Questions tagged [stopping-times]

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 $n$, $\{X_0, . . . , X_n\}$.

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1answer
25 views

Is the supremum of a right continuous process right continuous? Defining a stopping time as the first hitting time of a supremum process

This is a question that came to my mind while reading Lemma 13 of the notes here : https://almostsuremath.com/2009/12/23/localization/ In this lemma, the writer assumes $X$ is a right-continuous ...
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1answer
63 views

$\{\omega \ge \tau\} \in \mathscr{F}_{\tau -}$ for stopping times

Let $\tau, \omega$ be two stopping times. How do we show that $\{\omega \ge \tau\} \in \mathscr{F}_{\tau -}$? This result is used in the proof of Lemma 11 from : https://almostsuremath.com/2009/12/23/...
4
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1answer
72 views

Understanding proof of discrete optimal sampling theorem

Let $X = \{X_n\}_{n=0}^{\infty}$ be a closable submartingale. Then, for any stopping time $τ, X_τ$ is integrable and, for another stopping time $σ$, $E[X_\tau |\mathcal{F}_\sigma ]\ge X_{\sigma\wedge \...
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0answers
24 views

Why can we use limit inferior to calculate the expected value of a stopped process?

Consider ($\tau_n$) a diverging sequence of stopping times (e.g. $\inf\{t: X_t>n\}$). We can write the stopped local martingale $X_t^{\tau_n}$ = $X_{t\wedge \tau_n}$, which yields $\lim_{n\...
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1answer
91 views

Trouble with extending Doob's Optional Stopping Theorem

Let $\tau\geq0$ be a stopping time, $\mathbb{E}\tau<\infty$. Show $\{\tau\geq k\}\in\mathcal{F}_{k-1}$. Based on the identity $$ |X_{T\land n}|= \bigg|\sum_{k=1}^n(X_k-X_{k-1})\cdot\mathbf{1}\{\...
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2answers
50 views

Martingale and finite stopping time [closed]

Let $(X_n, \mathcal F_n)_{n\geq 0}$ be a martingale and $\tau$ be a a.s. finite stopping time and $\mathbb{E}(|X_\tau|) < \infty$. Could someone provide me an example where $\lim \inf\mathbb{E}(1_{...
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1answer
45 views

Show $\mathbb{E}_1\big[S_n\textbf{1}_{\{T_{\text{hit zero}} > n\}}\big] \not\to 0$ for SSRW.

Suppose we have a simple symmetric random walk starting at $1$, and define $$T_{\text{hit zero}} = \min\{n\geq 1 : S_n = 0\}.$$ I was trying to argue that $$\star =\mathbb{E}_1\big[S_n\textbf{1}_{\{T_{...
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0answers
41 views

Stopping times and convergence of indexes

Let $(\Omega,\mathcal F, P)$ be a filtered probability space. Let $n>0$ be a natural number, $T\in \mathbb R$ such that $T>0$, and $\delta=\frac{T}{n}$. Let $(\sigma^n_i)^n_{i=0}$, $(\tau^n_i)^...
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20 views

Bounds on expected stopping time

I have a continuous-time model and want to put a lower bound on the expected stopping time. The process stops when some random variable hits 0. Imagine sand pouring from an hourglass each second in a ...
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1answer
43 views

Possible example of stopped martingale not being in $L^1$

I am looking at Lawler's stochastic process book, and for the optional stopping theorem, under the assumption for the stopping time that $\mathbb{P}(T<\infty) = 1$, he had $\mathbb{E}[|M_T|]< \...
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49 views

Does inequality between stopping times depend on $\omega$?

We work on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in[0,T]},\mathbb{P})$, where $T$ is finite. Let $(\tau_n)$ be a sequence of stopping times increasing stationarly to $T$,...
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1answer
52 views

Analytical solutions to $E[f(X_\tau) e^{-\alpha\tau}]$

Let $X_t$ be a geometric Brownian motion: $$dX_t=\alpha X_tdt+sX_tdB_t,$$ where $\alpha,s>0$ are constants. Let $\tau=\inf_{u\geq t}\{X_u=K\}$ be the first time $X_t$ reaches the threshold $K$ (...
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1answer
93 views

Is it true $P(\sup_{k \in \mathbb{N}}X_k \geq \epsilon +x)=\dfrac{x}{\epsilon+x}$?

