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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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first-hitting-time and conditional probability

I have been struggling with the following problem. Let $\{X_{n}\}_{n\geq 1}$ be i.i.d. random variables with common distributional measure $\nu$. Let $B\subset\mathbb{R}$ be any Borel measurable set ...
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20 views

$P(T_2=T_{-3}), P(T_1<T_4<T_{-1})$ and $P(T_3<2)-P(T_{-3}<2)$

Let $W(t)$ be a Brownian motion and $T_x=\inf\{t:W(t)=x\}$. I need to calculate $P(T_2=T_{-3}), P(T_1<T_4<T_{-1})$ and $P(T_3<2)-P(T_{-3}<2)$. I'm not sure if I understand these ...
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Wiener process with drift $W_t+at$

Let $W$ be the standard Wiener process. The task is to establish for which $a$ and $b$ the stopping moment is finite almost surely. $$\sigma_{a,b}=inf\{t \geq 0\space W_t+at=b\}$$ I established that ...
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40 views

If $f:[0,\infty)\to\mathbb R$ is continuous and $\tau=\inf\left\{t>0:f(t)>\varepsilon\right\}$, then $f(\tau\wedge t)\le\varepsilon$ for all $t\ge0$

Let $f:[0,\infty)\to\mathbb R$ be continuous, $\varepsilon>0$ and $$\tau:=\inf\left\{t>0:f(t)>\varepsilon\right\}.$$ Are we able to show that $f(\tau\wedge t)\le\varepsilon$ for all $t\ge0$ (...
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30 views

If $T$ is a stopping time, what represent $\mathcal F_T$?

Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space and let $T:\Omega \to \mathbb N$ a stoping time. Let $(X_n)_{n\geq 1}$ a stochastic process and $\mathcal F_n=\sigma (X_1,...,X_n)$, and $\...
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Reference/suggestion needed - Hitting time of domain equal everywhere -> {domain = sphere, x0 = centre}

So, given a Brownian motion under drift $dX(t) = c dt + dB(t)$ where B(t) is a std Brownian process. Let the process start within some (convex, etc) domain $A$, i.e. $x_0\in A$. Does the following ...
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proof of stopping time proposition

enter image description here how to prove these statements? I only know how to prove part(a). but has no idea about the rest
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Question about stopping time : what would be $M_{\tau}$ if $\{\tau\leq t\}\notin \mathcal F_t$?

I really have difficulties by really understand in why stopping times are interesting. Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space, $(M_t)_{t\geq 0}$ a stochastic process and let $(\...
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key identities in the proof of Optional Sampling Theorem

Let $(\Omega,\mathscr{F},\mu)$ be a $\sigma$-finite measure space, let $(\mathscr{F}_n)_{n\in\mathbb{N}}$ be a filtration of sub-sigma-algebras of $\mathscr{F}$, let $(u_n)_{n\in\mathbb{N}}$ be a ...
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A new stopping time built from a stopping time

Let $T$ be a stopping time for the filtration $(\mathcal{F_n})_{n \in \mathbb{N}}.$ For all $n \in \mathbb{N} \cup \left\{+\infty \right\},$ we set $\phi(n)=\inf\left\{k \in \mathbb{N};\left\{T=n \...
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Convergence stopping time

If I have a sucession of continuous time stochastic processes, such that $L_t^N \xrightarrow{a.s} L_t$. Where $L_t^N$ is a jump process and $L_t$ is continuous (with respect to t). If $\tau^n = \inf\{...
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$\tau > \sigma$ is a stoppinig time when $\sigma$ is a stopping time and $\tau$ measurable regarding $\mathcal{F}_\sigma$

The problem is depicted in the title. I want to know to complete my proof. For any $t\ge0$ we want to show $$\{\omega;\tau(\omega)\ge t\}\in\mathcal{F}_t$$ Since we have $\tau\ge\sigma$. The left ...
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Minimal stopping time of brownian motion

Suppose $W$ is a Brownian motion, let $H_B$ be the hitting of $B \in \mathbb{R}$ and let $\tau$ be another stopping time that is taken to be minimal, i.e $(W_{t\wedge \tau})_{t \geq 0}$ is uniformly ...
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If a strong Markov process reaches a Borel set a.s., can it be restarted from that set?

Let $X$ be a strong Markov process on $E$, and $B\in \mathcal B(E)$. Suppose that, for some $x\in E$, $$ P_x(\exists t\ge0 \text{ such that } X_t\in B)=1. $$ My question: Does there exist a stopping ...
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Compute moments of Brownian motion stopped at exit time of $[a,b]$

Given $B_t$ a standard brownian motion and $a < 0 < b$ Set $T = \inf\{ t : B_t = a \vee B_t = b\} $ For any $\alpha \in \mathbb{Z}^+$, find $EB_T^\alpha$. I know I can use optional ...
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Sigma-algebra associated to a stopping time.

