This tag is for questions about stopping times. Let $X = \{X_n : n \geq 0\}$ be a stochastic process. A stopping time $\tau$ with respect to $X$ is a random time such that for each $n \geq 0$, the event $\{\tau = n\}$ is completely determined by (at most) the total information known up to time $n$, $...

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A proof by René Schilling that a continuous Lévy process is integrable

In his treatise "An Introduction to Lévy and Feller Processes" (arXiv link), Prof. Dr. René Schilling gives a short and seemingly straightforward proof for the claim that a continuous Lévy process is ...
3
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Finding g$^∗$, τ$^∗$ in 1-dimensional Brownian motion

How do I find g$^∗$, τ$^∗$ such that g$^∗$(s, x) = sup$_τ$E$^{(s,x)}$[e$^{−ρ(s+τ)}$B$_τ$$^2$] = E$^{(s,x)}$[e$^{−ρ(s+τ^{*})}$B$_{τ^*}^2$ ] , where B$_t$ is 1-dimensional Brownian motion, ρ > 0 is ...
3
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0answers
35 views

Show that the only nonnegative superharmonic functions in R are the constants

I am having trouble finding g$^∗$(x) when $$g(x) = \begin{cases} xe^{-x} & \text{for x > 0} \\[2ex] 0 & \text{for x $\leq$ 0}. \end{cases}$$ I would like to use the fact that the only ...
3
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0answers
30 views

Proving that only nonnegative superharmonic functions in R$^2$ are constants

How can I prove that the only nonnegative (B$_t$-) superharmonic functions in R$^2$ are the constants? So far, I know that u is a nonnegative superharmonic function and that there exist x, y ∈ R$^2$ ...
2
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0answers
73 views

Close-form solution for distribution of the stopping time for a path-dependent random process?

A time series $\{x_s\}_{s=1}^{\infty}$ is generated from $N(\bar{x},1/b)\ i.i.d.$. Parameter $\bar{x}$ is drawn from prior distribution $N(\phi_0,1/a)$. Define conditional expectation of $\bar{x}$ as ...
1
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1answer
41 views

Expected hitting time of Ornstein-Uhlenbeck process

If I recall correctly, it is known that for a standard brownian motion starting at $0$, that the expected time to hit some level $a>0$ is infinite. I'm curious if there's a proof of what the ...
3
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0answers
28 views

How to show that $S_k = \inf \{t \geq 0 | \|X(t)\| \geq k \} \to \infty$ as $k \to \infty$ a.s.

stack.exchangers! I am currently working my way through the proof given by Karatzas and Shreve (1988) of the Feynman-Kac Theorem (Theorem 5.7.6). However, I am missing out on the following problem: ...
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0answers
15 views

Predictable stopping time

Assume an increasing rightcontinuous $(X_t)_{t\geq 0}$ has the compensator $(A_t)_{t\geq 0}$. As saz pointed out, we want to assume that $A_t$ is continuous. Define the stopping time $\tau_s:=\inf\{t\...
1
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1answer
37 views

The hitting time $T-\tau^{l}$ has the same distribution as $\min\{\tau^{f},T\}$ regarding an Poisson Process.

Assume we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ where $\mathcal{F} =(\mathcal{F}_t)_{0 \leq t \leq T}$ is a Filtration, with $T < \infty$. On that prbability space we want to ...
4
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0answers
26 views

Comparison of stopped sub- and supermartingales when the future is discounted

Suppose we have a submartingale $X=\{X_t\}_t$ and a supermartingale $Y=\{Y_t\}_t$ which are adapted to the same filtration on a bounded set and have a common initial value $X_0=Y_0$. Suppose that $\...
1
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1answer
32 views

Continuity of the probability that a Brownian motion with drift hits an upper barrier before the lower barrier in the drift

Let $W$ be a Brownian motion and $u:\mathbb{R}_+ \to \mathbb{R}_+$ an upper barrier and $l:\mathbb{R}_+ \to \mathbb{R}_-$ a lower barrier. Let $$\tau_u(\mu) = \inf\{ t \colon \mu t + W_t \geq u(t)\}$$...
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11 views

What is the “time change” of an adapted finite-variation stochastic process?

