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-1
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1answer
49 views

Prove a thm on stopped processes given fundamental principle 'you can't beat the system'?

How does the principle below imply the thm below? From Williams' Probability w/ Martingales: Principle: Thm: What I tried: $$E[X_{T \wedge n} - X_0 | \mathscr{F_m}] =/ \le X_{T \wedge ...
2
votes
0answers
27 views

Optimal stopping time problem

I'm trying to solve a problem: $$ \sup_{0 \leq \tau \leq 1} E W_{\tau - \varepsilon}, $$ where $W$ is a Wiener process and $\varepsilon$ is a fixed real number. I've tried to approximate a Wiener ...
2
votes
0answers
35 views

An intuitive interpretation of stopping time

I have the following definition of exercise time. Let $T\in\mathbb{N}$ with $T>0$, let $(\Omega,\mathcal{F})$ be a probability space with the $\sigma$-algebra $\mathcal{F}=2^{\Omega}$ and let ...
0
votes
1answer
35 views

Continuity with respect to hitting time level

Let $\tau(x)$ be the first hitting time of a Lévy process $(X_t)_{t\geq 0}$ to level $x$. Let $f$ be a continuous function and $g(x)=\mathbb{E}[f(\tau(x))]$. Is it obviously true that $g$ is ...
1
vote
0answers
17 views

Exponential decay of a stopping time for an Ito diffusion process

Let $dX_t=dB_t + a \cot(X_t)dt$, with $X_0=x \in (0,\pi)$, where $a$ is a specific constant so that the lifetime of the process is infinite almost surely. The process has a transition density which ...
2
votes
1answer
51 views

Brownian motion: hitting times for closed sets are stopping times (and more).

Let $(B_t)$ be a $d$-dimensional Brownian motion, and consider the filtrations $(\mathcal{F_t^B}) = \sigma(B_0,...,B_t)$ and $\mathcal{F_t} = \cap_{\epsilon > 0}{\mathcal{F_{t+\epsilon}^B}}$ (the ...
0
votes
1answer
59 views

$X_T = \lim_{n \to \infty} X_{T \wedge n}$ if X is a supermartingale and T is a finite a.s. stopping time?

Given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}, \mathbb{P})$, let $X = (X_n)_{n \geq 0}$ be a $(\{\mathscr{F_n}\}, \mathbb{P})$-supermartingale and $T$ be a finite ...
-1
votes
1answer
77 views

Prove $X_T$ is integrable if $X$ is a supermartingale, $T$ is stopping time and other conditions

Let $X = ({X_n})_{n \ge 1}$ be a/an $(\{\mathscr{F_n}\}_{n \ge 1}, \mathbb{P})$-supermartingale in the filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}_{n \ge 1}, \mathbb{P})$. ...
0
votes
1answer
74 views

Prove $Y_S$ is integrable if $Y$ is a bounded supermartingale and $S$ is an a.s. finite stopping time. [closed]

Let $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}_{n \in \mathbb{N}}, \mathbb{P})$ be a filtered probability space, and let $Y = ({Y_n})_{n \in \mathbb{N}}$ be a/an $(\{\mathscr{F_n}\}_{n \in \mathbb{N}}, ...
0
votes
1answer
50 views

Prove $Y_S$ is integrable if $Y$ is a supermartingale and $S$ is a bounded stopping time.

Given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F_n}\}_{n \in \mathbb{N}}, \mathbb{P})$, let $Y = ({Y_n})_{n \in \mathbb{N}}$ be a/an $(\{\mathscr{F_n}\}_{n \in \mathbb{N}}, ...
1
vote
0answers
19 views

k-th hitting time is a stopping time

Could you check if my solution is correct? I find the filtrations quite tricky. Here is the problem: Let $\{X_n\}_{n \in \mathbb{N}}$ be a stochastic process and $B$ a borel set in $\mathbb{R}^N$. ...
1
vote
1answer
45 views

Stopping time $\max \{n: S_n \le t\}$

Let $\{X_n\}_{n \in \mathbb{N}}$ be iid and non-negative. Let $$N(t): = \max \{n: \ X_1 + \cdots + X_n \le t \}.$$ Is $N(t)$ a stopping moment with respect to the natural filtration ...
0
votes
0answers
29 views

Strong Markov property and stopping time

In the book by Jeanblanc, Yor & Cheney, "mathematical methods for financial markets", on page 17 above Prop.1.1.14.3, there is a strange identity of a strong Markov process $X$ that reads $${\bf ...
0
votes
1answer
71 views

Questions on Doob's Optional Stopping Theorem (a) and (b)

From Williams' Probability w/ Martingales: What is $X_T$ in red box above? I am fairly certain this was not defined previously in the book. There was this though: I have a feeling $X_T = ...
0
votes
1answer
16 views

