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

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30 views

Standard Brownian motion and stopping time

Let be $B$ standard Brownian motion and let $S \leq T$ two stopping times with $E(T) < \infty $ and $E(S) < \infty$. Prove that hold $$ E[(B_T - B_S)^2] = E[B_T^2 - B_S^2] = E(T-S).$$ Please ...
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19 views

Heads and tails with unlimited capital

Let $ S_{n} $ be the sum of won money by player 1 until moment $ n $ He gets 1 if he wins and loses 1 if he loses Let $ X = \inf\left\{ n : S_{n} = 1\right\} $ We assume that coin is asymetric and ...
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2answers
35 views

Proving that stopping time is finite a.s.

Let $$\tau_{a} = \inf\{t>0 : W_{t} + at = 5\}.$$ Prove that $\mathbb{P}(\tau_{a}<\infty) = 1$ for $a\ge0.$ My solution: We know that $W_{0} +a*0 < 5$. Furthermore, because $W_{t} \sim \...
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1answer
32 views

Bounding expectation of stopping time

Let $(X_{t})_{t\ge0}$ be adapted to $(\mathcal{F}_{t})_{t\ge0}$ with continuous trajectories. Assume that $X_{0} = 0$ and $X_{t}^{4} - 3t^{2}$ is a martingale with respect to $(\mathcal{F}_{t})$. Let ...
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26 views

Brownian Motion Absorbed at a Value [closed]

Let $\{B(t)\}$ be a Brownian Motion. For $a<0$, a Brownian Motion absorbed at a value $a$ is defined as $$B_a(t)= \begin{cases} B(t)& \text{ if } \min_{0\leq s\leq t}B(s)>a\\ a & \text{ ...
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6 views

Local martingale with constant stopping time

Let $M_t$ be a continuous local martingale (there exist an almost surely divergent sequence of stopping times $(T_n)$ such that $M^{T_n}$ is a square integrable martingale). Is it true that for each ...
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1answer
20 views

Compute $E(M_σ)$ when $σ = \inf(t ≥ 0: |B_t| = 1)$ and $M_t = 4B^2_t +e^{4B_t−8t}−4t$

Given $M_t = 4B^2_t +e^{4B_t−8t}−4t$ for $t ≥ 0$ and a Brownian motion $(B_t)_{t \geq 0}$. Compute $E(M_σ)$ when $σ = \inf(t ≥ 0: |B_t| = 1)$. I have tried to show that $E|M_σ|\leq K$ to apply Doob's ...
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43 views

Finding expected value of a stopping time dependent on a Poisson process and a variable $n$

Situation: We have that $\{W_t,t \geq 0\}$ is a Brownian motion and $\{N_t,t\geq 0\}$ is a Poisson process such that $N_t$ follows a Poisson distribution with parameter $t$. This process is ...
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1answer
40 views

Expected value of Brownian motion at a time decided by a rate one Poisson process.

Situation: We have that $\{W_{t},t\geq 0\}$ is a Brownian motion and $\{N_{t},t\geq 0\}$ is a Poisson process such that $N_{t}$ follows a Poisson distribution with parameter $t$. This process ...
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1answer
85 views

Show a Continuous Local Martingale is a Martingale

Let $B = (B_t)_{t≥0}$ be a standard Brownian motion started at zero, let $X=(X_t)_{t≥0}$ be a nonnegative stochastic process solving $$dX_t = 3 \, dt + 2\sqrt{X_t} \, dB_t \qquad(X_0 = 0)$$ and let $$...
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1answer
34 views

The distribution of a stopping time

Let $(X_n)_{n\geq0}$ be a sequence of real $i.i.d$ random variables and $\tau = \inf\{n\geq0 : X_n\in S\}$ with $S \in \mathcal{B}(\mathbb{R}) $ I am trying to find $\tau$'s distribution. Obviously, ...
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22 views

Intersection Exponent for one-dimensional Brownian Motion

We let $B^1,B^2$ be independent, one-dimensional Brownian Motions with $B^1(0)=1$ and $B^2(0)=-1$ and $T_n^i=\inf\{t\geq0:|B^i(t)|=n\}$. In Gregory Lawler's: Hausdorff Dimension of Cut-Points for ...
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Proof that thin sets are finely separated