Let $(\Omega, \mathcal F,\mathbb P)$ be a probability space equipped with a filtration $\{\mathcal F_t:t\in\mathbb R_+\}$ satisfying the usual conditions of completeness and right-continuity. Suppose ...
4
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0answers
44 views

Stopping-time sigma-algebra and the case at infinity, definition question.

Assume you have a probability space $(\Omega,\mathcal{F},P)$, and you have a filtration $\{\mathcal{F}_t\}$ and a stopping time $\tau$. Then all the books I have seen define the stopping time sigma-...
2
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33 views

A problem about the supremum of countable stopping times

Let $\left(\Omega\ ,\mathcal{F}\ ,\mathbb{P} \right)$ be a probability space with a countable filtration $F=\left\{ \mathcal{F}_0,\mathcal{F}_1,\cdots,\mathcal{F}_n,\cdots \right\}$. $\left\{ T_n \...
3
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0answers
54 views

Hitting times in two-dimensional case: expectation of Brownian motion at a hitting time

Consider two Brownian motions $$X_{1t}=\mu t+\sigma_1B_{1t}$$ and $$X_{2t}=\mu t+\sigma_2B_{2t}.$$ Here $B_{1t}$ and $B_{2t}$ are uncorrelated. Let $\tau_1$ and $\tau_2$ be the stopping times: \begin{...
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14 views

Reference for *optimal* stopping theorem for supermartingales

Can anyone introduce a good reference about optimal (not optional!) stopping times for submartingales / supermartingales? I am looking for some theorem like the one mentioned in this question. I ...
2
votes
0answers
35 views

Distribution of Markov Chain at a Stopping Time

Suppose $(X_t)_{t \geq 0}$ is a Markov chain on the state space $S$ with transition probability $p$, and that $\pi$ is a stationary distribution for $p$. If $X_0 \sim \pi$, then we know $X_t \sim \pi$ ...
2
votes
1answer
77 views

The jumping times of a càdlàg process are stopping times.

Protter first proves this theorem: Let $X$ be an adapted càdlàg stochastic process, and let $A$ be a closed set. Then the random variable: $T(\omega)=\inf\{t: X_t(\omega)\in A \text{ or } ...
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29 views

Independence between the first exit time from an interval and the value of Brownian motion at this first exit time

Suppose you have an arithmetic Brownian motion (or Brownian motion with drift ) called X, started at a level x such that a < x < b, where a and b are two real points . Define tau as the first ...
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1answer
19 views

On a condition for a.s. finite stopping times

Assume that $\{\tau_n: n \in \mathbb{N}\} $ is a sequence of stopping times with respect to some filtration such that $P[\tau_n < \infty] = 1. $ Is that true that there must exist a sequence of ...
0
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1answer
27 views

Prove $M_{S(k) \wedge n}$ is bounded in $\mathscr L^2$

Probability with Martingales: To prove $$\sup E[M_{S(k) \wedge n}^2] < \infty,$$ how can we use 12.12c? There aren't any stopping times there.
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Angle bracket (quadratic variation) process for martingales

Probability with Martingales: What is the relation between $\langle M_{S(k) \wedge n}\rangle \ = A_{S(k) \wedge n}$ and $\{N_n\}, \{ N_{ S(k) \wedge n } \}$ being martingales? It seems that $$\...
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0answers
68 views

Prove the stopped process $A^{S(k)}$ is previsible

Probability with Martingales: I have a different attempt in mind, but I'm guessing it's wrong because if it were right, the book would've used it. It seems that we must show that $$A_{S_k \...
0
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1answer
30 views

Prove $S_k$ is a stopping based on $A$ being previsible

Probability with Martingales: It looks like we have a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ where $A = \{A_n\}_{n \in \mathbb N}$ ...
0
votes
1answer
28 views

Check that stopping time is a.s. finite

I have the following situation. Let $(X_i)_{i\geq1}$ be a sequence of iid random variables in $\mathbb{Z}$ and consider the random walk $S_n=\sum_{i=1}^n{X_i}$, $S_0=x$. Let $y>x$ and consider ...
5
votes
1answer
241 views

Interlacing stopping times

This question is posed on a measurable space $(\Omega,\mathscr{F}$) equipped with a filtration $\{\mathscr{F}_t\}$. Recall that a random time $\tau\colon\Omega\rightarrow[0,\infty]$ is said to be a ...
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0answers
25 views

Is $B_{t\wedge H_a}$ bounded in $L^2$?