I was interested to understand the interpretation of the sigma field associated to a stopping time. Let $(X_n)_{n \ge 1}$ be a stochastic process and $(F_n)_{n \ge 1}$ its corresponding filtration. ...
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45 views

Expectation of a hitting time

I'm trying to find the expectation of a stopping time. Specifically, Let $T_1,...,T_n$ be i.i.d exponential random variables with mean $1$. Let $S_n = T_1 + ... + T_n$ denote their partial sum. ...
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stopping time almost surely finite

Let $(X_n)_n$ be a sequence of independent random variables and identically distributed such that $P_{X_1}=p\delta_1+q\delta_{-1}+r\delta_0$ where $0 \leq p,q,r<1$ and $p+q+r=1.$ Let $\alpha, \...
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2answers
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Càdlàg Feller process is quasi-left-continuous

I've been working in Chung's "Lectures from Markov Processes to Brownian Motion", and I got stuck at Exercise 1 from 2.4. The objective of the problem is to give a short proof of the quasi-left-...
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32 views

Stopping theorem counterexample

I have been thinking about the conventional counterexample to stopping theorems where the stopping time is not bounded. For example, flip a fair coin infinitely many times, representing heads with 1 ...
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Measurability of Last Exit Time of a Discrete Time Stochastic Process

Suppose we have a discrete-time stochastic process $\{X_t\}$ defined on a space $(\Omega,\mathcal{F})$ equipped with the probability measure $\mathbb{P}$. Suppose we know that $X_t \rightarrow 0$ ...
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Brownian motion independence from stopping time

Let $X_t$ be a standard one dimensional Brownian motion. Let $T = \inf\{t : X_t \in\{ 1,-1\} \} $ and $S = \inf\{ t : X_t \in\{ 1, -3\}\}$ a) Explain why $X_T$ and $T$ are independent. ...
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Understanding the proof of Snell Envelope Optimal Stopping Time

I am trying to understand the proof of the optimal stopping of Snell Envelope. (on Pg 22) A Snell Envelope is defined as follows: Let $Y_n$ be an adapted process to $\mathcal{F}_n$ with finite ...
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1answer
32 views

stopping time a.s. bounded?

let $(X_n)_n$ be a sequence of independent random variables and identically distributed $B(1,\frac{1}{2})$ (Bernoulli distribution). We set $Y_n=\sum_{k=1}^n(2X_k-1)$ and $\mathcal{F_n}=\sigma(X_0,...,...
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Prior state dependent transition probability ABRACADABRA problem

The power of the martingale trick for computing the expected stopping time is amply demonstrated in this question and this answer as an advanced version of the ABRACADABRA problem. However, it seems ...
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42 views

Showing that $ES_N=0$ for a random walk where $N$ is a stopping time and $EN^{1/2}<\infty$

Question Let $\xi_{1}, \xi_{2}, \dotsc$ be i.i.d random variables with $E \xi_i=0$ and $E\xi_i^2<\infty$. Let $S_n=\sum_{i=1}^n \xi_i$ and $N$ be a stopping time. If $EN^{1/2}<\infty$, then $...
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Shifted two-sided Brownian Motion

Let $(B_t)_{t\in\mathbb{R}}$ be a two-sided Brownian motion, defined as $B(t) = \begin{cases} B_1(t),\quad t >0 \\ 0, \quad t = 0 \\ B_2(-t), \quad t < 0 \end{cases}$. For some $a>0$ let $T:=\...
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Proof of Cramér-Lundberg inequality

I'm trying to prove the Cramér-Lundberg inequality, which deals with the probability of ruin for an insurance company given a certain initial capital. Specifically, if $Y_1, Y_2, \ldots$ are the ...
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50 views

Probability of Brownian Motion hitting -2 before 1?

Why is the probability of Brownian Motion hitting -2 before 1 is equal to 1/3? This is an interview question asked for Quant roles. I found a similar question was previously asked: Brownian motion ...
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generator of sigma algebra associated with stopping time

I am studying Martingales and Stopping Times from the 3rd edition of the book "Probability and Measure" by Patrick Billingsley. The following arose while I was reading page 465. Let $(\varOmega,\...
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Distribution of the stopping time for a size dependent process

Let $X(t)$ denote some stochastic process (which we assume is in steady state, i.e. the distribution of $X(t)$ is independent of $t$). We assume $X(t)$ decreases at a constant rate equal to one. ...
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1answer
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Optional sampling theorem with a.s. finite stopping time

I'm trying to prove the following generalized version of Doob's optional sampling theorem: Let $X$ be a square integrable martingale with respect to a filtration $\mathbb F = \left\{ \mathcal F_n \...
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1answer
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min & sum of discrete stopping times using alternate definition

Let $A, B$ be stopping times of a process $X_0, ... X_n$ where $A$ is a stopping time iff $I_{A = k}$ is a function of $X_0, ... X_k$ (where $I$ denotes an indicator variable). We want to show that: ...
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1answer
52 views

First exit time of a simple random walk

Let $S_n=\displaystyle\sum_{i=1}^{n}X_i$ be the simple random walk with $S_0=0$. Now define the stopping time $$\tau=\inf\{n>0| S_n\notin (a,b)\}.$$ I am trying to follow some steps in a book and ...
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1answer
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probability with martingales 12.2 sum of zero-mean independent variables in L^2

I am struggling with the following theorem from David Williams, Probability with Martingales: THEOREM Suppose that $(X_{k}:k\in\mathbb{N})$ is a sequence of independent random variables such that, ...
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1answer
34 views

Strong markov property vs usual markov property.