I would like to justify the following claim concerning thin sets found in Chung, Lectures from Markov Processes to Brownian Motion, page 112: Such a set is finely separated in the sense that each ...
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Hitting time distribution with exponential growth

Let $A_0=A>0$ and let $$dA_t = (rA_t - x)dt + \sigma dB_t,$$ where $B_t$ is standard Brownian motion and $r,x$ and $\sigma$ are positive constants. Let $T= \inf \{ t: A_t = 0 \}$ and $$G(A)=\Bbb{E}[...
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Is the following stopping time finite: $T:=\inf\{t\geq 0:B_t\geq \sqrt{t}+1\}?$

We have a Brownian motion process $B$ and a stopping time defined like this: $$T:=\inf\{t\geq 0:B_t\geq \sqrt{t}+1\}.$$ Is this stopping time almost surely finite, eg. $T<\infty$, and why? My ...
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2answers
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If $E(X_n^2)<\infty$, then for a Martingale $E(X_n^2)<M$ iff $\sum_{n=1}^\infty E[(X_n-X_{n-1})^2]<\infty$

Let $\{X_n\}_{n\geq0}$ be a martingale with $E(X_n^2)<\infty$ for all $n$. How to prove that: $E(X_n^2)<M$ for all $n$, if and only if $\sum_{n=1}^\infty E[(X_n-X_{n-1})^2]<\infty$. The ...
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1answer
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Why is $T:=\inf\{t\geq0:B_t\leq at^p-b\}$ a stopping time?

How can I prove that $T:=\inf\{t\geq0:B_t\leq at^p-b\}$ is a stopping time w.r.t. a natural filtration of $B$, where $B$ is a $BM$, $p>1/2$ and $a,b>0$? I can introduce a new random process, $...
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1answer
19 views

Conditions for Markov process not to reach point at infinity

My question concerns the book Lectures from Markov Processes to Brownian Motion by Kai Lai Chung, more precisely the remark at the bottom of page 76: We prove later in paragraph 3.3 that on $\{ t &...
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1answer
88 views

Probability on first hitting time of Brownian motion with drift

I am struggling with the following problem: Let $B$ be a one dimensional Brownian motion and $a,b>0$. Show that $$P[B_t=a + bt \text{ for some } t\geq 0] = e^{-2ab}.$$ The following hint is ...
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0answers
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The problem is about the expection of the exitpoint distance for the symmetric random walk.

Let $\nu(x)$ be a symmetric probability measure with respect to the origin on $x\in[-1,1]$ such that $\nu(\{0\})\neq 1$. Consider a random walk started at $S_0=0$, denoted $S_n=X_1+\cdots+X_n$, where ...
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1answer
41 views

Martingale of random walk and stopping time

Let $\{S_n\}$ be a symmetric random walk such that $S_0 = a$ for some $0 < a < K$. Let $T$ be the stopping time when the walk reaches $0$ or $K$. Show $$M_n = \sum_{i = 0}^n S_i - \frac{1}{3}...
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0answers
20 views

Convergence of expectation of hitting time of a symmetric random walk

Let $\{X_{n}: n\geq1\}$ be i.i.d random variables with the common distribution $\mathbb{P}(X_n=1)=p$ and $\mathbb{P}(X_n=-1)=q=1-p$ where $0<p<1$. Define $S_0:=0$ and $S_n = \sum_{j=1}^{n} X_j$ ...
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1answer
36 views

Stopping time of Feller process

Let $X$ be a Feller process on $\mathbb{R}$ with generator $Gf=\frac{1}{2}f''-f'$ on $C_c^2$. Let $\tau_b$ be the first time that $X$ hits $b\in\mathbb{R}$. Show that for $x > 0$, $bP_x(\tau_b < ...
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17 views

Conditional expectation of martingale and two bounded stopping times

I am trying to prove the following: Let $(X_n)$ be a martingale with respect to $(\mathcal{F}_n)$ and suppose $\tau_1$ and $\tau_2$ are bounded stopping times such that $\tau_1\le \tau_2 < B$, ...
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1answer
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A hitting time bigger than another hitting time

