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 $n$, $\{X_0, . . . , X_n\}$.
1,497
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Question about wide (broad) stopping time
Def.
Let a filtration $\left\{\mathcal{F}_{t}:t\ge 0 \right\}$ on a probability
space $(\Omega, \mathcal{F},\mathbf{P}).$
An $\left\{\mathcal{F}_{t}\right\}-{\color{BLUE} {\text{stopping time}}}$ is ...
- 187
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2
answers
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Can pointwise convergence in random variables guarantee some kind of convergence in a related stopping time?
Suppose we have a family (indexed by $n$) of discrete-time random process $\{X_{t}^n\}, {t\geq0} $, taking values in $\mathbb{R}$.
For each $n,t$, $X_{t}^n\geq0$. And $X_{t}^n$ converges pointwise ...
- 139
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2
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Expected number of strikes to kill a $3$-headed dragon
You want to slay a dragon with $3$ heads. There is $0.7$ chance of destroying a head and $0.3$ chance of missing. If you miss, a new head will grow. $X$ is a random variable for the number of rounds ...
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answers
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Simple random walk, Martingales, stopping time
Suppose $S_n = X_1 + \dots + X_n $ is a simple random walk starting at 0. For any K, let $$T = \min \{n: | S_n| =K\}. $$ $\bullet$ Explain why for every j, $$ \mathbb{P}\{T \leq j +K | T > j\} \geq ...
- 1,012
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2
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expected sum after rolling dice until getting 6 five times
We are rolling dice until we get 6 exactly five times. What is expected value of total sum? My approach: Lets denote $X_i$ - number rolled in $i$-th roll, and $S_n = \sum_{i=1}^{n}X_i$. So now i want ...
- 412
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0
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50
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Asymptotic Formula for a Version of the Secretary Problem
The version of the secretary problem to be considered here is the one where the hirer must pick the best or second-best secretary using exactly one choice. This answer provides the optimal strategy, ...
2
votes
1
answer
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Can we show a process has bounded mean?
Assume $X_0=0$. The process evolves with the following rules.
$X_{t+1} = X_{t} +1 $ if $X_t<10$;
If $X_t\ge 10$,
$$
X_{t+1} = \begin{cases}
X_t + 1 &\text{with probability } 1/2\\
X_t -3 &...
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vote
1
answer
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Hitting time of Brownian motion past a given point in time
The random variable whose distribution I am interested in is defined as follows:
$$\tau := \inf\{u > 1: W_u = 0\}$$
where $W$ is Brownian motion.
I derive the distribution below but it doesn't ...
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Optimal stopping: dice game
This problem has been asked lots of times, but I could not find a completely satisfactory answer.
The statement is as follows: we have a standard 6-sided die. Each time we roll it, if we get a number $...
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Expected hitting time for asymmetric random walk
Let $(X_i)$ be iid. random variables with:
$$\mathbb{P}(X_i = 1) = p > (1-p) = \mathbb{P}(X_i = -1)$$
and set $S_n = X_1 + \cdots X_n$. We will write $T_k$ for $\inf \{n : S_n = k\}$ i.e. the ...
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Uniqueness of optimal stopping time for a one-armed bandit to reach break-even?
This question is based on the derivation of the Gittins index in Weber's article On the Gittins index for multiarmed bandits.
Consider a one-armed bandit consisting of a Markov process $X$
and a ...
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Wiener Process Textbook or Reference Specifically Containing Ramp Intersection (or Exit Time) Analysis
I am looking for a reference (preferably a textbook so that additional preparation material is handy) that calculates the exit time of a Wiener process from a region bounded by sloped lines.
Thank you,...
- 1
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Martingale in a modified gambler's ruin problem
Let $\{Y_t\}_{t \in \mathbb{N}}$ be a random walk on $\mathbb{Z}$ defined as follows:
$\mathbb{P}(x,x+1) =
\begin{cases}
p & \text{ if } x \geq 1\\
1-p & \text{ if } x \leq 0
\end{cases}$
$\...
- 57
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Probability of a stopping time to be lower than another stopping time
I was wondering on how to compute the probability of a stopping time being lower than another stopping time.
More in details, just consider a drifted Brownian motion process $X_t$, and consider two ...
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Expectation of the minimum between two stopping times
I have some difficulties to answer a question about stopping time.
We consider $B$ a Brownian motion, $a<0<b$, $\tau_a=\inf\{t : B_t = a\}$ and $\tau_b=\inf\{t : B_t = b\}$. Then I put $\tau=\...
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From discrete time stopping theorem to continuous
I would like to generalize by dyadic discretization some theorem I have seen in finite time setting.
Here is the theorem
Let $(X_t)_t$ be a continuous martingale and $\tau$ a bounded stopping time ($\...
- 1,116
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0
answers
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Prove that for a martingale, any increasing stopping time is sufficient to be a local martingale
I would like to prove the following
Let $(X_t)_t$ be a continuous martingale. Prove that any sequence of increasing stopping times (with values in $\mathbb{R}_{+}$) is sufficient to get a local ...
