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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Strong Markov property and stopping time

In the book by Jeanblanc, Yor & Cheney, "mathematical methods for financial markets", on page 17 above Prop.1.1.14.3, there is a strange identity of a strong Markov process $X$ that reads $${\bf ...
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82 views

Questions on Doob's Optional Stopping Theorem (a) and (b)

From Williams' Probability w/ Martingales: What is $X_T$ in red box above? I am fairly certain this was not defined previously in the book. There was this though: I have a feeling $X_T = ...
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21 views

Optional Sampling Theorem - Martingales

I have problems with solving the following problem. Can anyone give me a hint how to solve it? Thanks in advance! Consider a contract that at time N will be worth either 100 or 0: Let S(n) be its ...
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Stopping times and hitting times for càdlàg processes

I can't find the proof of the following lemma in any book: LEMMA: If $X=\{X_t\}_{t\in T}$ is adapted and right continuous, then for every closed set $C \subset E $, the variable ...
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Solving Stochastic Differential Equation

Let $\beta > 0$, $0 < \gamma < 1$, and let $\tau$ be the first hitting time: $$\tau = \inf\{t:t \geq 0, |W_t| = \pi /4\}$$ Solve the SDE in the random interval $0 \leq t \leq \tau$ $$dX_t = ...
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53 views

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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29 views

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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1answer
61 views

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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64 views

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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89 views

Proof that the stopping time for a Brownian Motion is finite for given target levels

Given a standard brownian motion $W_t$ and defining $\tau$ as: $\tau :=\inf\{t\geq0:W_t=1$ or $W_t=-2\}$ The proof below shows that the stopping time is finite: $$\begin{align*} P(\tau < t) ...
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Is the following a stopping time?

If $\{X_n\}_{n\in\mathbb N_0}$ is a Markov chain is $T:=\{\inf n\ge1:X_{n}=X_{n-1}\}$ a stopping time ? $\{T=n\}=\{X_0\neq X_1, X_1\neq X_2\dots X_{n-2}\neq X_{n-1},X_{n-1}=X_n\}$, I would say ...
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Conditional expectation and stopping time $\mathbb{E}(X1_{T\leq m}|\mathcal{F}_{T\wedge m})=\mathbb{E}(X1_{T\leq m}|\mathcal{F}_T)$

Let $X$ be a random variable and $T$ a stopping time in a filtrated probability space. If $m > 0$ is it true that: $$\mathbb{E}\left(X1_{T\leq m}|\mathcal{F}_{T\wedge ...
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28 views

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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Application of Laplace transform to stopping times and expectations

Let $X_k$ be i.i.d. random variables such that $E[X_1]=m<\infty$. Consider $S_n = \sum_{k=1}^{n} X_k$. Let $\tau$ be a stopping time independent of $X_k$ with respect to the filtration $\{F_n\}_{n ...
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185 views

The expected value of stop-time for Brownian motion $\tau=\min_t\{B_t^2\geq t+1\}$.

Let $B_t,\;t\geq0$ be a standard Brownian motion. Define the stopping time $$\tau = \min_t\{B_t^2\geq t+1\}$$ Is the expected value $E(\tau)$ finite? Actually, my raw problem as following: $$\gamma ...
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1answer
33 views

Show submartingale property.

Let $\tau$ be a stopping time. Let $X_k$ be iid random variables such that $E[X_i] = m < \infty$. Also, $m>0$.Show that $\sum_{k=1}^{\min(\tau,n)} X_k$ is a submartingale. We need to show ...
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1answer
57 views

Prove $\tau=\inf\{t\in[0,T]:M_t=0\}\wedge T$ a stopping time for a continuous martingale $(M_t)_{t \geq 0}$

I have a question about a positive continuous martingale. Let $(M_t)_{t\in[0,T]}$ be a continuous martingale such that $P(M_t>0)=1$ for all $t\in[0,T]$. Set $\tau=\inf\{t\in[0,T]:M_t=0\}\wedge T$. ...
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2answers
96 views

Showing stopping is finite almost surely

Consider a discrete random walk taking values +1 or -1 with probabilities p and q, respectively. Let $S_n = \sum_{k=1}^{n}X_k$. Let $[-A,B]$ be an interval, $A,B \geq 1$. Now define $$\tau =\min(n:n ...
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Requesting references on the number of times a random process hit a target.