My question is : Show that if $\tau'$ is another hitting time for the filtration $\mathcal{F}_n$, then $$\tau := \inf\{n\geq\tau' : X_n \in B, \mbox{where $B$ is a Borel set}\}$$ is also a ...
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1answer
34 views

Stopping time with finite expectation

Given a filtered space $\left(S, \mathcal{\Sigma}, \{\mathcal{\Sigma_n: n\ge 0}\}, \mathbb{P}\right)$, and a stopping time $\tau: S \to \{0, 1,2, \ldots\}$ with respect to $\{\mathcal{\Sigma_n: n\ge 0}...
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1answer
42 views

Prove that $\mathbf{E}(X_{\tau_2}|\mathcal{F}_{\tau_1})=X_{\tau_1}$

Let $(X_n)$ be a martingale with respect to $(\mathcal{F}_n)$ and suppose $\tau_1$ and $\tau_2$ are bounded stopping times such that $\tau_1\leq \tau_2<B<\infty.$ Then $$\mathbf{E}(X_{\tau_2}|\...
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2answers
56 views

Show that $E(X|\mathcal{F}_\tau)=\sum\limits_{n\in\mathbb{N}}E(X|\mathcal{F}_n)\mathbf{1}_{\{\tau=n\}}$

If $\mathbf{E}X<\infty$ and $\tau$ is a stopping time, then $$\mathbf{E}(X|\mathcal{F}_\tau)=\sum_{n\in\mathbb{N}}\mathbf{E}(X|\mathcal{F}_n)\mathbf{1}_{\{\tau=n\}}.$$ My attempt: First assume ...
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0answers
38 views

Simulation of hitting time of brownian motion with drift

I want to generate the hitting time of brownian motion with drift (upper and lower depending on some binomial random variable $\delta = 0,1$). $\tau^{up} = inf(\tau : \mu \tau + \sigma W_{\tau}\ge h)...
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Proof verification: Application of optional stopping theorem

I am wondering if the proof of this equation below using the optional stopping theorem is correct. $\textbf{Proposition}$ Let $\tau : \omega \mapsto \{1,2,\dots,\infty\}$ be a stopping time w.r.t a ...
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0answers
15 views

Convergence in probability of stopping times

Let $(X^n)_n$ be a sequence of continuous real-valued stochastic processes in $[0,T]$, $T<\infty$ such that for every $t\in [0,T]$ $X^n_t$ converges in probability to $X_t$. Let $\tau^n=\inf\{t\in[...
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0answers
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Show that τ is a stopping time with respect to FX

https://i.stack.imgur.com/gvLDp.png Given the probability space in the attached image I would like to Show that τ is a stopping time with respect to FX. I have calculated the FX to be: F0X=σ({X0})=(...
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1answer
36 views

For a Brownian motion $(B_t)_{t \geq 0}$, do we have $E[(B_\tau - B_\sigma)^2]=E[B_\tau^2 - B_\sigma^2]$ for stopping times $\sigma \leq \tau$?

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0}, P)$ be a filtered probability space and let $(B_t)_{t \geq 0}$ be a Brownian motion on that space. The question is if the following is true: For ...
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1answer
30 views

Secretary problem - Is there an equation that allows one to have $r = 0$?

Secretary problem's equation I found this equation on wikipedia to resolve the secretary problem. I understand it but I have a small problem. Theoretically, if I would not want to reject any ...
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1answer
50 views

Expectation of a stopping time w.r.t Brownian motion

For a real-valued standard Brownian motion $B= (B_t)_{t\geq 0}$ we define the stopping times $ \tau_{a} := \inf \left\{ t> 0: B_t \leq a \right\},~a <0$, $ \tau_{b} := \inf \left\{ t&...
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On random integral

Excuse my question which is a little naive, but I want to know if the equality below is true Let $\tau$ be a finite stopping time ($\tau<T$) $$ \int_{\tau}^{T}du\overset{d}{=}T-\tau $$ where $\...
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1answer
48 views

