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3
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1answer
30 views

On the proof of lemma 1.2.4 of Stroock and Varadhan A question concerning stopping times

In the book Multidimensional diffusion processes, of Stroock and Varadhan one reads (page 23): This is the proof of $(i)$. Here the authors say Define $f_t$ on $(\{\tau \leq t\}, \mathcal{F}_t ...
1
vote
1answer
71 views

Does this game make you arbitrarily rich with probability one?

We toss a coin. If it's heads we win $\$ 1$, otherwise we lose $ \$ 1$. Fix some large sum. Will we be winning this amount with probability one at some point? We assume that we have infinitely many ...
3
votes
2answers
99 views

Stochastic variables independent given Tau

Say we have a filtration $(\mathbb{F}_s)$, and a stopping time $\tau$ w.r.t. to that filtration.Let $X_t$ be a continuous stochastic process (not required to be adapted to the mentioned filtration), ...
2
votes
1answer
24 views

Independence of a hitting time and the underlying stochastic process

While I was playing around with the Girsanov's Theorem I stumbled upon the following absurdity and I couldn't resolve it with the current knowledge of stochastic analysis that I have. $B$ being ...
1
vote
1answer
23 views

Show that $\mathbb{E}\left[c_{\tau\wedge n}X_{\tau\wedge n}-\sum_{i=1}^{\tau\wedge n}c_i\mathbb{E}(X_i-X_{i-1}\mid\mathcal{F}_{i-1})\right]\le 0$

I am trying to go through a past exam paper but I don't know how to deal with stopping times since we only did 2 exercises in class... I got stuck, so I would really appreciate if someone could help ...
0
votes
1answer
11 views

Expectations of stopping times in general

I have a very basic question: So for a stopping time $\tau$ with $E(\tau)<\infty$ we have $E(\tau)=\sum_{n=0}^\infty P(\tau>n)$, right? Why is that? Thanks!
3
votes
1answer
60 views

Markov and strong Markov properties

In my study of strong Markov property of an RCLL canonical Markov process I encounter the following definition: Suppose $Y_t:\omega\rightarrow \omega(t)$ is canonical Markov process with respect to ...
2
votes
1answer
43 views

Find $\mathbb{E}_{X_0 = x} X_\tau$ for an Ornstein-Uhlenbeck process $(X_t)_{t \geq 0}$ where $\tau = \inf\{t>0 \mid X_t \notin [a,b]\}$

Let $X_t$ satisfy the following SDE: $dX_t = X_t dt + \sigma dB_t$, $\sigma$ is a constant and $B_t$ is Brownian Motion. Find $\mathbb{E}_{X_0 = x} X_\tau$ where $\tau = \inf\{t>0 \mid X_t \notin ...
3
votes
0answers
61 views

A Markov Chain probability, conditioned on a random time.

My question: Upon reading theory about diffusion processes, i came across an argument which i believe simplifies to this: Say we have a Borel measurable set $A$ (if it matters you can set $A=\lbrace ...
1
vote
1answer
24 views

Rewriting probabilities as expectation

Consider the stopping time $\tau_a:=\lbrace{t>0| W_t >a\rbrace}$, where $W_t$ is a Brownian Motion. Define: $X_t:=W_{\tau_a+t}-W_{\tau_a}$. We have that $X_t$ is a Brownian Motion independent ...
0
votes
1answer
28 views

Simple Probability Inequality with Stopping Times

Suppose $U_1,...,U_n$ are independent random variable with $\mathbb{E}[U_i]=0$. Define $Z_k:=\sum_{i=1}^k U_i$. Set $T:=\inf \lbrace k \in N \mid |Z_k|>2\alpha \rbrace$. Clearly $\lbrace T=k ...
1
vote
1answer
30 views

An Algorithmic approach to the secretary problem with unknown n

I've been reading about the secretary problem these days and I got the idea for the case when we know the number of applicants $n$ in advance. I'd like to know what would be an algorithmic approach ...
1
vote
1answer
35 views

$E[|X(t)|]\leq K\implies E[|X(\tau)|]\leq K $?

Let $X(t)$ be a stochastic process. Assume that, for every $t\leq M$, it holds $$E[|X(t)|]\leq K, $$ for some constant $K$. Let now $\tau\in[0,M]$ be random (stopping time). Is it true that also ...
1
vote
1answer
38 views

Distribution Stopping time under Brownian motions

Considering $W$ the canonical process on $C([0,1],\mathbb{R})$ and the row filtration generated by the coordinate process of $W$, I want to prove that ...
0
votes
1answer
31 views

Illegal lottery problem (Merging dependent bernoulli trials)

Suppose I am in a town that playing lottery is illegal. If I buy a ticket for 1 dollar, I will win the lottery with probability $p$. Each time I buy a ticket, the police may catch me and confiscate ...
0
votes
0answers
53 views

Secretary Problem - Optimal algorithm for expected value of candidate.

