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1answer
54 views

Moving boundaries for Ornstein-Uhlenbeck processes

Let $\tau(X_t)$ be the first-passing time to the moving boundary $a(t)$ for an Ornstein-Uhlenbeck process $X_t$. I wonder how general an $a$ can be allowed in order to guarantee that $\tau$ becomes ...
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0answers
151 views

Brownian motion hitting probability

Let $B_t$ be a brownian motion and $g(t)$ a function of the time $t$. $B_0=0$. Let $\Phi$ be the c.d.f. of a normal distribution. At time $t$, the probability that $B_t > g(t)$ equals ...
2
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2answers
385 views

First hitting time for a brownian motion with a exponential boundary

Let $B_t$ be the standard Brownian Motion. Is the distribution/density of the first hitting time of $B_t$ for an exponential decaying boundary known? Trying to be more formal, if ...
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0answers
202 views

Hitting times and stopping times for cadlag processes

Let $X$ be a cadlag stochastic process. If $X$ is continuous, then I already know that $\inf\{t\geq 0: X_t \in C\}$ is a stopping time whenever $C$ is closed in $\mathbb{R}$. What if $X$ is only ...
2
votes
1answer
44 views

IID sequence and stopping time

Let $X_0=0, X_1, X_2,\dots, X_N$ be i.i.d. random variables, with Gaussian distribution $\cal N (0,1)$. For $k=0,\dots, N, S_k=\sum_{i=1}^k X_i$ and $\tau=\min\{k:S_k^2\geq N-k\}$. So $\tau$ is a ...
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1answer
33 views

Stoppingtimes: Why demand $\mathbb{E}[\tau]<\infty$?

I'm working with a discrete-time Markov Chain $\{Y_j, j \geq 0 \}$ that evolves untill a stoppingtime $\tau$ is reached. $X$ is een stochastic variable which depends on the state of the Markov Chain. ...
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0answers
116 views

Stopping time and martingale for random walks

Let $X_0=0, X_1, X_2,\dots, X_N$ be i.i.d. random variables, with Gaussian distribution $\cal N (0,1)$. For $k=0,\dots, N, S_k=\sum_{i=1}^k X_i$ and $\tau=\min\{k:S_k^2\geq N-k\}$. So $\tau$ is a ...
2
votes
1answer
256 views

Conditional hitting time distribution of a Brownian motion

This problem cropped up in some research I am doing. I imagine it is standard, but I cannot seem to find the answer. Let $W_t$ be a standard Brownian motion. Suppose there are four values $a < 0 ...
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2answers
57 views

Optional sampling

Let $(X_i)_{i\in\mathbb{N}}$ be iid random variables with $\mathbb{E}|X_1|<\infty$ and let $S_n \stackrel{\rm{}def}{=} X_1+\cdots+X_n$ for all $n\in\mathbb{N}$. If $T$ is a stopping time with ...
2
votes
1answer
455 views

proof that a stopped martingale is a martingale?

Defenition. $\mathcal{F}_{\tau}=\{F\subset \Omega: \forall n \in N \cup \{\infty\}, F\cap(\tau\leq n)\in \mathcal{F}_{n}$} is a sigma-algebra. Defenition. $\forall \omega \in \Omega: ...
1
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1answer
119 views

Optional Sampling Theorem in discrete setting

I have a question about proving the optional sampling theorem in discrete setting. I dont know if what I am doing is mathematical justified. Can someone help me with this? Defenition: Let $\tau$ be a ...
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2answers
168 views

Stopping times, Filtration, Martingales,

I am new here and I have a question. Defenition: Let $ \tau$ be a stopping time, then $\digamma_{\tau}=\{F\subset \Omega: \forall n \in N \cup \{\infty\} , F\cap(\tau\leq n)\in \digamma_{n}$} is a ...
1
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1answer
80 views

Probability that all events occur equally often in finite time

An experiment with $n$ equally likely events is repeated until all events have occurred equally often. What is the probability that the stopping time is finite ? The probability could be denoted by ...
0
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1answer
98 views

Show sequence of hitting times is sequence of stopping times.