Probability of hitting time as limit

Given a Markov Chain $(X_n)_{n \ge 0}$, the hitting time of a subset $A$ is defined as: $$T = \text{inf}\{n \ge 0, X_n \in A \}.$$ Many basic exercises ask to calculate: $$P(X_{T}=a | X_0=i), \quad ...
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1answer
37 views

Probability of sumprocess

Consider $(X_n)_{n\ge 1}$ iid with $P(X_1=1)=P(X_1=-1)=\frac{1}{2}$. Then there is the sumprocess $S_n:=\sum_{i=1}^n X_i$ and the stopping times $$T_{a,b}:=\min\{n\ge 1:S_n\in\{a,b\}\},$$ $$T_{a}:=\...
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18 views

Probability that a stopping time is smaller than another stopping time

Suppose an ant is undergoing Brownian Motion in $\mathbb{R}$. In each locality $x\in \mathbb{R}$, the rate of being killed by animal A is $f(x)$, and the rate of being killed by animal B is $g(x)$. ...
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1answer
32 views

show $\sup\{t \in \mathbb{N}_0 : S_t =1\}$ is a stopping time

Let $(X_n)_{n \in \mathbb{N}_0}$ be a sequence of independent and identically distributed random variables with $$\mathbb{P}(X_1 = 1) = \mathbb{P}(X_1 = -1) = \frac{1}{2}.$$ Define $S_t = \sum\...
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16 views

Common refinement for simple predictable processes

I'm trying to show that for a semimartingale $X$, the stochastic integral $\mathcal{I}_X : \mathcal{S}_{\text{ucp}} \rightarrow \mathbb{D}_{\text{ucp}}$ is cauchy continuous, since the idea is to ...
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0answers
13 views

Expected value of running minimum of BM

I need to calculate the following: $E[1_{\{Max_{0 \leq r \leq t} B(r) \ge 2\}}(-1 - min_{T_{2} \leq r \leq t}B(r))^{+}]$, where + means maximum of $0$ and the value in parenthesis. My attempt: The ...
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29 views

Set-theoretic minimum and maximum of two stopping times

I have a confusion about stopping times and hope you can help me there! Let $\sigma$ and $\tau$ be stopping times with values in $I=\big[0,\infty)$ with respect to the filtration $\mathbb{F} = (\...
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0answers
41 views

first exit time from the interval (a,b)(Brownian motion)

Let ${\left({B}_{t}\right)}_{t\ge 0}$ un $ BM^{1} $( Brownian motion),$\; a<0<b $, denote by $ \displaystyle\tau =\mathrm{inf}\left\{t\ge 0:{B}_{t}\notin (a,b)\right\} $ the first exit time from ...
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0answers
24 views

Find $P(\tau_a<\infty|X_0=x)$ given $dX_t = rdt+dW_t$ using PDE method

Let $dX_t = rdt+dW_t$ with $r$ constant, and $X_0=x$. Let $\tau_a$ be the hitting time of $X_t$ hitting $a$ for $a>0$. I want to compute $P(\tau_a<\infty|X_0=x)$ where $x<a$ using PDE method. ...
2
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1answer
73 views

Quick way for the expected first hitting time for a 2D Brownian Motion

Let $\{W_t\}_{t\ge 0}$ be a standard 2D Brownian motion starting at $(1,1)\in\Bbb R^2$. What's the probability that $W_t$ hits the positive half of $x$-axis before it hits the negative part? There ...
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1answer
34 views

Expected number of visits for a recurrent Markov chain

Let $(X_n)_{n \geq 1}$ be a recurrent irreducible Markov chain on a countable space. Let $a$ be a fixed point and $\tau$ be a stopping time such that almost surely $X_{\tau} = a$. For any $x,y$ let $G(...
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0answers
49 views

Brownian motion, first passage of time, strong markov property

So I need to find $P(max_{0\leq v \leq t} W(v) \geq 2$ and $min_{T_{2} \leq v \leq t} W(v) \leq -1)$. $T_{q}$ is the first passage time to level q My progress: If the first condition does not hold,...
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0answers
26 views

Counting processes with stationary and independent increments

I am reading a proof of the assertion that counting processes with stationary and independent increments are necessarily Poisson processes. There is some argument in it that I do not quite follow. I ...