Optional Sampling Theorem - Martingales

I have problems with solving the following problem. Can anyone give me a hint how to solve it? Thanks in advance! Consider a contract that at time N will be worth either 100 or 0: Let S(n) be its ...
3
votes
0answers
47 views

Stopping times and hitting times for cadlag processes

I can't find the proof of the following lemma in any book: LEMMA: If $X=\{X_t\}_{t\in T}$ is adapted and right continuous, then for every closed set $C \subset E $, the variable $\tau_{C}:=\inf\{t\in ...
2
votes
0answers
50 views

Solving Stochastic Differential Equation

Let $\beta > 0$, $0 < \gamma < 1$, and let $\tau$ be the first hitting time: $$\tau = \inf\{t:t \geq 0, |W_t| = \pi /4\}$$ Solve the SDE in the random interval $0 \leq t \leq \tau$ $$dX_t = ...
2
votes
1answer
45 views

Augmented filtration martingale proof.

Part a: Consider a Wiener process, $W_t$ and denote by ${\mathscr{F}_t}_{(t \geq 0)}$ the natural filtration generated by W. Let $\mathbb{R}_{+} = \{x : x \geq 0\}$ and $\mathscr{B}$ be a sigma ...
2
votes
0answers
27 views

Stopping time distribution and transforms with 1-dimension B-motion.

Let $W_t$ be a 1-dimensional Brownian Motion. For $x>0$, we define: $$\tau_{x} = inf \{ t \geq 0; |W_t| = x\}$$ Compute $E[e^{-s\tau_x}]$ and prove that $\tau_x$ is equal in distribution to ...
1
vote
0answers
32 views

Is this a stopping time?

Let $X_1, X_2, ...$ be a sequence of i.i.d. non-negative random variables. Let $\tau = \inf \{n : \{ \sum^{n-1} _{i=1} X_i \leq 1 \} \cap \{ \sum^{n} _{i=1} X_i \geq 1 \} \}$. Is $\tau$ a ...
2
votes
0answers
20 views

Verify argument about random time being measurable w.r.t. a reversed BM increment process.

Let $B$ be a brownian motion (assume for convenient notation that it is two-sided). Let $T:=\sup\{t<2 :\vert B_t-B_1\vert\geq 1\}.$ Let $Y$ be the process $s\mapsto (B_T-B_{T+s})$. I want to argue ...
2
votes
1answer
48 views

What is a good/extensive undergraduate level reference on random walks?

Random walks on graphs, expected times for different things, gambler's ruin. I seem to either stumble on some pretty advanced texts about group representation theory or texts that briefly mention it ...
1
vote
1answer
54 views

Proof that the stopping time for a Brownian Motion is finite for given target levels

Given a standard brownian motion $W_t$ and defining $\tau$ as: $\tau :=\inf\{t\geq0:W_t=1$ or $W_t=-2\}$ The proof below shows that the stopping time is finite: $$\begin{align*} P(\tau < t) ...
0
votes
0answers
24 views

Is the following a stopping time?

If $\{X_n\}_{n\in\mathbb N_0}$ is a Markov chain is $T:=\{\inf n\ge1:X_{n}=X_{n-1}\}$ a stopping time ? $\{T=n\}=\{X_0\neq X_1, X_1\neq X_2\dots X_{n-2}\neq X_{n-1},X_{n-1}=X_n\}$, I would say ...
3
votes
0answers
23 views

Conditional expectation and stopping time $\mathbb{E}(X1_{T\leq m}|\mathcal{F}_{T\wedge m})=\mathbb{E}(X1_{T\leq m}|\mathcal{F}_T)$

Let $X$ be a random variable and $T$ a stopping time in a filtrated probability space. If $m > 0$ is it true that: $$\mathbb{E}\left(X1_{T\leq m}|\mathcal{F}_{T\wedge ...
0
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0answers
24 views

Wald's Identity for Higher Moments

For the sum $S_N = \sum_{i=1}^N X_i$ where $N$ is a stopping time Wald's second identity tells us, if $var(X) = \sigma^2 < \infty$, that $\mathbb{E}[(S_N - N\mu)^2] = \sigma^2\mathbb{E}[N]$. I'm ...
2
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0answers
20 views

Application of Laplace transform to stopping times and expectations

Let $X_k$ be i.i.d. random variables such that $E[X_1]=m<\infty$. Consider $S_n = \sum_{k=1}^{n} X_k$. Let $\tau$ be a stopping time independent of $X_k$ with respect to the filtration $\{F_n\}_{n ...
2
votes
1answer
164 views

The expected value of stop-time for Brownian motion $\tau=\min_t\{B_t^2\geq t+1\}$.

Let $B_t,\;t\geq0$ be a standard Brownian motion. Define the stopping time $$\tau = \min_t\{B_t^2\geq t+1\}$$ Is the expected value $E(\tau)$ finite? Actually, my raw problem as following: $$\gamma ...
1
vote
1answer
30 views

Show submartingale property.

