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

1527 questions
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Racing Time Difference Formula

I am trying to develop an android app for a friend that uses the gps to tell you how many seconds ahead or behind you are from your target speed vs your actual speed. For example I could drive for 1 min at 10mph, with '60mph' selected on the app.…
Brady
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Stopping time that is not a reaching time?

For example, let $X=\left(X_t\right)_{t\ge0}$ a right-continuous random process, $x\in\mathbb{R}$ and $T_x = \inf\left\{t\ge0\;;\; X_t\ge x\right\}$. The random variable $T_x$ is a stopping time that is also a "reaching time". What would look like…
thomasb
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Show that $T\boldsymbol 1_B+M\boldsymbol 1_{B^c}$ is a stopping time if $T$ is a stoping time.

Let $S$ and $T$ two stopping time w.r.t. the filtration $(\mathcal F_t)_t$ such that there is $M>0$ s.t. $$S\leq T\leq M\quad a.s.$$ Show that if $B\in \mathcal F_S$, we have that $$\sigma =T\boldsymbol 1_B+M\boldsymbol 1_{B^c},$$ is a stopping time…
John
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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 hand side could be split into two disjoint part…
Apocalypse
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In what the fact that $\boldsymbol 1_{T<\infty}X_T$ is $\mathcal F_T-$measurable where $T$ is a stopping time is a good thing?

Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space and $T$ a stopping time. Let $(\mathcal F_t)$ a filtration and $$\mathcal F_T=\{A\in \mathcal F\mid A\cap\{T\leq t\}\in \mathcal F_t, \forall t\geq 0\}.$$ Let $(X_t)$ a stochastic process. In…
bob
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Proof of Optional sampling theorem

In the proof of the optional sampling theorem they define for a stopping time $\tau$ the sigma algebra $\mathcal{G}=\sigma(\cup_n \mathcal{F}_{\tau\wedge n})$. Then they use the fact that for the event $A\in F_\tau$ it holds that $A\cap…
user202723
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Stopping time, event, simple description

Let us suppose that we have two stopping times $T$ and $S$, where $T \leq S$. Can someone explain on a practical example why is event ${(T \leq n)} \subseteq {(S \leq n)}$?
Zenga
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