Let $(M_t)_{t\geq0}$ be a continuous process, $\epsilon >0$ and define a sequence $(S_n)_{n\geq 0}$ by $S_{i+1}=\inf\{t > S_{i} : M_t - M_{S_i} > \epsilon \}$ and $S_0=0$. Clearly $S_0$ is ...
2
votes
1answer
257 views

Example of a martingale and a stopping time with $E(T)<\infty$ but $E(X_T) \neq E(X_0)$

Is there an example of a martingale in discrete time $X_0, X_1, X_2,\ldots$ and a stopping time $T$ so that $E(T) <\infty$ but $E(X_T) \neq E(X_0)$? With added assumptions on how $X_n$ behaves, ...
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2answers
81 views

Racing Time Difference Formula

I am trying to develop an android app for a friend that uses the gps to tell you how many seconds ahead or behind you are from your target speed vs your actual speed. For example I could drive for 1 ...
1
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1answer
107 views

Arguing on stopping time probability

Consider the random walk where $X_t=\sum_{i=1}^t Y_i$, $Y_i$s are iid and take $\pm$1 with probabilities $p$ and $1-p$ respectively, where $0<p<0.5$. Define stopping time ...
0
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1answer
74 views

Implications between $\mathbb P [\tau < \infty] =1 $ and $\tau \in L_1 (\mathbb P)$

We've got the usual filtered stochastic basis $(\Omega, \mathcal F, (\mathcal F_n). \mathbb P), \space \tau : \Omega \to \mathbb{N}\cup \{\infty\}, [\tau \le n] \in \mathcal F_n$ ($\tau$ is an ...
1
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1answer
131 views

Stopping time $\tau_k$ measurable w.r.t. $\mathcal{F}_k$

We have $(\Omega,\mathcal{F},\mathbb{P})$ stochastic basis. Let $\tau: \Omega \to \mathbb{N}\cup \{\infty\}$ is a $(\mathcal{F}_k,k \in \mathbb{N} )$ be stopping time and define another stopping time ...
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2answers
233 views

$(S_n^2-n)_{n\ge 0}$ martingale and bounded stopping times

Consider the random walk $$S_n=\sum_{k}^{n}X_{k}$$ Where $X_k$'s are iid, $$\mathbb P(X_1=1)=\mathbb P(X_1=-1)=\frac{1}{2}$$ and $\mathcal{F}_{n}=\sigma(X_i,0\leq i\leq n)$. How do I prove that ...
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2answers
164 views

Martingale Stopping Time

Let $(S_n)_{n \ge 0}$ be a $(\mathcal F_n)$-martingale and $\tau$ a stopping time with finite expectation. Assume that there is a $c > 0$ such that, $\forall n, \mathbb E (|S_{n+1} - S_n | | ...
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0answers
103 views

Finite stopping times

I've come across two statements in a proof that I don't really understand. Let $X_{i}$ be iid with values in $\{-1,0,1\}$ all with positive probability. Define $S_{n}=\sum_{i=0}^{n}X_{i}$ and the ...
0
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1answer
51 views

What is the optimal stopping point for an experiment when expecting unknown event

Assume we notice that stock prices are rising and we can deduce we are in a bubble. Assume we start at $w(0)=0$ worth at time $t=0$ and the value grows linearly with time $(w(t)=t)$. We know that ...
3
votes
2answers
315 views

Stopping time proof

Let $\{X_t, t \ge 0\}$ be a continuous stochastic process and adapted to the filtration $\{\mathcal{F}_t,t\ge 0 \}$ and consider $$ \alpha = \inf\{t, |X_t|>1\}, $$ the first time the the process ...
0
votes
1answer
193 views

Stopping time for sum of iid random variables.

