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46 views

Why rational numbers in stopping times for continuous time processes

Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{\ge 0},P)$ be a filtered probability space. Let $X_t \in \mathbb{R}^n$ be a continuous stochastic process adapted to $\mathcal{F}_t$. Let $A \subset ...
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1answer
37 views

Meaning of $\mathcal A_{\tau}$ for stopping time $\tau$.

Let $(X_n)$ be a stochastic process, adapted to a filtration $\mathcal A_n$, and let $\tau$ be a stopping time, then $$ \mathcal A_{\tau} := \left\{ A \in \sigma\left(\bigcup_n A_n\right) : A \cap \{ ...
-1
votes
2answers
30 views

Brownian motion: first-hitting-time with double barrier [closed]

Let $(B_t)_t$ be a standard ($B_0=0$) Brownian motion , and $$ T_{a,b} = \inf\{t>0 : B_t \not\in(a,b)\} $$ where $a<0<b$. What is the expected first-passage time $\mathbf{E}[T_{a,b}]$?
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1answer
26 views

Optional Stopping Theorem for Stochastic Processes with Constant Mean (and not a Martingale)

Let $(X_n)_{n\geq 1}$ be a martingale with respect to $(Y_n)_{n\geq 1}$, i.e., the martingale condition $$ \mathbb{E}[X_n|Y_1, \ldots, Y_{n-1}] = X_{n-1} $$ holds. From this condition, it follows ...
3
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1answer
41 views

Uniformly integrable martingales and stopping time

I want to prove the statement below: Theorem: Let $(Y_n,\mathfrak{F})$ be a uniformly integrable martingale. Show that $(Y_{T\wedge n},\mathfrak{F})$ is a uniformly integrable martingale for any ...
1
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1answer
26 views

Unbounded stopping time

Suppose we have a sequence of i.i.d. random variables $(X_n)_{n \in \mathbb{N}}$ with $\mathbf{P}(X_n = -1) = \frac{1}{2}, \mathbf{P}(X_n = 0) = \frac{1}{3}, \mathbf{P}(X_n = 1) = \frac{1}{6}$. Denote ...
0
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0answers
18 views

Proof that a stopped continuous-time martingale is a martingale.

The proof for a stopped discrete-time martingale is shown as follows. Let $M=(M_n)_{n\ge0}$ be a discrete-time martinglae w.r.t. the filtration $(\mathcal F_n)_{n\ge0}$, and let $M^T=(M_{n\land ...
2
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0answers
24 views

Strong markov property in two dimensional Brownian motion

I don't understand the following claim from my book: Let $(B_t)$ be a standard Brownian motion. Let $u:\Omega \rightarrow \mathbb{R}$ be a continuous function, where $\Omega$ is a domain and $B(x, ...
0
votes
1answer
56 views

Expectation of hitting time for simple symmetric random walk

Assume there is a simple symmetric random walk $$S_n=X_1+...+X_n,\quad S_0=0$$ where $\mathbb P(X_i=\pm 1)=\frac{1}{2}$. Define $T=\inf\{n:S_n=1\}$. How to compute $\mathbb E(T)$? My idea: if ...
0
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1answer
19 views

Stopping time, event, simple description

Let us suppose that we have two stopping times $T$ and $S$, where $T \leq S$. Can someone explain on a practical example why is event ${(T \leq n)} \subseteq {(S \leq n)}$?
1
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1answer
24 views

independence of stopping time and a sigma algebra

Let $(B_t)$ be a standard Brownian motion and $\{ \mathcal{F}_t \}$ be the filtration generated by the Brownian motion. For a stopping time $\tau$, we know that $\{B_{\tau + t} - B_{\tau}\}_{t \geq ...
2
votes
0answers
43 views

