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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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Optimal permutation of transition probabilities in random walk to minimize expected stopping time

Background Consider the set of integers $\{1,\dots,n+1\}$ and a set of probabilities $p_1,\dots, p_n \in(0,1)$. We now define a random walk/Markov chain on these states via the following transition ...
whpowell96's user avatar
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1 vote
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Intuitive understanding of the definition of the $\sigma$-algebra of a stopping time $\tau$

I would like to better understand the basic intuition behind the definition of the $\sigma$-algebra of a stopping time $\tau$. Definition. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space ...
Quasar's user avatar
  • 5,450
0 votes
0 answers
45 views

Stopping time for uniform law

Let $X_1, X_2, \dots$ be IID Unif$(0,1)$ random variables and let $N=\min \{n : S_n=X_1 + \dots + X_n > \ln(2) \}$. Find the expectation of $N$. I've tried three approaches. First I showed that $...
Kilkik's user avatar
  • 1,952
1 vote
0 answers
26 views

Subsequence Process of Non-Markovian Stochastic Process

I have a problem that I haven't encountered before and would like to know if there is literature on the problem. Assume $X_t$ is a non-Markovian stochastic dynamical system and that $X_t \in S=\{1,2,...
E.S.'s user avatar
  • 11
0 votes
1 answer
44 views

conditional probability with respect to the stopped sigma algebra

Suppose that $\sigma$ is a almost surely finite stopping time with respect to some filtration $(\mathcal{F}_t)_{t\in\mathbb{R}}$, and let $X$ be a real walued random variable defined on the same ...
Niebla's user avatar
  • 454
5 votes
0 answers
178 views

Help needed findint the expected value of this stopping time

Let $\xi_i$ be iid random variables with $\mathbb{E}[\xi_i]=0$, and define: $$S_{(k)} = \sum_{i=1}^N \xi_{i+k}$$ Now, define: $$\tau = \min \left\{ k : S_{(k)} \notin (a,b) \right\}$$ How can I find $\...
user3141592's user avatar
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0 votes
0 answers
63 views

If $X_n$ is martingale, $N$ is a stopping time, is $X_{n+N}$ a martingale?

Is this true? If it is, can we change martingale to sub or super? My attempt (On submartingale): $\mathbb{E}[X_{n+N+1}\vert X_{n+N}]=\mathbb{E}[\mathbb{E}[X_{n+N+1}\vert N,X_{n+N}]\vert X_{n+N}]\ge \...
Ho-Oh's user avatar
  • 919
0 votes
0 answers
65 views

Gambler ruin's: Probability of $k$ consecutive win before j consecutive loss

Assume that a stock has a probability of $p$ to win, a probability of $q$ to lose, and a probability of $(1-p-q)$ to remain every day. What is the probability of $k$ consecutive wins occur before $j$ ...
Zhihao Xu's user avatar
0 votes
2 answers
59 views

If $S$ and $T$ are stopping times, is it true that $(S\leq T)\in \mathcal{F}_T$?

Suppose $S$ and $T$ are stopping times with respect to the continuous filtration $(\mathcal{F}_t)_{t\geq 0}$. Is it true that $(S\leq T)\in \mathcal{F}_T$? If the filtration is assumed to be discrete ...
No-one's user avatar
  • 667
1 vote
0 answers
52 views

Expected stopping time for biased random walk with increasing stepsize

Let $S_n$ be a stochastic process, with $$ S_{n+1} = S_n - 1 + \begin{cases} n^2 & \mathrm{with\;probability\;}\frac{1}{2}\\ -n^2 &\mathrm{else} \end{cases}. $$ Let now $T:=\inf\{n\geq0:S_n\...
lorenzw's user avatar
  • 73
0 votes
1 answer
40 views

Stopping time + constant for more general index-set

Let $(\Omega,\Sigma,\mathbb{P})$ be a Probabillity space, $I\subseteq [0,\infty)$ an Indexset which is closed under addition, i.e. for $a,b \in I $ holds $a+b\in I$ and let $\tau : \Omega \to I\cup \{\...
Leelia's user avatar
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1 vote
1 answer
115 views

Expected stopping time of summation of ranking supermartingale is finite

Given a stochastic process (that takes on real numbers) $X_n$, which is a ranking supermartingale, which we defined as $\mathbb{E}(X_{n+1}-X_n|\mathcal{F_n})\leq \epsilon\ < 0$. Let now $Y_n$ be ...
lorenzw's user avatar
  • 73
2 votes
1 answer
47 views

Is $\int_{\tau_0}^{\tau_1}f(W_t)\mathrm{d}t$ independent of $\mathcal{F}_{\tau_0}$ for some stopng times $\tau_0, \tau_1$ and Lebesgue integrable $f$?

