# 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 ...
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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 ...
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### 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 ...
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### 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 $j$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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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,...
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### 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 ...
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### 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 ...
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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 $t=1,2,...$ ...
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### 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 ...
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### 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 ...
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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 ...