# Tagged Questions

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 $n$, ...

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### Augmented filtration martingale proof.

Part a: Consider a Wiener process, $W_t$ and denote by ${\mathscr{F}_t}_{(t \geq 0)}$ the natural filtration generated by W. Let $\mathbb{R}_{+} = \{x : x \geq 0\}$ and $\mathscr{B}$ be a sigma ...
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### Stopping time distribution and transforms with 1-dimension B-motion.

Let $W_t$ be a 1-dimensional Brownian Motion. For $x>0$, we define: $$\tau_{x} = inf \{ t \geq 0; |W_t| = x\}$$ Compute $E[e^{-s\tau_x}]$ and prove that $\tau_x$ is equal in distribution to ...
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### Is $\tau = \inf \{n :X_1+\cdots+X_{n-1}\leq 1\leq X_1+\cdots+X_{n-1}+X_n \}$ a stopping time?

Let $X_1, X_2, ...$ be a sequence of i.i.d. non-negative random variables. Let $$\tau = \inf \{n :X_1+\cdots+X_{n-1}\leq 1\leq X_1+\cdots+X_{n-1}+X_n \}.$$ Is $\tau$ a stopping time for the ...
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### Verify argument about random time being measurable w.r.t. a reversed BM increment process.

Let $B$ be a brownian motion (assume for convenient notation that it is two-sided). Let $T:=\sup\{t<2 :\vert B_t-B_1\vert\geq 1\}.$ Let $Y$ be the process $s\mapsto (B_T-B_{T+s})$. I want to argue ...
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### What is a good/extensive undergraduate level reference on random walks?

Random walks on graphs, expected times for different things, gambler's ruin. I seem to either stumble on some pretty advanced texts about group representation theory or texts that briefly mention it ...
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### Wald's Identity for Higher Moments

For the sum $S_N = \sum_{i=1}^N X_i$ where $N$ is a stopping time Wald's second identity tells us, if $var(X) = \sigma^2 < \infty$, that $\mathbb{E}[(S_N - N\mu)^2] = \sigma^2\mathbb{E}[N]$. I'm ...
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### Does this game make you arbitrarily rich with probability one?

We toss a coin. If it's heads we win $\$ 1$, otherwise we lose$ \$1$. Fix some large sum. Will we be winning this amount with probability one at some point? We assume that we have infinitely many ...
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### Stochastic variables independent given Tau

Say we have a filtration $(\mathbb{F}_s)$, and a stopping time $\tau$ w.r.t. to that filtration.Let $X_t$ be a continuous stochastic process (not required to be adapted to the mentioned filtration), ...
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### Independence of a hitting time and the underlying stochastic process

While I was playing around with the Girsanov's Theorem I stumbled upon the following absurdity and I couldn't resolve it with the current knowledge of stochastic analysis that I have. $B$ being ...
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### Show that $\mathbb{E}\left[c_{\tau\wedge n}X_{\tau\wedge n}-\sum_{i=1}^{\tau\wedge n}c_i\mathbb{E}(X_i-X_{i-1}\mid\mathcal{F}_{i-1})\right]\le 0$

I am trying to go through a past exam paper but I don't know how to deal with stopping times since we only did 2 exercises in class... I got stuck, so I would really appreciate if someone could help ...
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### Expectations of stopping times in general

I have a very basic question: So for a stopping time $\tau$ with $E(\tau)<\infty$ we have $E(\tau)=\sum_{n=0}^\infty P(\tau>n)$, right? Why is that? Thanks!
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### Markov and strong Markov properties

In my study of strong Markov property of an RCLL canonical Markov process I encounter the following definition: Suppose $Y_t:\omega\rightarrow \omega(t)$ is canonical Markov process with respect to ...
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### Rewriting probabilities as expectation

Consider the stopping time $\tau_a:=\lbrace{t>0| W_t >a\rbrace}$, where $W_t$ is a Brownian Motion. Define: $X_t:=W_{\tau_a+t}-W_{\tau_a}$. We have that $X_t$ is a Brownian Motion independent ...