- 1,116
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votes
1
answer
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Stochastic process and reaching time to an interval
I have the following exercice to do and I would like to know if what I did is correct please.
Consider $X_t$ a continuous stochastic process and $\tau$ the reaching time to the interval $[a,b]\subset[...
- 1,116
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1
answer
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Gambler's Ruin and the expectation of a stopping time in the case the conditions of the Optional Stopping Theorem do not hold
In these lecture slides (pages 13-21) on optional stopping theorem (OST) and martingale I have found a great example of the gambler's ruin problem and what happens when the criterions of the OST are ...
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Stopping time for Brownian motion with time changing parameters
I am interested in computing the density of a stopping time $\tau^b = \inf\{ t>0 : X_t^b \leq 0 \}$ for some process $X^b = (X_t^b)_{t \geq 0}$ with initial condition $X_0^b =x_0 > 0$. More ...
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First ocurrence of a pattern
Let $X_0, X_1 , \dots$ be independent identically distributed Bernoulli random variables such that
$$ \mathbb{P} [ X_k = 0 ] = \mathbb{P} [ X_k = 1] = 1/2, \ k \geq 0 $$
Let us define
$$ \tau_{110} = \...
user1231900
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answers
25
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Show a stopping time is finite in probability
Define a stopping time $\tau:= \min\{t:W_{t} = 1 \text{ or } W_{t} = -1 \}$, I want to prove that $\mathbb{P}(\tau < \infty ) = 1$, where $W_{t}$ is a Brwonian motion. Here is my derivation:
$$
\...
2
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1
answer
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What is wrong with this proof about reaching time to a closed set being a stopping time?
At some point I wanted to prove that if we have a continuous (trajectory) stochastic process $X_t$ (we consider its natural filtration) and $A$ a closed set, then $\tau(\omega)=\inf\left\{t | X_t(\...
- 1,116
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About Martingale Stopping Theorem
Prove that:
A right-continuous process $X$ adapted to the filtration $\{\mathcal{F}_t\}$ is a martingale if and only if for any bounded stopping time $T$, $X_T \in L^1$ and $E[X_T] = E[X_0]$.
I know ...
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Computation of density function
Suppose (W i
t )t>0, i ∈ {1, 2} are two independent Wiener processes. Let $\tau = inf (t > 0 : |W_{t}2
| = a) $ for
a > 0. Show that the random variable W 1
τ has a density
f (x) = (2a cosh(...
3
votes
2
answers
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Measurability of indicator of two hitting times at the stopped $\sigma$-algebra
Let $\mathcal{F}$ be the complete filtration generated by the Brownian motion $B$, and let $a<0<b$. Define the stopping times
$\tau_a=\inf\{t\ge 0|B_t=a\}$ and $\tau_b=\inf\{t\ge 0|B_t=b\}$.
...
- 13k
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Conditional probability of hitting time when drifted Brownian motion hits a barrier
Let $a<0<b$ and $\mu\in \mathbb{R}$, and $X_t= \mu t+ B_t$ be a drifted brownian motion.
Is it possible to compute the following probability
$$
\mathbb{P}[{\tau_a<\tau_b}\mid \tau_a],
$$
...
- 13k
2
votes
1
answer
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Laplace transform of hitting time when drifted Brownian motion hits the lower barrier first
Let $a<0<b$ and $\mu\in \mathbb{R}$, and $X_t= \mu t+ B_t$ be a drifted brownian motion. Given a suitable $r>0$, is it possible to compute
$$
\mathbb{E}[e^{r\tau_a}1_{\tau_a<\tau_b}],
$$
...
- 13k
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0
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A detail in the proof of a martingale theorem about a.s. convergence under the stopping time condition
I was reading a theorem about martingale in a textbook by Yuan Shih Chow.However,I cannot understand a detail in the proof,and I will appreciate if you could explain me in the red frame why we can ...
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Stopping Time of a Sequence of Digits
Now I have a program which will generate a sequence of digits. Each digit will output a number uniformly randomly in $\{0,1,2,3,4,5,6,7,8,9\}$. However it will never print the same digit twice in a ...
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Dividing a cake into two pieces and/or choosing between them takes $(n^2-n) \over \log(n)$ steps!?
After reading the article of the cake cutting problem in the September 2023 issue of science news. I was thinking about using a divide and conquer strategy by having the cake eaters divide themselves ...
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Expectation of capped hitting time
Let's suppose we have a random walk $$Z_t=\sum_{i=0}^{t} X_i$$ where each $X_i$ can take values $\pm 1$ with symmetric probability, $P(X=1)=P(X=-1)=\frac{1}{2}$.
It's is known that the hitting time at ...
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Why are we interested in whether a given hitting time is a stopping time?