Let $a >0$, $(B_t)_{t\geq0}$ be a standard Brownian motion. Define the stopping time $$H_a := \inf\{t \geq 0 : B_t \geq a\}.$$ Then is the martingale $M_t$ where $M_t: = B_{t\wedge H_a}$ bounded ...
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0answers
23 views

Brownian Motion maximum process intuition

I am studying the maximum value of a Brownian Motion (BM) on an interval of time (as explained here between boxes 28 and 40) and I am having an issue aligning intuition with the mathematical result. ...
0
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2answers
56 views

The Second Hearts Problem

According to the last part of these lecture notes, if we have a standard deck of playing cards and turn cards until the first heart appears, the probability that the next card is a heart is $\color{...
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0answers
17 views

Finding optimal strategy in time series game

Let $P_t$ be a time series such that $P_{t+1} = \alpha P_t+S_{t+1}$, where $\forall t\geq0 : S_t \sim N(0,\sigma)$ Consider the following game: In each round $t$, a player sees $P_t$ and decides ...
0
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1answer
33 views

Expected time until pattern (1,0,0,1)

Let $(X_n)_{n\geq 0}$ be i.i.d. with $\mathbb P(X_n = 0 ) = \mathbb P(X_n = 1) = \frac{1}{2}$. Let $\tau_a$ be the stopping times defined as $$\tau_a = \inf\{n: (X_{n-3}, ... , X_n) = (1,0,0,1)\}$$ I ...
2
votes
1answer
45 views

Conditional independence of stopping times from i.i.d. stochastic processes

My question is somewhat arbitrary but I was thinking about independence of processes and stopping times. Say that we define two processes $X,Y$ on different probability spaces $(\Omega^i,\mathcal{F}^...
2
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0answers
38 views

Sigma algebra generated by the stopped process.

Let $(X_n)_{n \geq 0}$ be a sequence of random variables. Let $\mathcal{F}_n = \sigma (X_0, \dots, X_n)$ be a filtration and $T$ is a $(\mathcal{F}_n)_{n\geq 0}$-stopping time. I want to understand ...
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0answers
21 views

Understanding of Second Arcsine law for Brownian motion

Ok I'm trying to understand the second arcsine law which states: Let $g_t:=\sup\{s\leq t:W_s=0\}$, then $$\mathbb{P}(g_t\leq s)=\frac{2}{\pi}\arcsin \left(\sqrt{\frac{s}{t}}\right )$$ This won't be ...
0
votes
0answers
20 views

Conditional expectation of a geometric Brownian motion and stopping time

Let $X$ be a geometric Brownian motion, solution of $dX_t = \mu X_t dt + \sigma X_t dW_t, X_0 > 0$ and ${\cal F}$ its natural filtration. Let's consider $t \geq 0$ and $\tau_a$ the first hitting ...
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2answers
53 views

Where is my mistake in calculating stopping time?