Let $(X_k)_k$ be a non-negative martingale such that $\lim_{k \to \infty}X_k=0$ a.s. and $X_0=x\in ]0,\infty[.$ Prove or disprove the following: $$\forall\epsilon>0,P(\sup_{k \in \mathbb{N}}X_k \...
3
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1answer
70 views

Optimal stopping of a Poisson Process with a risky reward

I'm confident that there is a well-known solution to this problem, but I am having trouble finding a reference for it. I am also quite rusty on these kinds of problems, so I am having trouble solving ...
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32 views

Reference request: solving SDEs up to exit-time of the driving Brownian motion.

I wonder if someone out there has ever studied stochastic differential equations where the time-interval depends on the Brownian motion; i.e. \begin{align} \begin{cases} dX_t=f(t,X_t)dt+\sigma(t,X_t)...
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show that the limit of a continuous time process with discrete values is a martingale

Let $M_t$ an adapted, a.s. bounded process taking values on the integers. So it is a continuous time prcess with discrete values. Define: $P_t^i=\mathbb{E}[M_{\tau_i}|\mathcal{F}_t]$ Where stopping ...
5
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1answer
94 views

Conditions on stopping time being finite

Let $(Y_n)_{n \in \mathbb{N}}$ be independent random variables taking values in $\{-1, 0, 1\}$ such that $EY_n = 0$. Let the process $(X_n)_{n \in \mathbb{N}}$ with $X_n = \sum_{k = 1}^n Y_k$. Let $\...
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1answer
25 views

Why is this process bounded?

Let $X$ be a random variable such that $P(X=1) = p$ and $P(X = -1) =1-p$. Let $(X_j)_{j \in \mathbb{N}}$ be a sequence of independent copies of $X$. We define a process $M = (M_n)_{n \in \mathbb{N}}$ ...
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1answer
46 views

$\tau=s \mathbf{1}_{A^c}+t\mathbf{1}_A$, $A \in \mathcal F_s$ is a stopping time

Let $(\mathcal F_t)_{t\in T}$ be a filtration. Consider $s<t$ in T and $A \in \mathcal F_s$. I want to show that $\tau=s \mathbf{1}_{A^c}+t\mathbf{1}_A$ is a stopping time. My thoughts: In both ...
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0answers
30 views

biased simple random walk and optional stopping theorem

Suppose we have a biased simple random walk $S_K=X_1+\dots+X_k$ on integers which begins at $0$, where $X_l$s are i.d.d random variables such that $P(X=1)=p=1-P(X=-1)>1/2$. It's easy to check $EX=p-...
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strong stationary time

I would like to prove this two definitions of strong stationary time is equivalents. However, one of the sides has already managed to resolve. I want to prove: $\displaystyle P\{X_k = i |T \le k\} = \...
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0answers
56 views

Reference for a stopping-time problem

I am looking for a reference (perhaps an example in a text or an article) for the following stopping-time problem. I suspect the problem is well known. Random variables in an i.i.d. sequence $\langle ...
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1answer
33 views

First Passage Time distribution for a 2D Brownian particle with constant drift

Let's suppose we have an active Brownian particle whose overdamped equation of motion is given by \begin{equation} \begin{cases} \dot{x} = f + \sqrt{2D}\xi_x \\ \dot{y} = \sqrt{2D}\xi_y \end{cases} \...
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1answer
63 views

How to show that $n1_E + m1_{\Omega \setminus E}$ is a stopping time for $E\in\mathcal{F}_n$?

For $n,m\in\mathbb{N}_0$, $n<m$, and $E\in\mathcal{F}_n$, how to show that $n1_E + m1_{\Omega \setminus E}$ is a stopping time? Should I break it into cases, $\omega\in E$ and $\omega\in \Omega\...
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1answer
24 views

How does $\{S\wedge t \leq T \wedge t\}$ help here?

We want to prove the following implication : $$A\in \mathcal F_S \implies A\cap \{S\leq T\} \in \mathcal F_T$$ Since $A\in \mathcal F_S$, we have $A\in \mathcal F_\infty$ and $A\cap \{S\leq t\} \in \...
3
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1answer
66 views

Difference of two stopping times

Let $(X_{n})_{n \geq 0}$ be a sequence of random variables and $\tau,t$ stopping times with respect to the sequence $(X_{n})_{n \geq 0}$ $$\begin{align*}\{\tau+t =n\} = \{\tau+t = n\} \cap \{t \leq n\}...
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0answers
23 views

Is it possible to recover the fact that $\mathbb P(T_{a} < T_{b} ) = b/(b-a)$ using $\mathbb E[ e^{-\lambda T_{a}}\chi_{\{T_{a}< T_{b}\}}]$

