# 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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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 ...
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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 ...
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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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### 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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### 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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1 vote
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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 ...
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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 $[a,b]\subset[... 1 vote 1 answer 61 views ### Gambler's Ruin and the expectation of a stopping time in the case the conditions of the Optional Stopping Theorem do not hold In these lecture slides (pages 13-21) on optional stopping theorem (OST) and martingale I have found a great example of the gambler's ruin problem and what happens when the criterions of the OST are ... 4 votes 0 answers 46 views ### Stopping time for Brownian motion with time changing parameters I am interested in computing the density of a stopping time$\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$. More ... 0 votes 1 answer 46 views ### First ocurrence of a pattern Let$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 $$\tau_{110} = \... 0 votes 0 answers 25 views ### Show a stopping time is finite in probability Define a stopping time \tau:= \min\{t:W_{t} = 1 \text{ or } W_{t} = -1 \}, I want to prove that \mathbb{P}(\tau < \infty ) = 1, where W_{t} is a Brwonian motion. Here is my derivation:$$ \... 2 votes 1 answer 36 views ### 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(\...
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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(...
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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 $\tau_b=\inf\{t\ge 0|B_t=b\}$. ...
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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 $$\mathbb{P}[{\tau_a<\tau_b}\mid \tau_a],$$ ...
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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 $$\mathbb{E}[e^{r\tau_a}1_{\tau_a<\tau_b}],$$ ...
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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 ...
1 vote
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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 ...
1 vote
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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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### 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 ...
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1 vote
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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 ...
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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 ...