?

A gambler has $w$ dollars and in each game he loses or wins a dollar with equal $p=1/2$ probability. I want to calculate the average number of games that a gambler can play before he runs out of ...
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22 views

Geometric Brownian motion hitting time

Let $X$ be a geometric Brownian motion $dX_t = \mu X_t dt + \sigma X_t dW_t, X_0 > 0$ and ${\cal F}$ its natural filtration. Let $\tau_a$ be the first hitting time of $a$ by $X$. How can we relate ...
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1answer
58 views

Strong Markov property and time homogeneity

Let $X$ be a Markov chain with state space $\mathcal{S}$ and denote $\mathbb{N} := \{ 0, 1, \cdots\}$. We know that for any stopping time $\tau < \infty$ and any bounded measurable function $\phi : ...
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1answer
23 views

Recurrence of a state in a finite state space

Suppose $T_A := \inf\{ n \ge 1 : X_n \in A\}$ where $A \subset \mathcal{S}$ is finite. Assume $\mathbb{P}\{T_A < \infty \; | \; X_0 = x\}= 1$ for $\forall x \in \mathcal{S}-A$. I need to show that ...
3
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0answers
60 views

Strong Markov property proof

Let $X$ be a Markov chain with state space $\mathcal{S}$ and denote $\mathbb{N} := \{0,1, \cdots\}$. I need to show that for any stopping time $\tau < \infty$ and any bounded measurable function ...
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1answer
32 views

$T$ can be $\infty$ with positive probability

From Williams' Probability with Martingales How exactly do we know $T$ can be $\infty$ with positive probability or $$P(T = \infty) > 0 \text{ ?}$$ I'm guessing that that means there ...
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2answers
40 views

Expectation of stopping time on a random walk

Assume $X_1 , X_2 , \cdots$ are i.i.d. with distribution Bernouli$(\frac{1}{2})$, i.e., $P(X_i = 0)=P(X_i=1)=\frac{1}{2}$. Denote $S_0 := 0$, $S_n := \sum\limits_{i=1}^n X_i$, and $\tau_{1000} := ...
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1answer
27 views

Prove $A_{\infty} < \infty$?

From Williams' Probability with Martingales How do we know that $A_{\infty} < \infty$? If $T = \infty$, then $$E[A_{T \wedge n}] \le (K+c)^2$$ $$\to E[A_{n}] \le (K+c)^2$$ $$\to ...
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0answers
21 views

Prove $|M_{T \wedge n}| \le c + K$

From Williams' Probability with Martingales Is $\sigma_k^2$ random (and not constant)? How can that be? As far as I know unconditional variance and unconditional expectations are supposed ...
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1answer
14 views

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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20 views

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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28 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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36 views

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
35 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
18 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 ...
3
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1answer
28 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
51 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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26 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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1answer
62 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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0answers
23 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 ...
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1answer
41 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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1answer
35 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
26 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 ...
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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
47 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 ...
2
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1answer
39 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
60 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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0answers
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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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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1answer
39 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
48 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
23 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 ...
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2answers
107 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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0answers
45 views

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 ...
3
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1answer
63 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 ...
7
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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 ...
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1answer
50 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
34 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
31 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 ...