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34 views

New stochastic calculus

I am interested in Kagi and Renko approach and hope I can use it for a random walk process. I searched for it on internet but I couldnt find any basic material to read about it. Can someone please ...
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0answers
156 views

Essential supremum of a conditional expectation

Given the function \begin{equation} P(x,t) := \sup\limits_{t \le \tau \le T} E\left( g(X^{t,x}_{\tau}) \right) \end{equation} where $X^{t,x}$ is the unique solution to the SDE \begin{equation} X_u ...
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1answer
48 views

Probablity and Expected value

Suppose you are playing a fair coin game and you win a dollar if a head shows up and lose a dollar if tail. what is the expected value of rounds you played before you lose the first dollar from your ...
3
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1answer
79 views

how to prove $(X_{n})_{n\in \mathbb N}$ and $(Y_{n})_{n\in \mathbb N}$ are supermartingale.and $(Y_{n})_{n\in \mathbb N}$ is convergence to -7

Let $p \in [0 , \frac{1}{2}] $ and $\eta_{i}$ be i.i.d random variables and $P(\eta_{i}=1)=p$ and $P(\eta_{i}=-1)=1-p$ and $\mathcal F_{n}=\sigma(\eta_{1},\cdots,\eta_{n})$ and ...
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1answer
94 views

Stopping times of Markov chains

I have the following problem: Consider a state space $E$ and a Markov chain $X$ on $E$ with transition matrix $Q$ such that for every $x \in E$, $Q(x,x)<1$. Define: $\tau:=\inf\{n\geq 1:X_n\neq ...
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1answer
167 views

Conditional Expectation of martingale at stopping time

I am trying to understand the implications of the optimal stopping theorem, which is why I tought of the following problem. Consider the continuous-time Martingale $X = (X_t)_{t \geq 0}$ and the ...
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1answer
61 views

Strong Markov Property Brownian Motion Question

If $\tau$ is a stopping time and $\omega(t)$ is Brownian Motion then the Strong Markov Theorem states that $Z(t)=\omega(t+\tau) -\omega(\tau)$ conditioned on $\{\tau <\infty\}$ is distributed as ...
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52 views

Lower bound for stochastic process

Suppose the non-negative stochastic process $(X_t,Y_t)$ is such that $E\{X_t - X_a | Y_u \in A \,\,\forall u \in [a,t] \} \geq Z(A)(t-a)$. Let $T_{A}$ be the time of a visit to $A$. Assuming that the ...
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0answers
40 views

Time after last jump and waiting time before the next jump of Poisson process

Consider $N =(N_t)_{t\geq0}$ a Poisson process of intensity $\lambda > 0$ and $(T_n)_{n\geq 1}$ its jump instants. Then consider for all $t \geq 0$, $Z_t = t- T_{N_t} \mathbb 1 _{\{ t \geq ...
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43 views

Conditional distribution of compounded Poisson process

Consider a Poison a process $N = (N_t )_{t\geq 0}$ of intensity $\lambda >0$ whose instants of jumps are $(T_n)_{n\geq0} $ $(T_0 =0)$ and a process $\tilde{N} =(\tilde N_t )_{t\geq 0}$ defined as ...
3
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1answer
117 views

find the Law of probability Stopping time $T=inf\{n\ge 0: R_{n}\gt a\}$ for fixed number $a\gt 0$.

suppose $R_{n}=\sum_{i=1}^{n} X_{i}$ for $n\ge 1$ and $R_{0}=0$ , that $X_{i}\gt 0$ Random variables Are independent and distributed.find the Law of probability Stopping time $T=inf\{n\ge 0: ...
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2answers
86 views

Showing that a hitting time is $\mathbb P-\text{a.e.}$-finite

Let be $\alpha, \beta \in \mathbb R$ such that $\alpha < \beta $ and $x \in [\alpha, \beta ]$. Consider the random time $$T_x = \inf \{ t\geq 0 : x+ B_t \notin [\alpha, \beta]\},$$ where ...
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0answers
108 views

First hitting time on a element of $\mathcal B ( \mathbb R^d) $ a (right, left) continuous path stochastic process

It's known that, given $\Gamma \in \mathcal B (\mathbb R ^d)$ and $X = > (X_t)_{t\geq 0}$ with right-continuous path, the random time $$T_{\Gamma} = \inf \{ t\geq 0 : X_t (\omega) \in ...
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0answers
28 views

Probability of winding number in 2D Brownian motion

Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau ...
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1answer
53 views

question on Brownian Motion stopping time and end state

I came across this equation in my lecture notes, which states: $P(T_a < t , W_t \ge a) = P(W_t \ge a)$ where $T_a = \min\{t \ge 0, W_t \ge a\}$. I'm really confused by this equation: as far as I ...
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0answers
59 views

Bounding the expectation of monotone function of stopping times of Brownian motion

Let $X_t$ be a standard Brownian motion and let $Y_t:=X_t + \epsilon B_t$ where $B_t$ is an independent standard Brownian motion and $\epsilon>0$ is small. Let f be a monotone increasing function. ...
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0answers
67 views

Characterization of hitting time's law. (Proof check)

Under the same assumptions of this early question, consider also a the random time $T_a := \inf\{ t > 0: B_t \geq a\}$ which is a stopping time. Since $M^\lambda$ is a continuous martingale, Doob's ...
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1answer
50 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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128 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 ...
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2answers
259 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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169 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
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1answer
40 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
31 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
106 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
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1answer
210 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
54 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
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1answer
376 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: ...
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1answer
116 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
143 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 ...
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1answer
73 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 ...
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1answer
84 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 ...
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1answer
236 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
80 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 ...
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1answer
98 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 ...
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1answer
73 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 ...
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1answer
128 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
186 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
154 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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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 ...
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1answer
43 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 ...
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2answers
281 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 ...
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1answer
157 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{ ...
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2answers
151 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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194 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
550 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 ...
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2answers
205 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
200 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
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1answer
267 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
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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.