A gambler has $w$ dollars and in each game he loses or wins a dollar with equal $p=1/2$ probability. I want to calculate the average number of games that a gambler can play before he runs out of ...
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0answers
27 views

Geometric Brownian motion hitting time

Let $X$ be a geometric Brownian motion $dX_t = \mu X_t dt + \sigma X_t dW_t, X_0 > 0$ and ${\cal F}$ its natural filtration. Let $\tau_a$ be the first hitting time of $a$ by $X$. How can we relate ...
0
votes
1answer
59 views

Strong Markov property and time homogeneity

Let $X$ be a Markov chain with state space $\mathcal{S}$ and denote $\mathbb{N} := \{ 0, 1, \cdots\}$. We know that for any stopping time $\tau < \infty$ and any bounded measurable function $\phi : ...
0
votes
1answer
23 views

Recurrence of a state in a finite state space

Suppose $T_A := \inf\{ n \ge 1 : X_n \in A\}$ where $A \subset \mathcal{S}$ is finite. Assume $\mathbb{P}\{T_A < \infty \; | \; X_0 = x\}= 1$ for $\forall x \in \mathcal{S}-A$. I need to show that ...
3
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64 views

Strong Markov property proof

Let $X$ be a Markov chain with state space $\mathcal{S}$ and denote $\mathbb{N} := \{0,1, \cdots\}$. I need to show that for any stopping time $\tau < \infty$ and any bounded measurable function $\...
2
votes
1answer
39 views

$T$ can be $\infty$ with positive probability

From Williams' Probability with Martingales How exactly do we know $T$ can be $\infty$ with positive probability or $$P(T = \infty) > 0 \text{ ?}$$ I'm guessing that that means there ...
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votes
2answers
45 views

Expectation of stopping time on a random walk

Assume $X_1 , X_2 , \cdots$ are i.i.d. with distribution Bernouli$(\frac{1}{2})$, i.e., $P(X_i = 0)=P(X_i=1)=\frac{1}{2}$. Denote $S_0 := 0$, $S_n := \sum\limits_{i=1}^n X_i$, and $\tau_{1000} := inf\{...
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1answer
50 views

Prove $A_{\infty} < \infty$?

From Williams' Probability with Martingales How do we know that $A_{\infty} < \infty$? If $T = \infty$, then $$E[A_{T \wedge n}] \le (K+c)^2$$ $$\to E[A_{n}] \le (K+c)^2$$ $$\to \lim ...
0
votes
1answer
32 views

Prove $|M_{T \wedge n}| \le c + K$

From Williams' Probability with Martingales Is $\sigma_k^2$ random (and not constant)? How can that be? As far as I know unconditional variance and unconditional expectations are supposed ...
0
votes
1answer
15 views

Strong Markov Property Brownian Motion for Non-Stopping Time

Let $B$ be a Brownian motion and let $\mathcal{F}^B$ be its natural filtration. Define the random variable $$ \tau = \inf\{ t \ge 0 \mid B_t = \sup_{0 \le s \le 1} B_s \}.$$ Now, $\tau$ is not an $\...
3
votes
0answers
21 views

Expectation related to Wiener process using strong Markov property

Can you help me to understand a result I found in Krylov's book "Introduction to stochastic calculus". First, I will introduce some notations: $w_t,t\ge 0$ denotes a Wiener process. $\mathcal{B}(...
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0answers
28 views

expectation of stopping time in Wiener process

Let $(W_t)$ be a Wiener process and for $a>0$ define stopping time: $$\tau = \inf \left\{t>0: W_t + at = 5\right\}$$ a) show $\tau < \infty$ a.s; b) compute $\mathbb{E}\tau$. I have done ...
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vote
0answers
56 views

Brownian Motion hitting time is finite yet has infinite expectation?

I've read that a hitting time of a Brownian motion (defined as $T_a = \inf\{t\ge0:W_t=a\}$ where $W_t$ is a standard Brownian Motion, i.e. a Wiener process), has the following two properties, which I ...
1
vote
1answer
18 views

identity with Poisson process

I want to show (if it is true) that if $\tau$ is a stopping time and $\mathbb{E}\tau < \infty$ then $\mathbb{E}N_\tau = \mathbb{E}\tau$ for $N_t$ - Poisson process with parameter $1$. I started ...
2
votes
1answer
38 views

Brownian hitting time of a closed set

I am trying to prove that the first hitting time of a closed set H by a Brownian motion is a stopping time. I have found a proof that states: $$\{\mathcal{T}\leqslant t\} = \bigcap_{n=1}^{\infty}\...