Why are we interested in whether a given hitting time $\tau_B=\inf\{t \geq 0\,|\, X(t) \in B\}$ of a stochastic process $\{X(t)\}_{t \in [0,\infty)}$ is a stopping time,i.e. that $\{\tau_B \leq t\} \...
user727154
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1
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Optimal strategy to get maximum element of $n$ sequentially i.i.d uniform distribution $X_1, \cdots, X_n \sim \text{Uniform}(0,1)$?
This is a homework for probability. To warm-up let's consider the case of $3$.
Consider three random variables $X_1, X_2, X_3$ that are independently and identically distributed according to the ...
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1
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If $\tau$ is a stopping time, then $f(\tau)$ is also a stopping time.
I have been reading about stopping times and have a question about the composition of stopping times with a measurable function.
Let $f:\mathbb{R}_+\to\mathbb{R}_+$ be any measurable function with $f(...
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Expected value of a biased random walk given some stopping condition
I'm trying to work on this problem:
I have a biased random walk. With probability $0.7$, you gain $1$ and with probability $0.3$, you lose $1$. You start at $0$.
Set a stop loss at $-10$. Once the ...
- 1,341
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votes
2
answers
94
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Stopping independent random variables
Let $\{X_n\}$ be independent (and identically distributed if necessary) integrable random variables. Consider the stopping time, adapted to the natural filtration as usual, $\tau$ (that is $\{\tau=n\}\...
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Is every local martingale right continuous?
Is every local martingale right càdlàg (i.e. right continuous with left limits)?
At the university, in the definition of martingale we assume martingales to be right càdlàg processes.
We call an $X$ ...
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1
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Why $(X_0,\dots, X_\tau)$ is measurable w.r.t $\mathcal{F}_{\tau}$ for stopping time of a discrete time process?
Consider discrete time stochastic process $(X_0,\dots, X_i,\dots)$ adapted to filtration $\mathcal{F}_i$. Let $\tau$ be a stopping time. The following corollary is found in https://mathweb.ucsd.edu/~...
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First Passage Time of Discrete Inhomogeneous Poisson Process
Consider the discrete time stochastic process $m_j=\sum_{i=1}^jx_i$ for a finite time interval $j=1,\dots,n$, where the increments $x_i\sim\text{Pois}(\lambda_i)$ are all independent Poisson random ...
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Expectation of stopped process for nonfinite stopping time
Is there a standard general definition of expectation $\mathbb{E}(X_\tau)$ for a stopping time $\tau$ and a sequence of random variables $\{X_n\}$? Usually the assumption $\mathbb{P}(\tau<\infty)=1$...
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1
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Expected number of cards to draw before first ace using stopping times?
Consider the classic combinatorial problem that asks how many cards we should expect to have to draw on average before we see our first ace. There are several solutions. My first instinct was to write ...
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Reference for a hitting time (of a continuous SP) which is not a stopping time
I need a book reference for an example of a hitting time defined by a continuous stochastic process which is not a stopping time. I would be thankful for any recommendation!
user727154
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Stable under stopping implies stable under integration
Let $\mathfrak{H}_{0}^{2,c}$ denote the space of square-integrable continuous martingales starting from zero. Let $\mathfrak{A}$ be a closed linear subspace of $\mathfrak{H}_{0}^{2,c}$. Show that
$\...
- 399
1
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1
answer
128
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Probability of Brownian motion hitting $n$ integers and the expected hit within a time period
Given an Ito process $x(t):=\sigma B(t)$ with an unknown constant volatility $\sigma$. Is there a way to estimate $\sigma$ by simply counting the number of times the Ito process hits a lattice with ...
- 9,710
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0
answers
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Sigma algebra on the index set in the definition of a stopping time
Suppose that $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ is a filtered probability space with totally ordered index set $T$. A stopping time is a random variable $\tau: \Omega \to T$ (e.g. in ...
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Convergence of a sequence of sampled submartingales
I'd need a review, whether my proof is correct or if I forgot something.
Given:
$M$ is a right-continous submartingale (i. e. $\mathbb{E}[M_t | \mathcal{F}_s] \geq M_s$ for $s \leq t$)
$(\tau _n )_{n\...
- 59
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votes
1
answer
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The expected first exit time of Brownian motion from the closed ball
I want to ask about a reasoning part of calculation of the expected first exit time of Brownian motion from a ball, in Example 7.4.2 of Stochastic Differential Equations: An Introduction with ...
1
vote
1
answer
111
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Hitting time of a diffusion process
Consider a diffusion process (SDE)
$
dX(t) = \mu (X(t)) dt + \sigma(X(t)) dW(t), X(0) = x.
$
I am interested in a first hitting time of this process to a point $y < x$, more particularly I wonder ...
- 173
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0
answers
27
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Different interpretation of stopped sigma-algebras
I've seen a proof skatch of the discussed statement below, but I don't really understand it. Does anyone know what is the elaborated proof, or where can I find it?
$X$ is a progressive measurable ...
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