I was trying to understand the difference between strong Markov property and the usual Markov property for a discrete number of states. I think I understand why the strong Markov property implies the ...
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Crossing times of processes

Suppose that $X_t$ and $\tilde{X}_t$ are processes defined on the stochastic basis $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ such that for any time $t>0$, $1>p_t>0$, where $p_t$ is ...
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1answer
20 views

Image of a symmetric law

Assume I have a probability space $(\Omega, \mathcal{F}, P)$ that is mapped by a measurable function $X$ into $(E,\mathcal{E})$, moreover $P(X \in U)=P(-X \in U)$, now $Y$ maps this measurable space ...
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26 views

Probability of Ruin at the first claim

The number of claims $n \sim Po(\lambda)$, and let $X_n$ denotes the claim amounts of a claim which are all iid and they follow a $Exp(1)$ distribution. Assume the initial surplus is $U$, and the ...
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1answer
38 views

First passage time for Compound Poisson distribution

Let $S$ follows a Compound Poisson distribution $(S \sim CP(\lambda,F_x(x))$, i.e. $$S = \sum_{i=0}^{N}X_i,$$ where $N\sim Po(\lambda)$ and $X_i \stackrel{iid}{\sim} Exp(1)$. I know that the first ...
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1answer
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Doob's Optional Stopping Theorem: $\xi_\tau$ vs $\xi_{\tau\land n}$

I have some troubles in understanding the Optional Stopping Theorem by Doobs. I have a bit of confusion about the following (Brzezniak, Zastawniak - Basic Stochastic Processes p. 58-59): Let $\...
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Expected value of a Brownian motion before its first hitting time

Let $X_{t}$ be a Brownian motion with drift $\mu=0$ and variance $\sigma^{2}$. Also, let $X_{0} = a < b$. We know that the density of the first hitting time $H_{b} = inf \lbrace t: X_{t} = b \...
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1answer
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Computing $E(T_b^2)$ for asymmetric random walk

Let $S$ be an asymmetric random walk with $p=P(X_1=1)>1/2$. Define $T_b=\inf\{n:S_n=b\}$. Prove that $\text{var}(T_b)=\frac{4bpq}{(p-q)^3}$ where $q=1-p$. We know $$\text{var}(T_b)=ET_b^2-(ET_b)^...
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1answer
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Stopped random walk is not uniformly integrable

I know that in general Doob's Optional Stopping Theorem doesn't hold for unbounded stopping times, but that it does when the system up to the stopping time is uniformly integrable. One counter ...
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1answer
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Example of last hitting time failing to be a stopping time?

I'm trying to find an example where the last hitting time $\theta = \sup\{k \ge 0 : X_k \in B\}$ of a set $B$ by the stochastic process $(X_k)_{k\ge 0}$ is not a stopping time.
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Probability of max of dependent random variables

I am trying to compute the following probability: $$P(X-Y_k\leq f_k(D)-c, X-Y_i\geq f_i(D)-c\, \forall\,i: 0\leq i\leq k-1),$$ where $f_i$ are functions and $D,c$ are considered to be fixed, and $X,...
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What's the precise statement of the continuous-time optional stopping theorem?

I searched high and low in a number of probability / financial mathematics textbooks and surprisingly cannot find any precise statement of the continuous time optional stopping theorem. In particular, ...
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28 views

Does hitting time have scaling property or self-similar proerty

Consider the hitting time such that $\tau_x = \inf\{t>0: B_t = x\}$ where $B_t$ is a standard brownian motion. Can $\tau_x$ be scaled like that $\tau_{ax}=a^2\tau_x$ for $\forall a>0, a\in \...
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25 views

Stopping moment regarding the ratio of black balls to all balls.

The urn contains at start one black ball, one white ball and one green ball. In the next steps, we randomly pick a ball from the urn and put it back with one additional ball coloured as the one we ...
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1answer
27 views

Random Walk: Proving that $1 = \sum_{m=0}^{n}P_0(S_{n-m} = 0)P_0(\tau_0 > m)$

I would appreciate a further hint for this question: Let $S_n$ a random walk on $\mathbb Z$, with $S_0=0$. Let $\tau_0 = \inf\{n>0:S_n=0\}$, the hitting time of $0$. Show that $$ 1 = \sum_{m=0}...