Let $X_t$ be a continuous process, $A$ be some event, and $N(x)$ be the number of times $X_t(x)$ enter $A$ in a fixed time window $t \in [a,b]$. Are there results concerning $N(x)$? Like, what its ...
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How to find intersection of moving circle and line?

Say I have a point, with position (x1,y1) at time t=0, with velocity dx1 and dy1 in the x and y directions respectively, which may or may not collide with a circular entity with radius r, centered at ...
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If $(F_t)_t$ is a filtration, $T$ is a stopping time and $Y$ is $F_T$-measurable, then $1_{\left\{T=s\right\}}Y$ is $F_s$-measurable

Let $(\Omega,\mathcal A)$ be a measurable space $I\subseteq[0,\infty)$ $\mathbb F=(\mathcal F_t)_{t\in I}$ be a filtration on $(\Omega,\mathcal A)$ $\tau$ be a $\mathbb F$-stopping time $\mathcal ...
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1answer
36 views

If $X$ is an $\mathcal F_t$-adapted process with countable time domain and $\tau$ is a stopping time, then $X_\tau$ is $\mathcal F_\tau$-measurable

Let $(\Omega,\mathcal A)$ be a measurable space $I$ be an at most countable set $\mathbb F=(\mathcal F_t)_{t\in I}$ be a filtration in $(\Omega,\mathcal A)$ $X=(X_t)_{t\in I}$ be an $\mathbb ...
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1answer
21 views

Is there any intuition behind the statement $E[X_\tau \mid \mathcal{F}_\sigma]=X_\sigma$

Is there any intuition behind the statement $E[X_\tau \mid \mathcal{F}_\sigma]=X_\sigma$ a.s. I mean I know that the interpretation of the conditional expectation and how to visualize it somehow but I ...
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1answer
60 views

Why do we need optional stopping theorem?

For martingale,optional stopping theorem states: Let $(M_n)_{n\in \mathbb{N}}$ be adapted with $M_n\in L^1$ for all $n$ and if $(M_n)_{n\in \mathbb{N}}$ is a martingale, then $E[M_T]=E[M_0]$, for all ...
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1answer
50 views

Exist $\alpha < \infty$, $\beta > 0$ such that $\mathbb{P}\{T_\lambda > t\} \le \alpha e^{-\beta t}?$

Let $B_t$ be a standard one-dimensional Brownian motion. Suppose $\lambda > 0$ and let$$T_\lambda = \min\{t : |B_t| = \lambda\}.$$Do there exist $\alpha < \infty$ and $\beta > 0$ (which may ...
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1answer
38 views

A martingale characterization

I saw the following characterization of martingales (without proof) in some lecture notes I found on the web and I haven't been able to produce a proof it. Let $X$ be an adapted process. If ...
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1answer
46 views

Stopping time proof with discrete martingale

My professor gave me a very unclear proof of this theorem. It was so messy and unclear, I was unable to write down all the details of the proof. Theorem: Suppose $\tau \in T$, where $T$ is the set ...
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Time integral of Brownian motion's running maximum

Let $\mu \geq 0$ and consider $B_{\mu}(t) := B(t) + \mu t$ a one-dimensional BM with drift $\mu,$ and let $M_t := \max_{0 \leq s \leq t} B_{\mu}(t)$ be its running maximum. My question involves two ...
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1answer
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On the proof of lemma 1.2.4 of Stroock and Varadhan A question concerning stopping times

In the book Multidimensional diffusion processes, of Stroock and Varadhan one reads (page 23): This is the proof of $(i)$. Here the authors say Define $f_t$ on $(\{\tau \leq t\}, \mathcal{F}_t ...
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1answer
73 views

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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110 views

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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1answer
41 views

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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1answer
27 views

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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1answer
23 views

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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1answer
101 views

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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1answer
52 views

Find $\mathbb{E}_{X_0 = x} X_\tau$ for an Ornstein-Uhlenbeck process $(X_t)_{t \geq 0}$ where $\tau = \inf\{t>0 \mid X_t \notin [a,b]\}$

Let $X_t$ satisfy the following SDE: $dX_t = X_t dt + \sigma dB_t$, $\sigma$ is a constant and $B_t$ is Brownian Motion. Find $\mathbb{E}_{X_0 = x} X_\tau$ where $\tau = \inf\{t>0 \mid X_t \notin ...
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A Markov Chain probability, conditioned on a random time.

