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1answer
23 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
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1answer
25 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
48 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
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1answer
28 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
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1answer
28 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
49 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 ...
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0answers
20 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
46 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
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0answers
17 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
22 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
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1answer
33 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
39 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
33 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
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0answers
22 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
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0answers
18 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 ...
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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
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2answers
64 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
47 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 ...
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0answers
15 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
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1answer
46 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
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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
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0answers
48 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 ...
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0answers
19 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
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1answer
39 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
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0answers
37 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
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0answers
24 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
149 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
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0answers
45 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
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0answers
39 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
40 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 ...
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1answer
62 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
32 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 ...
0
votes
0answers
43 views

$E(X_T; T < \infty) \leq E(X_0)$ with $T$ stopping time

I'm doing this exercise: $(X_n)$ is a non-negative supermartingale and $T$ a stopping time, then $$E(X_T; T < \infty) \leq E(X_0)$$ My attempt: $(X_n)$ is a negative supermartingale, and so ...
1
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0answers
49 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 ...
-1
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1answer
78 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 \{ ...
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votes
2answers
68 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
45 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
votes
1answer
112 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
vote
1answer
57 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 ...
1
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0answers
38 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
votes
0answers
28 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
159 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
votes
1answer
22 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 ${(T \leq n)} \subseteq {(S \leq n)}$?
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vote
1answer
30 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
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0answers
49 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
52 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
61 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
54 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 : ...