I recently encountered the secretary problem and there are essentiall two problems: Maximizing the probability of choosing the best candidate. Gnedin proved in his paper ...
3
votes
0answers
30 views

Charaterize the $\mathcal{F}_\tau$ a sigma algebra for the stopping time $\tau$

Consider a stochastic process $X: [0, \infty) \times \Omega \to \mathbb{R}^d$ We define $\mathcal{F}_t = \sigma(X(s), 0 \leq s \leq t)$ The sigma algebra generated by the sets $\{\omega: X(s,\omega) ...
-1
votes
1answer
32 views

Martingale: Show $p\{T<+\infty \}=1$.

Let $(X_i)$ i.i.d. such that $p\{X_i=+1\}=p\{X_i=-1\}=\frac{1}{2}$ and let $(S_n)$ the martingale define by $S_0=0$ and $S_n= X_1+...+X_n$. Moreover, let $$T=\begin{cases}\inf\{n\geq 0\mid ...
1
vote
1answer
25 views

How to show that consecutive hitting times are a sequence of stopping times?

Let $\{X_n:n=0,1,\ldots\}$ be a martingale with respect to a filtration $\{\mathcal F_n\}$. Let $A,B$ be nonempty, disjoint Borel sets and define $T_0=0$, \begin{align} S_n &= \inf\{m\geqslant ...
2
votes
2answers
50 views

“The first time a continuous local martingale grows in absolute value beyond $n$” is a localizing sequence

How can it be shown that, for a continuous local martingale $X$ defined w.r.t. the filtered probability space $(\Omega, \mathcal{A}, P; \mathcal{F})$, the stopping times $\tau_n := \inf \{t \geq 0 ...
0
votes
0answers
18 views

Expected difference between a process and a stopped process

Let $\tau$ be a stopping-time such that for some $t\geq 0$ the probability $P(\tau<t)=\epsilon$ is small. I am interested in quantifying the following difference $$E[f(X_t)] - E[f(X_{t\wedge\tau})] ...
2
votes
0answers
52 views

Brownian Motion Hitting Times

I am reading through Walsh's Knowing the Odds book and came across this problem. Let $B_t$ be Brownian motion. Find the probability that $B_t$ hits plus one and then minus one before time one. I am ...
2
votes
0answers
28 views

Limit of decreasing sequences of markov time (stopping time) is markov time?

Let $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t \geqslant 0}, \mathbb{P})$ be a filtered probability space and let $\tau_n \geqslant \tau_{n+1}$ be a markov time (stopping time) with respect to ...
1
vote
0answers
42 views

Proving Galmarino's Test

Galmarino's Test gives a condition equivalent to being a stopping time. It says: Let $X$ be a continuous stochastic process with index set $\mathbb{R}_+$ (i.e. each sample path is a continuous ...
1
vote
0answers
22 views

jump-diffusion hitting time

Suppose I have a stochastic process $dS_t= rS_t dt + \sigma S_t dW_t + dJ_t$ where $W_t$ is a brownian motion and $J_t$ a compound poisson process of parameter $\lambda$ with lognormal jump size, ...
2
votes
0answers
92 views

Brownian motion stopped at the hitting time of an independent Brownian Motion

While I was working on the exit time of planar BM out of a square I came across the following observation, which I cannot grasp. I define this exit time as $$\tau = \inf\{t \geq 0: \lvert B(t)\rvert ...
0
votes
1answer
53 views

Escape probabilities in a random walk.

So, a lot of theory in symmetric random walks seems to concentrate on 'hitting/stopping times' and things like that. So I started wondering... How would I go about calculating the probability of ...
2
votes
0answers
32 views

2D Brownian Motion — Does this argument work?

Consider a 2D Brownian Motion $(X_1(t),X_2(t))$ starting at $(x_1,x_2) \in \mathbb{R}^2$. For every $s\geq0$, let $$\tau_s = \inf \left\{t \geq 0 \mid X_1(t) - x_1 > s \right\}\qquad Y_s = ...
1
vote
1answer
24 views

Properties of Stochastic Interval

I'm reading "Limit Thoerem for Stochastic Processes" and finding it hard to calculate the Stochastic interval.For example : In proposition 2.10,$T$ is a stopping time: If $A\in\mathcal F_0$,I need ...
1
vote
1answer
33 views

Does Brownian Motion return to the origin infinitely soon? [closed]

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
44 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 ...
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votes
3answers
854 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
34 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
48 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
23 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
54 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
77 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
20 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
20 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
31 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
31 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
60 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
24 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
25 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
32 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
70 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
62 views

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

$S,T$ 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
71 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
45 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 ...