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2
votes
1answer
34 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 : ...
2
votes
1answer
17 views

Hitting time process of Brownian motion [on hold]

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
22 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 ...
0
votes
0answers
15 views

Question about upcrossings of Brownian motion

I am very stuck on this problem: Given a Brownian motion $(B_s)$, write $S_t = \sup_{0 \leq s \leq t} B_s$, for each $t>0$. All stopping times and martingales are considered w.r.t the filtration ...
1
vote
0answers
37 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?? ...
4
votes
1answer
40 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 ...
4
votes
1answer
36 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 $n \rightarrow \infty$. ...
-1
votes
1answer
32 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 $\mathbb{E} (W_{T_{-a,b}}) = 0$?
0
votes
1answer
9 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
votes
1answer
28 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. ...
3
votes
1answer
58 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
31 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
vote
0answers
18 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 = ...
0
votes
1answer
26 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
votes
2answers
38 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 ...
1
vote
2answers
41 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
vote
1answer
27 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
vote
0answers
31 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
vote
2answers
41 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 ...
1
vote
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 ...
1
vote
0answers
27 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 $ ...
2
votes
0answers
56 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 ...
1
vote
0answers
16 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 ...
1
vote
0answers
8 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 ...
0
votes
1answer
20 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= ...
0
votes
0answers
29 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 ...
5
votes
1answer
129 views

How to get closed form solutions to stopped martingale problems?

Way back when, I took a course in stochastic processes in college. I remember being frustrated by the plethora of abstract proofs without much in the way of how to use them to get actual results. It ...
2
votes
1answer
31 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 ...
1
vote
1answer
103 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 ...
1
vote
1answer
40 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 ...
4
votes
0answers
97 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 ...
1
vote
0answers
21 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
vote
0answers
16 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 ...
0
votes
1answer
30 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 ...
0
votes
1answer
37 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 ...
2
votes
0answers
121 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
votes
0answers
27 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
votes
1answer
33 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: ...
0
votes
1answer
28 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
36 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 ...
0
votes
0answers
24 views

Hitting time and its distribution

÷I'm reading an italian book about casual process (Probabilità e modelli aleatori of Enzo Orsingher). At pag 105 there's the probability of the stopping time $T_\beta$. $$P\{T_\beta \leq ...
1
vote
1answer
55 views

Jumping times of a Lévy Process

If one has a Levy-process, are the times when the process has a jump of size exceeding a positive $\varepsilon$ actually stopping times w.r.t. the canonical filtration? In more detail: Let ...
2
votes
1answer
38 views

Verifying stopping times…

Let $m$ be a natural number, $$g_m:=\sup\left\{ {n\leq m: S_n\leq 0}\right\}$$ and $$d_m:=\inf\left\{ {n\geq m: S_n \leq 0}\right\}$$ I have to check if they are are stopping times. It's still a new ...
1
vote
1answer
28 views

Distribution of two-sided boundary stopping time of Brownian motion.

If $B_t$ is a Brownian motion, and a one-sided boundary stopping time is given by: $\tau_a=\inf\{t:B_t=a\}$ the distribution of $\tau_a$ is given by: $f_{\tau_a}(t)=\frac{|a|}{\sqrt{2\pi ...
3
votes
1answer
138 views

A martingale with bounded increments either converges or diverges to both infinities a.s.

I am reading page 236 "Probability : theory and examples" by R. Durrett. Theorem 31. Let $X_1, X_2,\ldots$ be a martingale with $|X_{n+1}-X_n|\leq M<\infty$. Let $C=\{\lim X_n \;\;\; \text{exists ...
2
votes
2answers
187 views

Expectation of a stopping time of a Wiener process

How can we calculate $\mathbb{E}(\tau)$ when $\tau=\inf\{t\geq0:B^2_t=1-t\}$? If we can prove that $\tau$ is bounded a.s. (i.e. $\mathbb{E}[\tau]<\infty$), then we can use the fact that ...
1
vote
0answers
1k views

Minimum of two stopping times is a stopping time.

So far I've already shown that the sum and the maximum of two stopping times is a stopping time, but the minimum is giving me some problems which I just can't get around. This is what I've tried: Let ...
1
vote
1answer
34 views

Do we need $\tau \leq \nu$ to show $E(X_\tau)=E(X_\nu)$?

My lecture notes claim that if $(X_n)$ is a martingale and $\tau$ is a stopping time bounded by $N$ then $$E(X_\tau)=E(X_{\tau \wedge N})=E(X_{\tau \wedge 0})=E(X_0)$$ and then remarks that if $\tau$ ...
0
votes
1answer
42 views

Stopping times problem: $ \tau_+ = \inf \{t \ge 0 \mid W_t>0\}$

Stopping times problem, $\tau_+ = \inf \{t \ge 0 \mid W_t>0\}$ I can not prove the following : P/S: When I look at the stopping time, I feel that $\{W_0 > 0\} = \{\tau_+ = 0\}$ , is that ...
2
votes
0answers
19 views

Comparing hitting time of two random walks

There are two random walks, $S^t_i=S^{t-1}_i+ X_i^t$ for $i=1,2$, $X^t_i$ i.i.d they have boundaries $h_1$ and $h_2$ respectively. I'm wondering if it's possible to calculate the probability that one ...