The tag has no wiki summary.

learn more… | top users | synonyms

1
vote
0answers
9 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
25 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
25 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
16 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 ...
2
votes
1answer
38 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
41 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
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 $\mathbb{E} (W_{T_{-a,b}}) = 0$?
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 $n \rightarrow \infty$. ...
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. ...
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
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 = ...
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
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
42 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
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
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
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 $ ...
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?? ...
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 ...
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
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 ...
1
vote
1answer
104 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 ...
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
123 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 ...
5
votes
1answer
130 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 ...
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 ...
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 ...
0
votes
0answers
60 views

Ito formula proof for bounded functions using stopping time

I'm self studying with the Oksendal book "Stochastic differential equations" and trying to do some exercises by myself. P.57 the exercise asks for the following (a screenshot will save us typing ...
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 ...