Laplace transform of stopping times

I am nearly done with a question: Let $(B_t)$ be a Brownian motion on $\mathbb{R}$. For a fixed $x >0$, let $\tau$ be a stopping time defined by $$ \tau = \inf \{t \geq 0 : B_t \not \in (-x,x) ...
3
votes
1answer
35 views

Hitting time process of Brownian motion [closed]

I am stuck with this problem: Let $(B_t)$ be a standard Brownian motion in $\mathbb{R}$. For $t \geq 0$, let $$ H_t = \inf \{ s \geq 0 : B_s = t \}, \quad S_t = \inf \{ s \geq 0 : B_s > t \}. $$ ...
4
votes
1answer
33 views

Stopped process of Brownian motion

I am baffled about the following problem: Let $(B_t)$ be a standard Brownian motion. Let $$ \tau:= \inf\{ t \geq 0 :B_t = x \} \wedge \inf\{ t \geq 0 :B_t = -y \}$$ be a stopping time, where $x,y ...
3
votes
1answer
48 views

Determining if some random variable is a stopping time

I am stuck on this issue: Let $(B_t)$ be a Brownian motion. We know that since $\{0\}$ is a closed set in $\mathbb{R}$ and that $(B_t)$ is a continuous adapted process, $$ \tau:= \inf \{ t\geq 0 : ...
4
votes
1answer
49 views

Bounding an expected hitting time

Consider a stochastic differential equation: $$dX_t = dW_t + \sin(X_t) dt, \, X_0 = x$$ where $W_t$ is a Wiener process. Define $$\tau_1 = \inf \{ t : X_t \in 2 \pi \mathbb{Z} \} \\ \tau_2 = \inf ...
-1
votes
1answer
35 views

How to show the expected value of a hitting time Brownian motion?

We have $W_t$ as a Brownian motion and $$T_{−a,b} = \inf \{t ≥ 0 : W_t \not\in [−a, b]\}\qquad a, b > 0$$ How do you show $\mathbb{E} (W_{T_{-a,b}}) = 0$?
4
votes
1answer
47 views

Uniformly integrable martingale in a finite time horizon

Let $\{ M (t) \mid t \in [0,T] \}$ be a martingale and $\{ \tau_n \mid n = 1, 2, \ldots\}$ be an increasing sequence of stopping times such that $\tau_n \rightarrow \infty$ as $n \rightarrow \infty$. ...
0
votes
1answer
31 views

If $X^T(t)=X(t\land T)$ is said to be the process $X$ stopped at $T$. I want prove following statment

Let $X$ be a stochastic process defined on a probability space $(\omega ,\mathcal F,P)$ endowed with a filtration $(\mathcal F)_{t \ge0}$ and let $T$ , $T^\prime$ be $\mathcal F_{t}-$stopping times. ...
1
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1answer
16 views

Irreducible Markov chain and finite sets

Let $(X_n)_{n\geq 0}$ be a irreducible Markov chain defined on a countable state space $S.$ Let $F \subset S$ a finite set and $\tau=inf\{n \geq 1; X_n \notin F\}$. If $x \in F$ how to prove that ...
0
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1answer
32 views

Simple question about an equality of a stopped process

Let $T$ be stopping time, and $X_n$ be stochastic process. Then the stopped process $X_{n \wedge T}$ can be written as $$X_{n \wedge T} = X_n 1_{T \ge n} + X_T 1_{T < n}$$ where $1_{(\cdot)}$ is ...
1
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0answers
21 views

iid random variables and stopping time

This is Exercise 14.30 from Probability for Statistics and Machine Learning. Let $X_i$ be iid with $E|X| < \infty$, and let $T$ be a stopping time adapted to $\{ X_i \}$. Let $S_n = ...
3
votes
1answer
74 views

Density of first hitting time of Brownian motion with drift

I just started learning about Brownian motion and I am struggling with this question: Suppose that $X_t = B_t + ct$, where $B$ is a Brownian motion, $c$ is a constant. Set $H_a = \inf \{ t: X_t =a ...
0
votes
1answer
29 views

