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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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Distribution of the stopping time for a size dependent process

Let $X(t)$ denote some stochastic process (which we assume is in steady state, i.e. the distribution of $X(t)$ is independent of $t$). We assume $X(t)$ decreases at a constant rate equal to one. ...
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1answer
24 views

Optional sampling theorem with a.s. finite stopping time

I'm trying to prove the following generalized version of Doob's optional sampling theorem: Let $X$ be a square integrable martingale with respect to a filtration $\mathbb F = \left\{ \mathcal F_n \...
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33 views

min & sum of discrete stopping times using alternate definition

Let $A, B$ be stopping times of a process $X_0, ... X_n$ where $A$ is a stopping time iff $I_{A = k}$ is a function of $X_0, ... X_k$ (where $I$ denotes an indicator variable). We want to show that: ...
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28 views

First exit time of a simple random walk

Let $S_n=\displaystyle\sum_{i=1}^{n}X_i$ be the simple random walk with $S_0=0$. Now define the stopping time $$\tau=\inf\{n>0| S_n\notin (a,b)\}.$$ I am trying to follow some steps in a book and ...
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21 views

probability with martingales 12.2 sum of zero-mean independent variables in L^2

I am struggling with the following theorem from David Williams, Probability with Martingales: THEOREM Suppose that $(X_{k}:k\in\mathbb{N})$ is a sequence of independent random variables such that, ...
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18 views

Super martingale and stopping time

I am a little confused about stopping times and martingales. Suppose I have a super-martingale $f(X_t)$ where $X_t$ is a martingale. Also, $f()$ is increasing in $X_t$. Consider a person looking to ...
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1answer
14 views

Strong markov property vs usual markov property.

I was trying to understand the difference between strong Markov property and the usual Markov property for a discrete number of states. I think I understand why the strong Markov property implies the ...
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Crossing times of processes

Suppose that $X_t$ and $\tilde{X}_t$ are processes defined on the stochastic basis $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ such that for any time $t>0$, $1>p_t>0$, where $p_t$ is ...
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1answer
17 views

Image of a symmetric law

Assume I have a probability space $(\Omega, \mathcal{F}, P)$ that is mapped by a measurable function $X$ into $(E,\mathcal{E})$, moreover $P(X \in U)=P(-X \in U)$, now $Y$ maps this measurable space ...
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17 views

Probability of Ruin at the first claim

The number of claims $n \sim Po(\lambda)$, and let $X_n$ denotes the claim amounts of a claim which are all iid and they follow a $Exp(1)$ distribution. Assume the initial surplus is $U$, and the ...
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21 views

First passage time for Compound Poisson distribution

Let $S$ follows a Compound Poisson distribution $(S \sim CP(\lambda,F_x(x))$, i.e. $$S = \sum_{i=0}^{N}X_i,$$ where $N\sim Po(\lambda)$ and $X_i \stackrel{iid}{\sim} Exp(1)$. I know that the first ...
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1answer
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Doob's Optional Stopping Theorem: $\xi_\tau$ vs $\xi_{\tau\land n}$

I have some troubles in understanding the Optional Stopping Theorem by Doobs. I have a bit of confusion about the following (Brzezniak, Zastawniak - Basic Stochastic Processes p. 58-59): Let $\...
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30 views

Expected value of a Brownian motion before its first hitting time

Let $X_{t}$ be a Brownian motion with drift $\mu=0$ and variance $\sigma^{2}$. Also, let $X_{0} = a < b$. We know that the density of the first hitting time $H_{b} = inf \lbrace t: X_{t} = b \...
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1answer
21 views

Computing $E(T_b^2)$ for asymmetric random walk

Let $S$ be an asymmetric random walk with $p=P(X_1=1)>1/2$. Define $T_b=\inf\{n:S_n=b\}$. Prove that $\text{var}(T_b)=\frac{4bpq}{(p-q)^3}$ where $q=1-p$. We know $$\text{var}(T_b)=ET_b^2-(ET_b)^...
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25 views

Stopped random walk is not uniformly integrable

I know that in general Doob's Optional Stopping Theorem doesn't hold for unbounded stopping times, but that it does when the system up to the stopping time is uniformly integrable. One counter ...
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22 views

Example of last hitting time failing to be a stopping time?

I'm trying to find an example where the last hitting time $\theta = \sup\{k \ge 0 : X_k \in B\}$ of a set $B$ by the stochastic process $(X_k)_{k\ge 0}$ is not a stopping time.
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Probability of max of dependent random variables

I am trying to compute the following probability: $$P(X-Y_k\leq f_k(D)-c, X-Y_i\geq f_i(D)-c\, \forall\,i: 0\leq i\leq k-1),$$ where $f_i$ are functions and $D,c$ are considered to be fixed, and $X,...
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29 views

What's the precise statement of the continuous-time optional stopping theorem?