Let $\tau$ be a stopping time. Let $X_k$ be iid random variables such that $E[X_i] = m < \infty$. Also, $m>0$.Show that $\sum_{k=1}^{\min(\tau,n)} X_k$ is a submartingale. We need to show ...
0
votes
1answer
46 views

Prove $\tau=\inf\{t\in[0,T]:M_t=0\}\wedge T$ a stopping time for a continuous martingale $(M_t)_{t \geq 0}$

I have a question about a positive continuous martingale. Let $(M_t)_{t\in[0,T]}$ be a continuous martingale such that $P(M_t>0)=1$ for all $t\in[0,T]$. Set $\tau=\inf\{t\in[0,T]:M_t=0\}\wedge T$. ...
1
vote
2answers
79 views

Showing stopping is finite almost surely

Consider a discrete random walk taking values +1 or -1 with probabilities p and q, respectively. Let $S_n = \sum_{k=1}^{n}X_k$. Let $[-A,B]$ be an interval, $A,B \geq 1$. Now define $$\tau =\min(n:n ...
0
votes
0answers
16 views

Requesting references on the number of times a random process hit a target.

Let $X_t$ be a continuous process, $A$ be some event, and $N(x)$ be the number of times $X_t(x)$ enter $A$ in a fixed time window $t \in [a,b]$. Are there results concerning $N(x)$? Like, what its ...
0
votes
0answers
37 views

How to find intersection of moving circle and line?

Say I have a point, with position (x1,y1) at time t=0, with velocity dx1 and dy1 in the x and y directions respectively, which may or may not collide with a circular entity with radius r, centered at ...
2
votes
0answers
27 views

If $(F_t)_t$ is a filtration, $T$ is a stopping time and $Y$ is $F_T$-measurable, then $1_{\left\{T=s\right\}}Y$ is $F_s$-measurable

Let $(\Omega,\mathcal A)$ be a measurable space $I\subseteq[0,\infty)$ $\mathbb F=(\mathcal F_t)_{t\in I}$ be a filtration on $(\Omega,\mathcal A)$ $\tau$ be a $\mathbb F$-stopping time $\mathcal ...
0
votes
1answer
29 views

If $X$ is an $\mathcal F_t$-adapted process with countable time domain and $\tau$ is a stopping time, then $X_\tau$ is $\mathcal F_\tau$-measurable

Let $(\Omega,\mathcal A)$ be a measurable space $I$ be an at most countable set $\mathbb F=(\mathcal F_t)_{t\in I}$ be a filtration in $(\Omega,\mathcal A)$ $X=(X_t)_{t\in I}$ be an $\mathbb ...
2
votes
1answer
20 views

Is there any intuition behind the statement $E[X_\tau \mid \mathcal{F}_\sigma]=X_\sigma$

Is there any intuition behind the statement $E[X_\tau \mid \mathcal{F}_\sigma]=X_\sigma$ a.s. I mean I know that the interpretation of the conditional expectation and how to visualize it somehow but I ...
1
vote
1answer
48 views

Why do we need optional stopping theorem?

For martingale,optional stopping theorem states: Let $(M_n)_{n\in \mathbb{N}}$ be adapted with $M_n\in L^1$ for all $n$ and if $(M_n)_{n\in \mathbb{N}}$ is a martingale, then $E[M_T]=E[M_0]$, for all ...
3
votes
1answer
48 views

Exist $\alpha < \infty$, $\beta > 0$ such that $\mathbb{P}\{T_\lambda > t\} \le \alpha e^{-\beta t}?$

Let $B_t$ be a standard one-dimensional Brownian motion. Suppose $\lambda > 0$ and let$$T_\lambda = \min\{t : |B_t| = \lambda\}.$$Do there exist $\alpha < \infty$ and $\beta > 0$ (which may ...
3
votes
1answer
33 views

A martingale characterization

I saw the following characterization of martingales (without proof) in some lecture notes I found on the web and I haven't been able to produce a proof it. Let $X$ be an adapted process. If ...
0
votes
1answer
38 views

Stopping time proof with discrete martingale

My professor gave me a very unclear proof of this theorem. It was so messy and unclear, I was unable to write down all the details of the proof. Theorem: Suppose $\tau \in T$, where $T$ is the set ...
0
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0answers
22 views

Time integral of Brownian motion's running maximum

Let $\mu \geq 0$ and consider $B_{\mu}(t) := B(t) + \mu t$ a one-dimensional BM with drift $\mu,$ and let $M_t := \max_{0 \leq s \leq t} B_{\mu}(t)$ be its running maximum. My question involves two ...
3
votes
1answer
39 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
73 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
108 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
35 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
26 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
21 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
91 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
50 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
68 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 ...