Approximation of a unbounded stopping time and convergence of respective $\sigma$-algebras

Fixed a filtration $(\mathscr{F_n})_n$ in a probability space and given a stopping time $\tau$ w.r.t $(\mathscr{F_n})_n$ that is finite almost surely we can construct a non-decreasing sequence of ...
0
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2answers
42 views

Stopping Time Subset Proof

My probability textbook has a really crappy proof for the following result. Suppose $S$ and $T$ are stopping times, with $S(\omega) \le T(\omega)$ for all $\omega$. Prove that $\mathcal{F}_S \subset ...
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2answers
54 views

Stopping time intuition

Let $(X_n)_{n \geq 1}$ be independent and identically distributed random variables with $P(X_n=1)=P(X_n=-1)=\frac {1}{2}$ for all $n \geq 1$ and let $S_n = X_1+X_2+ \cdots +X_n$. If we define a ...
1
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0answers
32 views

Bivariate stopped processes

Take two dependent Levy processes $L_1(t)$ and $L_2(t)$ with law $\mathcal{L}(L_1(1),L_2(1)$. If we stop the first process at a general time $t=s_1$ and stop the second process at another general time ...
1
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1answer
31 views

Probability of Wiener process hitting a particular point at an independent stopping time

Assume we have a stopping time $T$ that is independent of a Wiener process $W$. If $T$ were taking discrete values (let's say in $\mathbb{N}_0$), one can easily show (using the independence and the ...
1
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2answers
49 views

Is this a stopping time or not?

Let $(\xi_n)_{n\in\mathbb{N}_0}$ be a sequence of independent identically distributed random variables that take values in $\left\{-1,1\right\}$ with equal probabilities. Define ...
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0answers
20 views

Is the following rule a stopping time in regards to reverse filtration?

Let $X_1, \dotsc, X_n \sim F$, where $F$ is a distribution function with support in $[0,1]$. For $t \in [0,1]$, define the sigma-algebra: $$ \mathscr{F}_t = \sigma(1_{\{X_i \leq s\}}\;,\; 1 \geq s ...
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0answers
34 views

Prove that $\text{Var} \tau = \frac{1 − (p − q)^2}{(p-q)^3} $ where $\tau$-stopping time

Let $S_n = \xi_1 + \dots + \xi_n$ be asimetric random walk such that $P(\xi_i = 1) = p > \frac{1}{2}$ and $P(\xi_i = -1) = q $. Let $\sigma^2 =1-(p-q)^2$ and let $X_n=(S_n-n-(p-q)n)^2 - \sigma^2n $ ...
1
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0answers
40 views

What is the distribution of the area between a Brownian Bridge and the x-axis?

Lets say that we have a Standard Brownian Bridge ($\sigma=1$) with endpoints $(0,0),(1,0)$ Is there a way to derive the distribution of the area between a sample path of this bridge and the x-axis?? ...
2
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0answers
61 views

Optimal Stopping for One-Armed Bandit with a Fixed, Known Payout.

I'm very new to bandit problems (apologies if I've formatted my question incorrectly), but I have to solve the optimal stopping of what I think is a very simple case. I have a bandit problem with one ...
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0answers
17 views

Looking for resources: Generalizations of martingales to $\mathbb R^2$

In most introductory courses, a martingale $Y$ is defined as a stochastic process $$Y: T \times \Omega \to S$$ ,which satisfies certain conditions. ($\Omega$ is a probability space and a filtration ...
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0answers
9 views

Time changes conditions to be adapted

Given a process $X_t$ and another process $T_t$ which is increasing, what conditions should we require such that the process $X_t$ is adapted to the time change $T_t$, that is such that $X_t$ is ...
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1answer
22 views

question on a stopping time problem.

