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2
votes
1answer
26 views

Does Brownian Motion return to the origin infinitely soon?

Consider a standard unidimensional Brownian Motion $B_t$ (Wiener process). Fact: This process returns to the origin infinite number of times with probability one. Consider a stopping time $\tau = ...
2
votes
0answers
19 views

Why do we use an exponential Martingale for the stopping time of a BIASED random walk?

The following is a passage from the lecture notes: Let a simple random move to the right with probability $p$ and to the left with probability $q = 1 − p$. We want the probability that it hits ...
10
votes
3answers
828 views

What is meant by a stopping time?

TL;DR: is a stopping time some sort of event, or is it a point in discrete time, or something else entirely what is an example of something which is not a stopping time? is my understanding of the ...
0
votes
1answer
21 views

Brownian Motion Hitting Time Distribution

Define $\tau_a = \inf \left\lbrace t \geq 0 | B(t) \geq a \right\rbrace $ for some $a>0$. The problem is to show that $ \tau_a \stackrel{d}{=} \sqrt a\tau_1 $. What I've done so far: $$P(\tau_a ...
1
vote
1answer
32 views

Gambling Game martingale

State the optional sampling theorem for martingales and bounded stopping times. You start with a capital of £100 and bet repeatedly on the toss of a coin. On each toss you may bet any whole number of ...
0
votes
1answer
19 views

Expected Stopping Time for BM

I'm working on this homework problem for Brownian Motion. Suppose we define a stopping time $\tau_a = inf \left\lbrace t \geq 0 : B(t) = a \right\rbrace$ for some $a>0$. I already showed in a ...
3
votes
1answer
29 views

Brownian motion proof of Dirichlet problem

I am reading the proof of the Dirichlet theorem stated in the following form: Theorem: Let $D$ be a bounded domain in $\mathbb{R}^d$ such that every boundary point satisfies the Poincare cone ...
3
votes
1answer
72 views

Tower Property for Expectations and Stopping Times

Let $(\Omega,(\mathcal{F_t})_{t\geq0},P)$ be a filtered proability space with $X\in L^1(P)$ and two stopping times $S$ and $T$. Show that \begin{equation*} ...
1
vote
1answer
19 views

Showing that a set is included in a filtration at a stopping time

The title may sound strange. Sorry for that but the question is short and easy to understand. I have a set $A \in \mathcal{F}_t$ where $(\mathcal{F}_t)$ is a filtration on some probability space. For ...
1
vote
1answer
14 views

Stopped brownian motion

Assume $B_t$ is a standard complex (or 2D if you wish) brownian motion and $\tau$ is a stopping time relative to $B_t$. I want to know if it is possible to construct another brownian motion $W_t$ such ...
2
votes
1answer
25 views

First hit of a martingale

I came across this result somewhere and I don't grasp its proof in its entirety. Let $M$ be a continuous martingale such that $M_0 = 0$. Define $\tau_x = \inf\{t\geq 0: M_t =x \}$. Then, $$P\{\tau_a ...
0
votes
0answers
18 views

Cramér Lundberg Risk Model - exponential distribution of claim sizes

I am studying the classical ruin model, which express the insurance company free surplus at time $t$ as $C_t=u+ct-\sum_{i=1}^{N(t)}Y_i $ where: $ct$ is the premium income up to time t $u$ is the ...
2
votes
1answer
38 views

Density of running supremum of Brownian motion until a stopping time

I am stuck on an exercise in my book: The question relies on the following fact: Let $M$ be a continuous, non-negative local martingale such that $M_0=1$ and $M_t \rightarrow 0$ almost surely as ...
0
votes
1answer
19 views

Expectation of a Wiener process at a Stopping Time - 2

I am working through an answer to the following question and I do not understand a statement given towards the end of the solution, specifically why $\tilde{W}(\sigma) = 1$. (This question is related ...
0
votes
0answers
24 views

Levy process of argument in the complex plane

I am stuck on this question: Let $B$ be a Brownian motion in $\mathbb{C}$ started at $1$. Let $\theta_t$ be a continuous determination of the argument of $B_t$, i.e. $B_t = |B_t| e^{i \theta_t}.$ ...
0
votes
1answer
27 views

Expectation of a Wiener process at a Stopping Time

I am working through an answer to the following question and do not understand an expectation which takes place at the end. $\textbf{Question:}$ Define the following stochastic process \begin{align} ...
0
votes
1answer
29 views

Calculate $\mathbb{E}(T^2)$ and $\mathbb{E}(\int_0^T X_s \,d s)$ for exit time $T$ of Brownian motion $(X_t)_{t \geq 0}$

Let $T$ be the exit time of from the interval $[-b,a]$ of a standard Brownian Motion $X_t$, then how would we go about calculating the following two expectations: $E[T^2]$ (and) $E[\int_0^T X_tds]$? ...
1
vote
1answer
67 views

Supremal distribution of positive continuous martingale, which converges to zero a.s.