Consider two hitting times $T_{a},T_{b}$ for a Brownian motion $B$ such that $a < 0 < b$. Is it possible to recover the fact that $\mathbb P(T_{a} < T_{b} ) = b/(b-a)$ using the fact that I ...
4
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1answer
44 views

A Markov Chain ($Y_n$) based on the stopping time of another Markov Chain $X_n$

Problem Setting: If $(X_n)^{\infty}_{n=0}$ is a homogeneous Markov chain with state space $S$ and a transition matrix $P = (p_{ij})$ where $p_{ii} < 1$ for all $I \in S$ and consider $(Y_n)^{\infty}...
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1answer
32 views

Given a homogeneous $X_n$ Markov chain, how can we show stopping time with respect to $X_n$ [closed]

Problem Setting: If $(X_n)^{\infty}_{n=0}$ is a homogeneous Markov chain with state space $S$ and a transition matrix $P = (p_{ij})$ where $p_{ii} < 1$ for all $I \in S$ and consider $(Y_n)^{\infty}...
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1answer
39 views

Stopping time build by 'filtration'-Brownian motion

So I got following problem: Let $B_t$ be a $\{H_t\}_{t \in \mathbb{R}_+}$ Brownian motion where $\{H_t\}_{t \in \mathbb{R}_+}$ is a right-continuous complete filtration and consider $$S_t := \inf \...
4
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1answer
71 views

Show that $\mathbb E[\exp(-\lambda(T_{a}\land T_{b}))]=\cosh(\frac{a+b}{2}\sqrt{2\lambda})/\cosh(\frac{a-b}{2}\sqrt{2\lambda})$

Let $a < 0 < b$ and further for a standard Brownian motion $(B_{t})$ define the stopping times $T_{x}:=\inf\{t \geq 0: B_{t}=x\}$. Show that: $$\mathbb E[\exp(-\lambda(T_{a}\land T_{b}))]=\cosh(\...
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0answers
24 views

strong uniform time

I would like to understand the definition below. A strong uniform time $T$ is a randomized stopping time for $\{X_n\}_{n \ge 0}$, where $X_n$ is a markov chain, such that (i) $\displaystyle P\{X_k = ...
2
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1answer
47 views

Interpretation of Stopped $\sigma$-algebra

I'm currently learning the definition of a stopped $\sigma$-algebra: Let $T$ be a stopping time. We define the stopped $\sigma$-algebra $\mathcal{F}_T$ as $$ \mathcal{F}_T := \{A \in \mathcal{F}\mid\...
3
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2answers
184 views

Showing that a stopping time is finite for a biased random walk.

If we consider $X_i$ iid with $\mathbb{P}(X_i=1) = p$ and $\mathbb{P}(X_i=-1)=1-p$. Where $p \in (1/2,1)$. The random walk is then given by, $$S_n=\sum_{i=1}^n X_i. $$ We also define the stopping time ...
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0answers
70 views

Is $\inf\{t>0:\Delta\omega(t)\in B\}$ a stopping time on the canonical path space?

Let $E$ be a normed $\mathbb R$-vector space and $$\Omega:=\{\omega:[0,\infty)\to E\mid\omega\text{ is càdlàg}\}.$$ Moreover, let $$\pi_t:\Omega\to E\;,\;\;\;\omega\mapsto\omega(t)$$ and $$\mathcal ...
0
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1answer
36 views

Distribution of a stochastic process at a stopping time.

Suppose I have a continuous-time stochastic process $\{X(t)\}$ defined on a filtered probability space $(\Omega, \mathcal{F},\{\mathcal{F}_t\},\mathbb{P})$ and that I know the distribution of a ...
3
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2answers
157 views

Prove that $ P(T>n)=P\left(X_{n}>Y_{n}\right)-P\left(X_{n}<Y_{n}\right) $

I am having a very hard time proving the below statement. I keep getting the wrong result so I feel like I am using the wrong probability identities. I would appreciate any help! $\operatorname{Let}\...
1
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1answer
51 views

How do we show that the jumping times $\inf\{t:\Delta X_t\in B\}$ of a Lévy process $X$ are stopping times?

We can show the following: If $E$ is a normed $\mathbb R$-vector space, $x:[0,\infty)\to E$ is càdlàg, $B\subseteq E\setminus\{0\}$, $\tau_0:=0$ and $$\tau_n:=\inf\underbrace{\{t>\tau_{n-1}:\Delta ...
3
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1answer
63 views

Probability of hitting time conditional on the last step of Simple random walk.