I borrowed some lecture notes on stochastic calculus, which contained the following exercise: Let $(X_n)_{n>0}$ be a sequence of random variables with $X_n: \Omega \to [0,\infty)$. We set $S_n= ...
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0answers
30 views

Processes adapted to time changes

I have a question regarding a passage in Chapter X of "Calcul Stochastique et Problèmes de Martingales"J.Jacod(1979). In (10.13) they define an adapted process $X$ to the time change $\tau(t)$ as a ...
2
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1answer
32 views

Predictable process with stopping time

I would be very gratefull if someone could help me with my question below. Intuitivly I can see that it is correct but I am unsure of how to prove it. Let T be a stopping time in $\mathcal{F}_t$ for ...
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1answer
42 views

Stopping rule for house selling problem

We have a house to sell. Each day an offer of $X_n$ comes for the house. Each offer costs an amount $k$ to observe. You may think of $k$ as advertisement costs. When you receive an offer you must ...
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1answer
109 views

Laplace transform stopping time

Consider a stochastic differential equation: $$\frac{dX}{dt} = b + \sigma \frac{dW}{dt}, X(0) = x$$ where $b,\sigma$ are constant, $x \in [0,1]$, and $W$ is a Wiener process. Let $\tau = \inf \{ t ...
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0answers
106 views

Dose “optional stopping theorem” imply “optional sampling theorem”?

Suppose $X$ is a martingale,$\tau$ and $\sigma$ are two stopping times which satisfy (a)$\sigma\le\tau$ and (b)the "optional stopping theorem" holds,that is to say: $$\mathbb E[X_\sigma]=\mathbb ...
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0answers
23 views

Infinitesiman generator of Time dipendent process

I'm trying to find the infinitesiman generator of this process $dY_{t}=\dfrac{b-Y_{t}}{1-t}dt+dB_{t}$ $0\leq a <1$, $Y_{0}=a$ where $B_{t}$ is a brownian motion; and I've found the solution: ...
1
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1answer
22 views

Application of Strong Markov Property

Theorem SMP (Strong Markov Property) Let $X$ be a time homogenous Markov process with $T=\mathbb R_+$ or $\mathbb Z_+$ and let $\tau$ be a stopping time taking countably many values. Then ...
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1answer
31 views

The proof of first exist time is a stopping time.

Here is a proof of verifing the hitting time is a stopping time :(the last part of the web page) https://lecturenotes.math.cmu.edu/mediawiki/index.php/Stochastic_Calculus_(Fall_2012)/Lecture_1 the ...
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1answer
48 views

Last hit before random time s in Poisson point process - expected value.

I'm stuck computing the expected value of the last hitting time before a time $s$ in the waiting time paradoxon. Suppose we come to a bus stop at a time $s \in \mathbb{R}$, where buses are randomly ...
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0answers
155 views

Law of a geometric brownian motion first hitting time (proof checking)

I need to use it in a small step in the middle of a simulation and I think I'm not getting correct results to this probabilities and so for my all subsequent simulation. Could someone ...
0
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0answers
30 views

About measurability of a stopping time.

If $S,T$ are two stopping time w.r.t. $\mathcal F_t$ define $R=S\wedge T$.Then $R$ is a stopping time .How to prove $R$ is measurable w.r.t $\mathcal F_T$? Is there something wrong with this ...
0
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1answer
37 views

A step in verifying a stopping time.

Suppose $X$ is a cadlag process adapted to $\{\mathcal F_t\}$ and $H$ is a closed set.Verify $\sigma_H\triangleq\inf\{t\ge0:X_t(\omega)\in H\}$ is a stopping time . The first step is: ...
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1answer
29 views

An equality in stopping time.

In a proof,I need the following equality: Suppose $\tau,\sigma$ are two stopping time and $A$ is a event.Then: $$(A\cap\{\sigma\le\tau\})\cap\{\tau\le t\}=(A\cap\{\color{red}{\sigma}\le ...
0
votes
1answer
39 views

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 ...