Suppose we have $m$-sided biased die. Let $X_i$ be the outcome of the $i$'th roll with the die. Furthermore let $\mathbb{P}[X_i=k]=p_k$ with $k \in \{1,...,m\}$. We define $T=\min\{n\text{ ...
3
votes
2answers
174 views

coupon collector problem for different number of copies of each coupon type

I would like to pose a question on a variation on the classical coupon collector's problem: coupon type $i$ is to be collected $k_i$ times. What is the expected stopping time or the expected number of ...
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0answers
223 views

stopping time expectation for gambler's ruin

2 players A and B start with x & y dollars respectively, and they bet against each other 1 dollar each time by tossing a fair coin. I let $X_n = x + \sum_{i=1}^{n}\xi_i$ where $\xi_i$ are i.i.d. ...
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0answers
51 views

Expectation related to renewal measure

Let $X_1,X_2,\ldots$ be i.i.d. random variables, and $S_n=X_1+\cdots+X_n$. Assume that $0 < \mathbf{E}(X_1) < \infty$ (but don't assume that the $X_i$ are $>0$). Let $N$ be the almost surely ...
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3answers
803 views

Why is stopping time defined as a random variable?

I've been given a crash course in stochastic processes and martingales for the purposes of a semester project on them. The guy I'm working with has been, I feel, a little vague in the definition of ...
1
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2answers
215 views

Stopping time for a martingale

Let $X_1,X_2,\ldots$ be iid random variables where $X_i\in\{-1,0,1,2,...\}$, $P(X_i=0)<1$ and $E(X_1)=\mu$. Let $S_n=1+X_1+\cdots+X_n$ and $T=\inf \{n:s_n=0\}$. Show that $E(T)=\infty$ if $\mu=0$ ...
2
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1answer
227 views

Doob's stopping time theorem with unbounded stopping time

Let $(X_t)_{t\geq0}$ be Brownan motion on $\mathbb R$, and $\tau$ is a stopping time adapted with the natural filtration generated by the Brownian motion. If $X_0=0$, $E(e^{\tau/2})<+\infty$. ...
4
votes
1answer
290 views

Stopping time and Brownian motion (specific example)

Let $B$ be a Brownian motion. I want to show that $$ \inf\{t\geq0 \mid B(t)=\max_{x\in [0,1]}B(s)\} $$ is not a stopping time w.r.t. the standard filtration. How can one intuitively see that this ...
1
vote
1answer
57 views

What is $1_{\{\tau_n>0\}}X^{\tau_n}$ process saying?

As title says, what is $1_{\{\tau_n>0\}}X^{\tau_n}$ process? I do have understanding of what stochastic processes are, but not sure what is this specific process saying.
3
votes
1answer
248 views

Favourable modification of “Double or Nothing”

I am working through 'Great expectations: the theory of optimal stopping' by Y.S. Chow, H. Robbins, D. Siegmund and cannot fill in the gap in reasoning regarding the existence of optimal stopping ...
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 ...
3
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1answer
183 views

Optional sampling exercise

I came across the following exercise in Stochastic Calculus: Let $B=(B_t)_{t\geq0}$ be a standard Brownian motion. Let also $M$ be the following process: $M_t=B^4_t-6t(B^2_t-\dfrac{t}{3})$ for ...
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2answers
1k views

Sum of two stopping times is a stopping time?

Let $\sigma$ and $\tau$ be two stopping times in $\mathscr{F}_t$ and let this filtration satisfy all the usual conditions. Question: Is $\sigma + \tau$ a stopping time? Attempt at a solution: I ...
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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 ...
3
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1answer
245 views

Stopping time on Wiener Process

Let $W_t$ be a Wiener process and for $a\geq0$ $$\tau_a:=\inf \left\{ t\geq0: |W_t|=\sqrt{at+7} \right\}.$$ Is $\tau_a<\infty$ almost everywhere? What about $E(\tau_a)$ then?
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0answers
170 views

Sufficient condition in terms of stopping times for a stochastic process to be a local supermartingale

(Question edited in response to Nate's comment) Let $(X_t)_{t\geq 0}$ be a continuous (or càdlàg), real-valued process, and define stopping times $\tau_{s,a,b}=\inf~ [s,\infty)\cap\{t:X_t\notin ...