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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253 views

Stopping times, Filtration, Martingales,

I am new here and I have a question. Definition: Let $ \tau$ be a stopping time, then $\mathcal F_{\tau}=\left\{F\subset \Omega: \forall n \in N \cup \{\infty\} , F\cap(\tau\leq n)\in \mathcal F_{n}\...
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0answers
29 views

A problem about the supremum of countable stopping times

Let $\left(\Omega\ ,\mathcal{F}\ ,\mathbb{P} \right)$ be a probability space with a countable filtration $F=\left\{ \mathcal{F}_0,\mathcal{F}_1,\cdots,\mathcal{F}_n,\cdots \right\}$. $\left\{ T_n \...
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2answers
50 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 $\color{...
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0answers
42 views

Hitting times in two-dimensional case: expectation of Brownian motion at a hitting time

Consider two Brownian motions $$X_{1t}=\mu t+\sigma_1B_{1t}$$ and $$X_{2t}=\mu t+\sigma_2B_{2t}.$$ Here $B_{1t}$ and $B_{2t}$ are uncorrelated. Let $\tau_1$ and $\tau_2$ be the stopping times: \begin{...
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11 views

Reference for *optimal* stopping theorem for supermartingales

Can anyone introduce a good reference about optimal (not optional!) stopping times for submartingales / supermartingales? I am looking for some theorem like the one mentioned in this question. I ...
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0answers
39 views

$\langle M_{S(k) \wedge n}\rangle = A_{S(k) \wedge n}$ - definition?

Probability with Martingales: What is the relation between $\langle M_{S(k) \wedge n}\rangle \ = A_{S(k) \wedge n}$ and $\{N_n\}, \{ N_{ S(k) \wedge n } \}$ being martingales? It seems that $$\...
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0answers
23 views

Prove $\lim M_{S(k) \wedge n}$ exists a.s. if $S(k) = \infty$. Is $N_n \ge 0$?

Probability with Martingales: Why does $\lim M_{S(k) \wedge n}$ exist a.s.? Is it connected to $$\sup E[M_{S(k) \wedge n}^2] < \infty$$ ? What I tried: My approach is to use: If $\lim ...
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0answers
52 views

Prove $A^{S(k)}$ is previsible

Probability with Martingales: I have a different attempt in mind, but I'm guessing it's wrong because if it were right, the book would've used it. It seems that we must show that $$A_{S_k \...
2
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0answers
31 views

Distribution of Markov Chain at a Stopping Time

Suppose $(X_t)_{t \geq 0}$ is a Markov chain on the state space $S$ with transition probability $p$, and that $\pi$ is a stationary distribution for $p$. If $X_0 \sim \pi$, then we know $X_t \sim \pi$ ...
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0answers
35 views

If a process is previsible, is the stopped process previsible? [closed]

Assume we have a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $A = \{A_n\}_{n \in \mathbb N}$ is an $\{\mathscr F_n\}_{n \in \mathbb N}$-...
2
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1answer
74 views

The jumping times of a càdlàg process are stopping times.

Protter first proves this theorem: Let $X$ be an adapted càdlàg stochastic process, and let $A$ be a closed set. Then the random variable: $T(\omega)=\inf\{t: X_t(\omega)\in A \text{ or } ...
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1answer
27 views

Prove $M_{S(k) \wedge n}$ is bounded in $\mathscr L^2$

Probability with Martingales: To prove $$\sup E[M_{S(k) \wedge n}^2] < \infty,$$ how can we use 12.12c? There aren't any stopping times there.
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1answer
48 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 \lim ...
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0answers
27 views

Independence between the first exit time from an interval and the value of Brownian motion at this first exit time

Suppose you have an arithmetic Brownian motion (or Brownian motion with drift ) called X, started at a level x such that a < x < b, where a and b are two real points . Define tau as the first ...
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1answer
19 views

On a condition for a.s. finite stopping times

Assume that $\{\tau_n: n \in \mathbb{N}\} $ is a sequence of stopping times with respect to some filtration such that $P[\tau_n < \infty] = 1. $ Is that true that there must exist a sequence of ...
5
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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 ...
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1answer
29 views

Prove $S_k$ is a stopping based on $A$ being previsible

Probability with Martingales: It looks like we have a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $A = \{A_n\}_{n \in \mathbb N}$ ...
0
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1answer
27 views

Check that stopping time is a.s. finite

I have the following situation. Let $(X_i)_{i\geq1}$ be a sequence of iid random variables in $\mathbb{Z}$ and consider the random walk $S_n=\sum_{i=1}^n{X_i}$, $S_0=x$. Let $y>x$ and consider ...
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0answers
25 views

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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0answers
23 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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0answers
17 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 ...
2
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1answer
45 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 $(\Omega^i,\mathcal{F}^...
1
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1answer
49 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}\ \ B_t=-\beta\...
2
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0answers
38 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 ...
1
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0answers
19 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 ...
0
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1answer
59 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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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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0answers
19 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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0answers
23 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
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
63 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 $\...
2
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1answer
39 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
42 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} := inf\{...
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0answers
26 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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235 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 ...
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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. $\mathcal{B}(...
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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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1answer
42 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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43 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\} = \bigcap_{n=1}^{\infty}\...
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
19 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
65 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 ...
3
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1answer
30 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
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 $E[X_N]< \infty$?
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0answers
33 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 ...
1
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0answers
24 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 ...