# Tagged Questions

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### Prove of Stopping time

Let $(X_k)_{k\in\mathbb{N}}$ be iid random variables with $\mathbb{P}(X_1=1)=\mathbb{P}(X_1=-1)=\frac{1}{2}$. Let $Z_n=\prod_{k=1}^n(1+X_k)$, so $Z_n$ a martingale. Consider ...
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### Optional sampling

Let $(X_i)_{i\in\mathbb{N}}$ be iid random variables with $\mathbb{E}|X_1|<\infty$ and let $S_n \stackrel{\rm{}def}{=} X_1+\cdots+X_n$ for all $n\in\mathbb{N}$. If $T$ is a stopping time with ...
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### Example of a martingale and a stopping time with $E(T)<\infty$ but $E(X_T) \neq E(X_0)$

Is there an example of a martingale in discrete time $X_0, X_1, X_2,\ldots$ and a stopping time $T$ so that $E(T) <\infty$ but $E(X_T) \neq E(X_0)$? With added assumptions on how $X_n$ behaves, ...
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### Arguing on stopping time probability

Consider the random walk where $X_t=\sum_{i=1}^t Y_i$, $Y_i$s are iid and take $\pm$1 with probabilities $p$ and $1-p$ respectively, where $0<p<0.5$. Define stopping time ...
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### Implications between $\mathbb P [\tau < \infty] =1$ and $\tau \in L_1 (\mathbb P)$

We've got the usual filtered stochastic basis $(\Omega, \mathcal F, (\mathcal F_n). \mathbb P), \space \tau : \Omega \to \mathbb{N}\cup \{\infty\}, [\tau \le n] \in \mathcal F_n$ ($\tau$ is an ...
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### Stopping time $\tau_k$ measurable w.r.t. $\mathcal{F}_k$

We have $(\Omega,\mathcal{F},\mathbb{P})$ stochastic basis. Let $\tau: \Omega \to \mathbb{N}\cup \{\infty\}$ is a $(\mathcal{F}_k，k \in \mathbb{N} )$ be stopping time and define another stopping time ...
Assume we notice that stock prices are rising and we can deduce we are in a bubble. Assume we start at $w(0)=0$ worth at time $t=0$ and the value grows linearly with time $(w(t)=t)$. We know that ...