My question: Upon reading theory about diffusion processes, i came across an argument which i believe simplifies to this: Say we have a Borel measurable set $A$ (if it matters you can set $A=\lbrace ...
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1answer
31 views

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 ...
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31 views

Simple Probability Inequality with Stopping Times

Suppose $U_1,...,U_n$ are independent random variable with $\mathbb{E}[U_i]=0$. Define $Z_k:=\sum_{i=1}^k U_i$. Set $T:=\inf \lbrace k \in N \mid |Z_k|>2\alpha \rbrace$. Clearly $\lbrace T=k ...
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1answer
75 views

An Algorithmic approach to the secretary problem with unknown n

I've been reading about the secretary problem these days and I got the idea for the case when we know the number of applicants $n$ in advance. I'd like to know what would be an algorithmic approach ...
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1answer
43 views

$E[|X(t)|]\leq K\implies E[|X(\tau)|]\leq K $?

Let $X(t)$ be a stochastic process. Assume that, for every $t\leq M$, it holds $$E[|X(t)|]\leq K, $$ for some constant $K$. Let now $\tau\in[0,M]$ be random (stopping time). Is it true that also ...
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1answer
45 views

Distribution Stopping time under Brownian motions

Considering $W$ the canonical process on $C([0,1],\mathbb{R})$ and the row filtration generated by the coordinate process of $W$, I want to prove that ...
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1answer
40 views

Illegal lottery problem (Merging dependent bernoulli trials)

Suppose I am in a town that playing lottery is illegal. If I buy a ticket for 1 dollar, I will win the lottery with probability $p$. Each time I buy a ticket, the police may catch me and confiscate ...
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113 views

Secretary Problem - Optimal algorithm for expected value of candidate.

I recently encountered the secretary problem and there are essentiall two problems: Maximizing the probability of choosing the best candidate. Gnedin proved in his paper ...
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Charaterize the $\mathcal{F}_\tau$ a sigma algebra for the stopping time $\tau$

Consider a stochastic process $X: [0, \infty) \times \Omega \to \mathbb{R}^d$ We define $\mathcal{F}_t = \sigma(X(s), 0 \leq s \leq t)$ The sigma algebra generated by the sets $\{\omega: X(s,\omega) ...
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42 views

Martingale: Show $p\{T<+\infty \}=1$.

Let $(X_i)$ i.i.d. such that $p\{X_i=+1\}=p\{X_i=-1\}=\frac{1}{2}$ and let $(S_n)$ the martingale define by $S_0=0$ and $S_n= X_1+...+X_n$. Moreover, let $$T=\begin{cases}\inf\{n\geq 0\mid ...
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1answer
34 views

How to show that consecutive hitting times are a sequence of stopping times?

Let $\{X_n:n=0,1,\ldots\}$ be a martingale with respect to a filtration $\{\mathcal F_n\}$. Let $A,B$ be nonempty, disjoint Borel sets and define $T_0=0$, \begin{align} S_n &= \inf\{m\geqslant ...
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2answers
90 views

“The first time a continuous local martingale grows in absolute value beyond $n$” is a localizing sequence

How can it be shown that, for a continuous local martingale $X$ defined w.r.t. the filtered probability space $(\Omega, \mathcal{A}, P; \mathcal{F})$, the stopping times $\tau_n := \inf \{t \geq 0 ...
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28 views

Expected difference between a process and a stopped process

Let $\tau$ be a stopping-time such that for some $t\geq 0$ the probability $P(\tau<t)=\epsilon$ is small. I am interested in quantifying the following difference $$E[f(X_t)] - E[f(X_{t\wedge\tau})] ...