I have a trouble proving the below result: Let $\left(S_{n}\right)_{n \geq 0}$ be a simple random walk defined by $p(1)=p$ and $p(-1)=q$ where $p+q=1$ Let $k$ be a positive integer. Prove: $$ P_{0}\...
3
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2answers
105 views

If $x:[0,\infty)\to E$ is càdlàg and $\tau_0:=0$, show that $\inf\left\{t>\tau_{n-1}:\Delta x(t)\in B\right\}\xrightarrow{n\to\infty}\infty$

Let $E$ be a normed $\mathbb R$-vector space and $x:[0,\infty)\to E$ be càdlàg with $x(0)=0$. Moreover, let $x(0-):=x(0)$, $$x(t-):=\lim_{s\to t-}x(s)\;\;\;\text{for }t>0$$ and $$\Delta x(t):=x(t)-...
6
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2answers
118 views

Toss a coin until you get $n$ heads in a row. Expected number of tails tossed?

You toss a fair coin until you get $n$ heads in a row. What is the expected number of tails you tossed? It can be well shown that the expected number of tosses required to get $n$ heads in a row is $2^...
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1answer
45 views

Whether the optional stopping theorem is true for a positive continuous martingale and integrable stopping times

We know that for a uniformly integrable continuous martingale $M$ there holds the optional stopping theorem for all stopping times: $E[M_T|\mathscr{F}_S]=M_S$ for stopping times $S\leq T$. I'm ...
0
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1answer
16 views

If the jumps of a path $x$ are bounded by $c$, how do we show that implies $\|x(t)\|\ge2c$ implies $\inf\{s:\|x(s)\|\ge c\}<t$

Let $E$ be a normed $\mathbb R$-vector space and $x:[0,\infty)\to E$ be càdlàg with $x(0)=0$. Moreover, let $x(0-):=0$, $$x(t-):=\lim_{s\to t-}x(s)\;\;\;\text{for }t>0$$ and $$\Delta x(t):=x(t)-x(t-...
2
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2answers
101 views

If $X$ is a Lévy process, is $\tau_n:=\inf\left\{t\ge\tau_{n-1}:\left\|X_t-X_{\tau_{n-1}}\right\|_E\ge c\right\}$ independent identically distributed?

Let $E$ be a normed $\mathbb R$-vector space and $(X_t)_{t\ge0}$ be an $E$-valued Lévy process on a filtered probability space $(\Omega,\mathcal A,(\mathcal F_t)_{t\ge0},\operatorname P)$, i.e. $X_0=...
0
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1answer
32 views

Show that the hitting time of a closed set is stopping time

Let $(E,d)$ be a metric space and $B\subseteq E$ be closed, let $x:[0,\infty)\to E$, $I:=\{t\ge0:x(t)\in B\}$ and $\tau:=\inf I$. If $I$ is nonempty and $\tau\in I$, then we easily see that $$\tau\le ...
0
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1answer
43 views

Are $1+\exp(-B_{1}^2)$ and $\inf\{t \geq 0 : B_t \geq W_t + \exp(-t)\}$ stopping times?

I have got difficulties with an exercise on stochastic processes. Let $B$ and $W$ be two independent Brownian motions on filtration $(\mathcal{F}_t)_{t\geq 0}$ Are $\lambda$ = $1+\exp(-B_{1}^2)$ and $...
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0answers
21 views

Iterated optional times

In the following proposition of Kallenberg's book Foundation of Modern Probabilities, where $\theta_t$ is the shift operator defined by $\theta_t(\omega)(s)=\omega(t+s)$, Proposition 11.8 (iterated ...
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2answers
47 views

Expected value inequality for $X_n^T$

Let $X_n$ be a bounded random process and let $T$ be a stopping time that is finite almost surely. Suppose we have the inequality $E(X_n) \geq c$, where $c$ is a constant. Does it follow that $E(X_n^T)...
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0answers
24 views

Interpertation $X_{\tau}$

Let's say $X_i$ iid with $P(X_i=1)=P(X_i=-1)$ and $S_n=\sum_{i=0}^nX_i$. We also define the stopping time $\tau= \inf\{k:X_k=1 \}$. How do you interpret the random variable $X_\tau$?
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0answers
13 views

Computational complexity based on stopping time

I need help determining the asymptotic computational complexity of a Gibbs sampling algorithm. I believe this algorithm is related to stopping time, but I know very little about stopping time problems....

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