Let $W=(W_t)_{t\geq 0}$ be the standard Brownian motion, let $X_t=W_t+X_0$ with initial distribution $X_0\sim\delta_x$ for $x\in\mathbb{R}$, let $\tau_1^1=\tau^1=\inf\{t>0:X_t=1\}$, write $\tau_0^1=...
Daan's user avatar
  • 362
1 vote
1 answer
47 views

Decomposing a general stopping time into stopping components

Let $(X_n)_{n \geq 0}$ be a discrete-time Markov chain taking values in a finite state space $S$, with transition matrix $P$. Let $(\mathcal F_n)_{n\geq 0}$ be the natural filtration and let $\tau \...
Jeffrey Jao's user avatar
1 vote
1 answer
35 views

Expression of a stochastic process at a stopping time: $X_{\tau}$

Let $(X_t)_{t \in [0,\infty)}$ be a real valued stochastic process, let $\mathcal{F}$ be a filtration on $[0,\infty)$. Let $\tau$ be a stopping time of $\mathcal{F}$. Is it possible to give an ...
Fran712's user avatar
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3 votes
1 answer
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Prove a Characterization for Localization of Stochastic Processes

In Exercise 3.4.16 in Stochastic Calculus and Applications by Cohen and Elliott, it states: Let $C$ be a set of processes such that if $X$ is a process with $X^T, X^S \in C,$ for $S,T$ stopping times,...
Yung-Hsiang Huang's user avatar
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0 answers
38 views

Circular Breakout Game: time or collisions needed to reach nth layer

Shower thoughts... A tiny ball starts inside a unit circle, surrounded by fixed concentric circles of increasing integer radius length. So it starts in is the 0-th "level". The ball moves in ...
vallev's user avatar
  • 406
2 votes
1 answer
50 views

Sum of hitting times and hitting times of sum of Brownian Motion

I have came across a question that asks to prove that for a brownian motion $B$, the first hitting time of $T_a=\inf\{t≥0 : B_t=a\}$ has a $1/2-$stable distribution, in that if we have $n$ independent ...
R.V.N.'s user avatar
  • 767
1 vote
1 answer
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How can I show that $\mathbb{E}(T)=\frac{a^2}{\sigma^2}$?

Let $(X_i)_{i=1,2,...}$ be a sequence of iid random variables such that $\mathbb{P}(X_i=\pm 1)=\frac{1}{2}$ and with $\operatorname{Var}(X_i)=\sigma^2>0$. Set $S_t=\sum_{k=1}^t X_i$ for $t=1,2,...$ ...
user123234's user avatar
  • 2,935
2 votes
1 answer
65 views

$\mathbb E(\max(X_1,...,X_{t+1})|\mathcal{F}_t)$ where the $X_i$ are iid uniform

Let $X_1,...,X_T$ be independent and identically distributed uniform random variables on $[0,1]$. Let $$M_t:=\max\{X_1,...,X_t\},$$ $L_t=M_t-ct$ for a $c>0$ and $L_0:=-\infty$. If $\mathcal{F}_t=\...
Analysis's user avatar
  • 2,482
2 votes
1 answer
47 views

Meaning of $\mathcal{F}_{\min\{n,\tau\}}$

Suppose $\mathcal{F}_n$ is a filtration and $\tau$ is a stopping time. What does $\mathcal{F}_{\min\{n,\tau\}}$ mean in this context? I am struggling to grasp what that should mean since $\tau$ ...
MathMaestro's user avatar
4 votes
1 answer
78 views

Expected value of stopping time of non symmetric random walk$E[\tau_{a,b}]$ is finite

Suppose we have a random walk $S_n$ that increases by $1$ with probability $p \ne \frac{1}{2}$ and decreases by $1$ with probability $1-p$. And let $a,b \in \mathbb{N}$. How can I show that the ...
user007's user avatar
  • 615
2 votes
1 answer
40 views

Prove that for symmetric random walk $S_n$ it holds that $S_{min\{n,\tau\}}^2 - \min\{n,\tau\}$ is a uniform integrable martingale

I want to show that for a stopped symmetric random walk $S_n$ we have that $S_{min\{n,\tau\}}^2 - \min\{n,\tau\}$ is a uniform integrable martingale. Where $\tau = \inf\{n \ge 0 : S_n \in \{-a,b\}\}$. ...
user007's user avatar
  • 615
0 votes
0 answers
33 views

Hitting closed set for right-continuous process

It is known that the random variable $$\tau = \inf \lbrace t \ge 0 : X_t \in A \rbrace$$ is a stopping time with respect to $\left(\mathcal{F}_t^X\right)$ if $A$ is a closed set and $X_t$ is a ...
Barabara's user avatar
  • 704
1 vote
0 answers
62 views

Poisson process and a stopping time

Let $(N_t)$ be a Poisson process. For $a \in \mathbb{N}$, is the random variable $$\tau = \inf \lbrace t \ge 0 : N_t = a \rbrace$$ a stopping time with respect to $\left(\mathcal{F}_t^N\right)$? I ...
Barabara's user avatar
  • 704
2 votes
1 answer
41 views

Is stopping time for martingale with continuous trajectories finite almost surely?