I searched high and low in a number of probability / financial mathematics textbooks and surprisingly cannot find any precise statement of the continuous time optional stopping theorem. In particular, ...
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26 views

Does hitting time have scaling property or self-similar proerty

Consider the hitting time such that $\tau_x = \inf\{t>0: B_t = x\}$ where $B_t$ is a standard brownian motion. Can $\tau_x$ be scaled like that $\tau_{ax}=a^2\tau_x$ for $\forall a>0, a\in \...
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Stopping moment regarding the ratio of black balls to all balls.

The urn contains at start one black ball, one white ball and one green ball. In the next steps, we randomly pick a ball from the urn and put it back with one additional ball coloured as the one we ...
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27 views

Random Walk: Proving that $1 = \sum_{m=0}^{n}P_0(S_{n-m} = 0)P_0(\tau_0 > m)$

I would appreciate a further hint for this question: Let $S_n$ a random walk on $\mathbb Z$, with $S_0=0$. Let $\tau_0 = \inf\{n>0:S_n=0\}$, the hitting time of $0$. Show that $$ 1 = \sum_{m=0}...
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1answer
29 views

Expressing $\mathbb{P} \left( \sup_{s \leq t} B_s>a \right)$ in terms of stopping times

In this video lecture the professor is proving the theorem that For a Brownian motion $(B_t)_{t \geq 0}$ it holds that $$P(M(t)>a) = 2 P(B(t)>a)$$ where $M_t := \sup_{s: s \leq t} B(s)$. ...
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65 views

Ornstein Uhlenbeck process hits zero with probability 1 in finite time

I am looking for a reference or a proof which shows that $P(\tau_0^Y<\infty)=1$ for an ornstein uhlenbeck process $Y$ given by $$ dY(t)=-\frac{1}{2}\alpha Y(t)dt+ \frac{1}{2} \sigma dW(t),Y(0)=y&...
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19 views

Two valued stopping time gives a martingale

Let $T:S\to\mathbb{N}$ be a stopping time. Let $(X_n)_n$ be an $(F_n)_n$-adapted process that is integrable. How do we prove that $X_n$ is a martingale $\iff$ for all $T$ that take maximally two ...
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1answer
39 views

How to prove that the exit time of a Brownian motion is a stopping time?

Given the following setting: Let $\{W_t:t\geq 0\}$ be a Brownian motion. for arbitrary $a>0$, define the exit time of the interval $[-a,a]$ as $$\tau=\inf\{t\geq 0:|W_t|>a\}$$ The question ...
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1answer
36 views

stopping time and quadratic variation process

Let $\tau$ be a stopping time and $(M_n)_{n \in \mathbb{N}_0}$ be a martingale with $\mathbb{E}(M_n^2)<\infty$ for any $n \in \mathbb{N}_0$. Show that, if $\langle M \rangle_{\tau} = 0$ (where it ...
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1answer
51 views

Conditional expectation of Brownian motion given stopping-time sigma algebra

Let $W$ be a Brownian motion with filtration $(F_t)$. Let $\tau$ be a stopping time. It is well-known by the strong Markov property that the law of $W_{\tau+t}-W_\tau$ given $F_\tau$ is normal with ...
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121 views

Conditional expectation for Brownian motion

Consider two Brownian motions $(W_t)_{t\ge 0}$ with starting point $x$ and $(W'_t)_{t\ge 0}$ with starting point $y$. Define $T:=\inf\{t\ge 0:W_t=0\}$, the first time when $W_t$ is equal to $0$. Show ...
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1answer
32 views

Cauchy density function for Brownian motion

Let $\{W(t):t\geq0\}$ be a Brownian motion, and let $\{\mathcal{F}_{t},t\geq0\}$ be its natural filtration. Let $\{W_{2}(t):t\geq0\}$ be a Brownian motion, independent of $\{W(t):t\geq0\}$. Denote, ...
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69 views

Convergence of $\sigma-$algebra for converging stopping time

Given a filtration, ${\mathcal{F}_t},t\in[0,\infty).$ Let $T_n$ be a sequence of stopping time that converges to $T$ and $T_n\le T_{n+1}.$ We have correpsonding $\sigma-$algebra, ${\mathcal{F}_{T_n}}$ ...
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90 views

Collection: Results on stopping times for Brownian motion (with drift)

The aim of this question is to collect results on stopping times of Brownian motion (possibly with drift), with a focus on distributional properties: distributions of stopping times (Laplace ...
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38 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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How to find the first crossing time of an Ornstein-Uhlenbeck process across a boundary?

If we have an Ornstein-Uhlenbeck process defined on a radial coordinate, 0$\leq r $, by $dr=-\alpha r \cdot dt +\sigma \cdot dW$, which has a value $r(t=0)=r_0<S$, how can we find the distribution ...
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37 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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1answer
20 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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52 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
43 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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0answers
10 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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25 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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64 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
73 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
110 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
45 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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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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82 views
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80 views

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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29 views

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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25 views

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
25 views

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
21 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 &...