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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Is the supremum of a right continuous process right continuous? Defining a stopping time as the first hitting time of a supremum process

This is a question that came to my mind while reading Lemma 13 of the notes here : https://almostsuremath.com/2009/12/23/localization/ In this lemma, the writer assumes $X$ is a right-continuous ...
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$\{\omega \ge \tau\} \in \mathscr{F}_{\tau -}$ for stopping times

Let $\tau, \omega$ be two stopping times. How do we show that $\{\omega \ge \tau\} \in \mathscr{F}_{\tau -}$? This result is used in the proof of Lemma 11 from : https://almostsuremath.com/2009/12/23/...
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Does inequality between stopping times depend on $\omega$?

We work on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in[0,T]},\mathbb{P})$, where $T$ is finite. Let $(\tau_n)$ be a sequence of stopping times increasing stationarly to $T$,...
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Analytical solutions to $E[f(X_\tau) e^{-\alpha\tau}]$

Let $X_t$ be a geometric Brownian motion: $$dX_t=\alpha X_tdt+sX_tdB_t,$$ where $\alpha,s>0$ are constants. Let $\tau=\inf_{u\geq t}\{X_u=K\}$ be the first time $X_t$ reaches the threshold $K$ (...
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Is it possible to recover the fact that $\mathbb P(T_{a} < T_{b} ) = b/(b-a)$ using $\mathbb E[ e^{-\lambda T_{a}}\chi_{\{T_{a}< T_{b}\}}]$

Consider two hitting times $T_{a},T_{b}$ for a Brownian motion $B$ such that $a < 0 < b$. Is it possible to recover the fact that $\mathbb P(T_{a} < T_{b} ) = b/(b-a)$ using the fact that I ...
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Iterated optional times

In the following proposition of Kallenberg's book Foundation of Modern Probabilities, where $\theta_t$ is the shift operator defined by $\theta_t(\omega)(s)=\omega(t+s)$, Proposition 11.8 (iterated ...