So the question is as follows: Let $M$ be a positive continous martingale, converging a.s. to zero as $t \rightarrow \infty$. Prove that for every $x>0$: $\mathbb{P}\{\sup_{\{t \geq 0 \}} M_t > ...
1
vote
1answer
53 views

Justifying a step in proving $M_{S\wedge T} = \mathbb{E}[M_T | \mathcal{F}_S ]$

$S,T$ are stopping times and $M$ is a (right) continuous martingale. My lecturer set this as an exercise and I am given a solution(essentially split $M_T = M_T \mathbf{1}_{S≤T} + M_T ...
1
vote
1answer
60 views

Martingales and stopping times question

Let $X_n$ be iid r.v.s such that $P(X_n=1)=P(X_n=-1)=1/2$, and $S_n=\sum_{k=0}^{n}X_k$. Define $S_0=0$ a.s. . Prove that for all $k,n \in \mathbb{N}$, $\mathbb{E}[S^2_{n \wedge T_k}]=\mathbb{E}[{n ...
0
votes
1answer
41 views

Proof of Optional sampling theorem

In the proof of the optional sampling theorem they define for a stopping time $\tau$ the sigma algebra $\mathcal{G}=\sigma(\cup_n \mathcal{F}_{\tau\wedge n})$. Then they use the fact that for the ...
0
votes
1answer
60 views

Prove that discrete first hitting time is a stopping time

I have problems with the proof that a first hitting time is a stopping time: Let $\tau$ be the first hitting time into the set A, for a process $\{ X_n \}$ adapted to a filtration $\mathcal F_n$. I ...
0
votes
0answers
26 views

Optimal stopping strategy

I try to solve the following problem : Given a series of random variables : X1,X2,... such that each one can get either -1 or 1 with probability 0.5, give a strategy to maximize the expected value of ...
2
votes
1answer
57 views

Proof of stopping theorem for bounded stopping times

Let $\tau$ be a bounded stopping time and $X=X_n$ a martingale. Then $X_\tau$ is integrable and $E(X_\tau)=E(X_0)$. I need help with the proof at discrete time, at one step I am not sure I ...
1
vote
0answers
26 views

Expected value of Brownian Bridge evaluated at a stopping time

Denote by $B$ a Brownian bridge process, by $B(\omega)$ a realization of it and by $B_t$ the projection to the time point $t \in [0,1]$. Now let $c < 0$ and $$t^*(\omega) = \sup\{t \in [0,1]: ...
1
vote
1answer
29 views

Upper bound for martingale at a stopping time

This seems like a simple question, but I cannot figure out the following. Let $\{M_i\}_{i\geq 0}$ be a martingale adapted to a filtration $\mathcal{F}_i$, with the following conditions: ...
1
vote
1answer
48 views

Distribution of Brownian motion before stoping time.

Let $B_{t}$ be a standard Brownian motion. Stopping time $\tau_{a} = \inf \{t \ge 0: |B_{t}| = a\}$. How to find $E[B_{\frac{\tau_{a}}{2}}]$? Or where is it possible to read about it? Thanks in ...
1
vote
1answer
37 views

Are the following Stopping Times?

I've been working through the following list of stopping time questions. I am have problems with the final two (e and f). I appreciate any assistance offered. $\textbf{Question:}$ Let $S,T : ...
0
votes
1answer
41 views

Stopping times and typing monkeys

This is a question about the "standard solution" in this question: Let $(X_t)_{t\in\mathbb{N}}$ be the stochastic process modeling a monkey who types a random letter (uniform distribution) of the 26 ...
1
vote
1answer
39 views

Definition of $\sigma$-algebra $\mathcal{F}_\tau$ with $\tau$ a stopping time

If $\tau$ is a stopping time and $(\mathcal{F_t})_{t\in I}\subset \mathcal{F}$ is a filtration, then the $\sigma$-algebra of the $\tau$-past is defined as $$\mathcal{F}_\tau := \{A\in\mathcal{F} : ...
1
vote
0answers
24 views

Good reference on stopping times and continuous time change

I've been trying to look at stopping times and continuous time change in martingales but have trouble understanding without some concrete examples. Anyone knows of any good references that might be ...
2
votes
0answers
20 views

Formal argument on independence of consecutive hitting times of a Markov chain.

I'm refering to the question: Differences of consecutive hitting times. I'm interested in the independence of consequtive hitting times of certain values of a Markov chain. And I do "understand" the ...
0
votes
0answers
12 views

Is Markov Chain sampled at stopping times a Markov chain?

