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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 $n$, ${X_0, . . . , X_n}$.

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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Density of first hitting time of Brownian motion with drift

I just started learning about Brownian motion and I am struggling with this question: Suppose that $X_t = B_t + ct$, where $B$ is a Brownian motion, $c$ is a constant. Set $H_a = \inf \{ t: X_t =a \}$ for $ a >0$. Show that for $c \in \mathbb{R}$,…
Richard
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Collection: Results on stopping times for Brownian motion (with drift)

The aim of this question is to collect results on stopping times of Brownian motion (possibly with drift), with a focus on distributional properties: distributions of stopping times (Laplace transform, moments,..) distributional properties of the…
saz
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Does this Condition on Exit Times imply $X_t$ is a Local Supermartingale?

Let $(X_t)_{t\geq 0}$ be a continuous (or càdlàg), real-valued process, and define stopping times $$\tau_{s,a,b}=\inf~ [s,\infty)\cap\{t:X_t\notin (a,b)\}.$$ We can interpret $\tau_{s,a,b}$ as the first time after time $s$ that the process hits $a$…
Ben Derrett
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'Intuitive' difference between Markov Property and Strong Markov Property

It seems that similar questions have come up a few times regarding this, but I'm struggling to understand the answers. My question is a bit more basic, can the difference between the strong Markov property and the ordinary Markov property be…
Debreu
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What is meant by a stopping time?

TL;DR: is a stopping time some sort of event, or is it a point in discrete time, or something else entirely what is an example of something which is not a stopping time? is my understanding of the concepts and definitions below correct? I am…
piman314
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Why is stopping time defined as a random variable?

I've been given a crash course in stochastic processes and martingales for the purposes of a semester project on them. The guy I'm working with has been, I feel, a little vague in the definition of stopping times, and I can't seem to find anything…
Jack M
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Definition of $\sigma$-algebra $\mathcal{F}_\tau$ with $\tau$ a stopping time

If $\tau$ is a stopping time and $(\mathcal{F_t})_{t\in I}\subset \mathcal{F}$ is a filtration, then the $\sigma$-algebra of the $\tau$-past is defined as $$\mathcal{F}_\tau := \{A\in\mathcal{F} : A\cap \{\tau \leq t\} \in \mathcal{F}_t \text{ for…
Qyburn
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Expected number of iterations till a walk on unit line exceeds a distance

Problem Each timestep $i$, a uniformly distributed next target $X_i$ is sampled in the range $0$ to $1$. We start out at point $X_0$ on the number line. Each timestep, we go from our current point to the next sampled point. What is the expected…
user1145925
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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 matrix $$ M = \begin{pmatrix} 1-p_1 & p_1 & 0 &…
whpowell96
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Proof that a stopped continuous-time martingale is a martingale.

The proof for a stopped discrete-time martingale is shown as follows. Let $M=(M_n)_{n\ge0}$ be a discrete-time martinglae w.r.t. the filtration $(\mathcal F_n)_{n\ge0}$, and let $M^T=(M_{n\land T})_{n\ge0}$ be the stopped martingale, where $T$ is a…
Zhanbin Du
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How to get closed form solutions to stopped martingale problems?

Way back when, I took a course in stochastic processes in college. I remember being frustrated by the plethora of abstract proofs without much in the way of how to use them to get actual results. It seems that every theorem in intro stoch. processes…
user76844
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Sum of two stopping times is a stopping time?

Let $\sigma$ and $\tau$ be two stopping times in $\mathscr{F}_t$ and let this filtration satisfy all the usual conditions. Question: Is $\sigma + \tau$ a stopping time? Attempt at a solution: I need to demonstrate that $\{ \sigma + \tau \leq…
Jase
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Showing that the first hitting time of a closed set is a stopping time.

I found this exercise online: I am stuggling with the last part of the second exercise, that is I am not able to show that $\tau = \sup_i \tau_i$. Obviously we have that $\tau \ge \sup_i \tau_i$, but I struggle to show the other inequality. Any…
user119615
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Proving Galmarino's Test

Galmarino's Test gives a condition equivalent to being a stopping time. It says: Let $X$ be a continuous stochastic process with index set $\mathbb{R}_+$ (i.e. each sample path is a continuous function of time). Let $\mathscr{F}$ be the filtration…
TYS
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Expected hitting time of given level by Brownian motion

I've been looking at this for some time now and still have no sensible solutions, can somebody help me out please. Say I define the stopping time of a Brownian motion as followed: $$\tau(a) = \min (t \geq 0 : W(t) \geq a)$$ (first time the random…
Vol_Smile
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