Let $(M_t,F_t)_t$ be martingale with continuous trajectories such that $M_t - t^{3/2}$ is also a martingale, and $M_0=3$. Let $T=\inf\{t : \lvert M_t \rvert = 8\}$. Show that $T$ is finite almost ...
romperextremeabuser's user avatar
0 votes
0 answers
31 views

Markov and Supermartingale with stopping time $\tau$

Let $X_n$, $n\geq 0$, be a non-negative integer valued integrable markov chain with transition probabilities $$ p(j|i) = \begin{cases} p_n & j=(i-1)\vee 0 \\ 1-p_n & j=i+1 \\ 0 &\text{...
Kumar's user avatar
  • 1,167
3 votes
1 answer
50 views

Expected stopping time for Brownian motion conditional on lower barrier being hit first

Suppose we have a Brownian motion $B(t)$, with $B(0) = 0$, and boundaries $a < 0 < b$. Define $\tau_a = \min\{t : B(t) = a\}$ and $\tau_b = \min\{t : B(t) = b\}.$ Conditional on the fact that $a$...
Andrew Ferdowsian's user avatar
2 votes
2 answers
45 views

Show that this stopped process converges ucp to the original process

Question Let $M$ be a continuous local martingale with null at zero. Let $\tau_n=\inf\{t:|M_t|>n\}$ be a stopping time. Does $M^{\tau_n}\to M$ u.c.p (uniformly on compacts in probability)? Here, a ...
Mingzhou Liu's user avatar
2 votes
0 answers
26 views

Proof of Theorem 5.38 in Morters and Peres' Brownian Motions book

In the proof of Theorem 5.38 which states that if $\tau_{a}$ is the hitting time of $a>0$ and if $\sigma_{a}=\inf\{t\geq 0:B(t)\geq |a|\}$, then $\int_{0}^{\tau_{a}}\mathbf{1}_{\{0\leq B(t)\leq a\}}...
Dovahkiin's user avatar
  • 1,285
0 votes
0 answers
27 views

Ratio of expectations of hitting times to the same boundary by Brownian Motions with different drifts

Consider two processes on a finite time interval $[0, T]$: $$ X_1(t) = -x + at + W(t), $$ $$ X_2(t) = -x - at + W(t), $$ where $x > 0$ is some starting position, and $W(\cdot)$ is a standard ...
george's user avatar
  • 198
2 votes
1 answer
78 views

Optional Stopping Theorem for martingales bounded except at the stopping time

One form of the Optional Stopping Theorem is as follows. Let $X$ be a submartingale, $T$ a stopping time, and $a$ a constant such that for all $n$ we have $|X_{n \wedge T}| \le a$ almost surely; then ...
lily's user avatar
  • 3,769
4 votes
0 answers
60 views

Ratio of Expected Hitting Times of Brownian Motion with Drift

Suppose $W_t$ is Brownian motion and consider the following two stopping times: $$\tau_a \equiv \inf \{t \ge 0 : W_t + at \ge b(t) \} \wedge T$$ and $$\tau_{-a} \equiv \inf\{t \ge 0: W_t - at \ge b(t)\...
qp212223's user avatar
  • 1,662
0 votes
0 answers
42 views

Expected number of steps to get "ABRACADABRA" [duplicate]

Suppose you have a 26-sided die, each face is labelled from A-Z, what is the expected number of steps to observe the sequence "ABRACADABRA" for the first time? ANS = $26^{11} + 26^4 + 26$ A ...
Harsh's user avatar
  • 378
3 votes
1 answer
59 views

Is $\inf\left\{t\in\left[0,1\right]\vert t+B^2_t=1\right\}$ a stopping time?

Problem Let $\left(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P}\right)$ be a filtered probability space such that $(\mathcal{F}_t)_{t\geq 0}$ is a complete and right-continuous filtration ...
Wilfred Montoya's user avatar
0 votes
1 answer
38 views

Optional Stopping Theorem and Stopped $\sigma$-fields

This is a simple exercise needed to prove the Optional Stopping Theorem that I'm working on. Suppose $(X_n)$ is a supermartingale and we have stopping times $T, S$. Then we already know in general ...
mtcicero's user avatar
  • 529
3 votes
1 answer
45 views

The distribution of the first hitting time for the Constant Elasticity of Variance process.