Given a Markov hain $\{X_n\}$ and $T_n$ be an increasing sequence of stopping times, is $\{X_{T_n}\}$ Markov chain ?
4
votes
2answers
70 views

Localisation in the proof of Ito's formula

I am reading Karatza's and Schreve's book "Stochastic Calculus and Brownian Motion" and I don't understand a strange thing as follows: Let $X=X_0 + A +M $ be a semimartingale, where $A$ is a ...
0
votes
1answer
52 views

Existence of localizing stopping times that reduce a local martingale to a square integrable martingale

Something is weird from a proof that I am reading: The well-known theorem of characterization of quadratic variation states that: Suppose $X$ is a continuous local martingale and $A$ is a continuous ...
0
votes
0answers
20 views

Proving that the indicator function of an interval with stopping times as endpoints is a predictable process

Let $S \leq T$ be two finite stopping times. I would like to show from first principles that $$ X: [0, \infty) \times \Omega \rightarrow \mathbb{R} ; \quad (t, \omega) \mapsto \mathbf{1}_{(S(\omega), ...
1
vote
1answer
58 views

Probability that Brownian Motion hits $t+1$ before $t-1$

Compute the probability that a brownian motion starting at $0$ hits the line $t+1$ before the line $t-1$. Here is what I did: I figured it has to do with optional stopping theorem. The ...
1
vote
0answers
20 views

Are these two inequalities equivalent?

We have worked in a lecture (about the optional stopping theorem) with the following two inequalites: $\mathbb{E}[T \mid X_0] \leq \frac{X_0}{c}$ and $\mathbb{E}[T ] \leq ...
4
votes
0answers
78 views

Using Girsanov theorem to prove density of stopping time

Let $B$ be a standard Brownian motion and for $a>0$ and $b>0$, and set $$\sigma_{a,b} = \inf\{t\,:\, B_t + bt = a\}.$$ There are at least two ways to solve the following problem (the other one ...
1
vote
0answers
20 views

Floor function of scale of stopping time with translation is non-increasing

Oké, so this question was one we had with a course of Stochastic Integration, it is however part of bigger proof, but I'll formulate the part I am uncertain about. The question is as follows: $T$ is ...
2
votes
1answer
42 views

$E(S_T^2)\not=E (\sum_{i=1}^T \sigma_i^2) $ when $E|T|<\infty$

I am currently learning random walk and come across a problem concerning stopping time. The question asks to give an example that $X_1,X_2,...$ independent r.v. with mean $0$ and variance ...
1
vote
0answers
40 views

Ito's lemma applied to functions involving stopping times

Recently, I come across an exercise in my book that asks us to apply Ito's formula to $$Y_t = e^{rt} \mathbf{1}_{ \{ \tau \leq t \} },$$ where $\tau$ is a stopping time. However, this is an inherent ...
0
votes
0answers
45 views

Sum of two stopping times

This question has been asked here before but I came up with a different answer than the ones given there. So I would like to post it here to get my answer checked. Question: Let $\sigma$ and $\tau$ ...
2
votes
1answer
157 views

The expected time until reaching a specified set in a Markov chain

I am reading an article in which they discuss a specific Markov chain in an example, and it turns out I need to sharpen up my Markov knowledge. First the setup. I have a continuous time Markov chain ...
0
votes
0answers
48 views

Define a maximization problem as an optimal stopping problem

We work over $\mathbb{R}_+^L$. Let $V$ be the set of vectors whose coordinates take values $0$ or $1$. Let $\mathbf{w}(t)$ (in $\mathbb{R}_+^L$) a vector that changes each time slot. To each vector ...
2
votes
0answers
41 views

Ito's formula applied to a stochastic function

The Ito's formula stated in my book is in the form $F(t,X_t)$, where $F: \mathbb{R}^{d+1} \rightarrow \mathbb{R}$ is a $d+1-$dimensional deterministic $C^{1,2}$ function and $(X_t)_{t \geq0}$ is a ...
0
votes
1answer
9 views

Modification of a local martingale

I am quite curious to know if the following is true, which comes up to my mind when reading a paper on SLE: For any local martingale $(X_t)_{t \geq 0}$ and stopping time $\tau$, is it true that $$ ...
1
vote
1answer
46 views

Verifying a proof of martingales.

I am trying to prove the following: Let $T$ be a stopping time bounded by $c$, and let $(X_n)$ be a martingale, then $E(X_T)=E(X_0)$. Here is what I did: $\int ...
1
vote
1answer
70 views

Intuition behind Stopping Times

I'm attending a stocahstic processes course. I have some trouble with the intuition behind a stopping time. I will consider the discrete case to make it simpler. a stopping time is given by ...
0
votes
1answer
48 views

Exponential of Brownian motion with negative drift

I am reading a text on Brownian motion and don't understand the following: Let $X_t = \exp \{ W_t - \frac{t}{2} \}$, where $W$ is a standard Brownian motion on $\mathbb{R}$. Let $T_n = \inf \{ t \geq ...