The Constant Elasticity of Variance (CEV) process is a one dimensional diffusion process given by the following stochastic differential equation. \begin{equation} d X_t = \mu X_t \cdot dt + \sigma X_t^...
Przemo's user avatar
  • 11.5k
3 votes
0 answers
47 views

Brownian motion and inverse hitting time

Given a standard one-dimensional Brownian motion $B$, we define the hitting time as \begin{align*} \tau_a:=\inf\{t\geq0 \, : \, B_t = a\}, \end{align*} with $a \in \mathbb{R}$. From the continuity of ...
Otsuaf's user avatar
  • 109
2 votes
1 answer
73 views

Expected value of the square of a stopping time

Problem Let $a>0$ and $B$ be a standard $\mathbb{R}$-valued Brownian motion. Define the stopping time $S_a:=\inf\{t\geq 0\ \vert \left\lvert B_t\right\rvert = a\}$. Compute $\mathbb{E}\left[S_a^2\...
Wilfred Montoya's user avatar
5 votes
1 answer
103 views

Expected value of the exponential of a stopping time

Problem Let $a>0$ and $B$ be a standard $\mathbb{R}$-valued Brownian motion. Define the stopping time $S_a:=\inf\{t\geq 0\ \vert \left\lvert B_t\right\rvert = a\}$. Compute $\mathbb{E}\left[e^{-\...
Wilfred Montoya's user avatar
0 votes
0 answers
24 views

If the stopping time is finite on the origin, it is finite for any other point, why?

In the context of a simple random walk, a claim that I find on some proofs: When proving that a stopping time at any point $x$ is finite, it suffices to prove that the stopping time at the origin is ...
phi's user avatar
  • 409
0 votes
1 answer
48 views

First hitting time of biased random walk in 2 dimensions

I am trying to find the distribution of first hitting times of a 2D biased random walk to reach a set of points in $\mathbb{R}^2$. I found the answer for the 1D case of brownian motion here https://en....
dguerrero's user avatar
2 votes
1 answer
63 views

Stirling approximation of the probability that the stopping time is finite

Let $\left(Y_n\right)_{n \geq 1}$ a sequence of i.i.d random variables such that $\mathbb{P}\left(Y_1=1\right)=\mathbb{P}\left(Y_1=\right.$ $-1)=1 / 2$. Define $S_0=0, \mathcal{F}_0=\{\Omega, \...
phi's user avatar
  • 409
0 votes
1 answer
35 views

Product of stopping times in discrete time

Let $T$ and $S$ be stopping times wrt a filtration $(\mathcal{F}_n)$ and assume that $S \leq T<\infty$ a.s. The question is whether $TS$ is also a stopping time. In continuous time, I believe it ...
RBN's user avatar
  • 1
0 votes
1 answer
46 views

Definition of stopping time

I am having a hard time for understanding stopping time, especially on notations. My book says: Def. A stopping time for the filtration $(g_n)$ is a random variable $T:\Omega \rightarrow \mathbb{N} \...
JAEMTO's user avatar
  • 695
3 votes
1 answer
70 views

Expected number of picks from normal distribution such that the sum exceeds 𝑟? Does the value converge?

I have found the similar question for Uniform Distribution i.e, U(0,1) but not for Normal distribution, So basically the problem is, using a standard normal distribution i.e., 𝑋∼𝑁(𝜇,$𝜎^2$) random ...
Sunny Chaturvedi's user avatar
2 votes
1 answer
53 views

Probability of Some Discrete Walk Hitting a Boundary?

A sequence $M_i$ of length 100 is consisting of 50 "-1"s and 50 "+1"s randomly permuted (like balls drawn from an urn without replacing). Then a path $S_i$ is generated by the ...
Gabriella Chaos's user avatar
3 votes
1 answer
251 views

Why is the Brownian Motion related to the Normal Distribution?

I am trying to learn more about the Inverse Gaussian Distribution and its applications (e.g. First Passage Time). First, define a Brownian Motion with Drift: If $X(t) = \mu t + W(t)$ represents a ...
Uk rain troll's user avatar
3 votes
1 answer
136 views

Maximising Score in Dice Game

We are playing a game with a die where we roll and sum our rolls. We can choose to stop at any time and take the sum as our score, but if we roll the same face twice in a row (consecutively), we lose ...
Harsh's user avatar
  • 378
0 votes
1 answer
59 views

Theorems invoked to evaluate the finiteness of a stopping time

I would like to know what are the theorems and/or results invoked in this kinds of equalities: $$ P_x[\tau_r < \infty] = \lim_{R\to\infty}P_x[\tau_{r,R}=\tau_r] $$ where, to give an example ...
Enrico's user avatar
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