All Questions
14
questions
1
vote
0
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44
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From discrete time stopping theorem to continuous
I would like to generalize by dyadic discretization some theorem I have seen in finite time setting.
Here is the theorem
Let $(X_t)_t$ be a continuous martingale and $\tau$ a bounded stopping time ($\...
- 1,381
2
votes
1
answer
91
views
Stochastic process and reaching time to an interval
I have the following exercice to do and I would like to know if what I did is correct please.
Consider $X_t$ a continuous stochastic process and $\tau$ the reaching time to the interval $[a,b]\subset[...
- 1,381
0
votes
0
answers
59
views
How can I show that $(\chi_{\operatorname{loc}}(\Bbb{F},\Bbb{P}))_{\operatorname{loc}}(\Bbb{F},\Bbb{P})=\chi_{\operatorname{loc}}(\Bbb{F},\Bbb{P})$
If $\chi$ is a family of random processes, then $\chi_{\operatorname{loc}}(\Bbb{F},\Bbb{P})$, the localized class of $\chi$, will denote the family of processes wich are locally in $\chi$ for $(\Bbb{F}...
- 299
4
votes
1
answer
140
views
Probability of seeing HTTH before THTH in coin flips
I'm doing a past paper question and trying to use Doob's Optional Stopping to find the probability that for independent identical $(X_n)$ uniform on $\{0,1\}$ we see the pattern $a = (1,0,0,1)$ before ...
- 846
1
vote
1
answer
71
views
Let $M$ be a continuous martingale, $r >0$, and $\tau := \inf \{t \ge 0 : \langle M \rangle_t \ge r \}$. Then $M_\tau$ is square-integrable
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and $\mathcal G = (\mathcal G_t, t \ge 0)$ a filtration. Let $M$ be a real-valued continuous martingale w.r.t. $\mathcal G$ such that
$$
(\...
- 5,817
0
votes
1
answer
71
views
How do I prove that $\Bbb{P}_x(\tau_1<\infty)=1$, where $\tau=\inf\{n\geq 1:X_n\neq X_0\}$?
Let $(X_n)$ be a Markov chain on a countable space $E$ and let $T$ be it's transition matrix. We assume $T(x,x)<1$ for all $x\in E$. Let $\tau=\inf\{n\geq 1:X_n\neq X_0\}$. I want to show that $\...
- 2,018
3
votes
1
answer
196
views
About continuous local martingales, question on Le-Gall's book
Background
Hello, I'm working on question 4.24 on Le-Gall's Brownian motion(...) and I would ask you to check if my ideas are correct. The question is as follows:
$(M_t)$ is a cont. local martingale ...
- 1,162
1
vote
0
answers
27
views
Recursively defining stopping times for Doob's upcrossing estimate
Suppose $u_n$ is a submartingale and define $\tau_0:=0$ and recursively for $k=1,2,3,\dots, $
$$\sigma_k:= \inf\{n > \tau_{k-1}: u_n \le a\} \wedge N\; \text{and}\; \tau_k:= \inf\{n>\sigma_k:u_n ...
- 5,611
1
vote
1
answer
313
views
Almost surely finite stopping time and the limit of a martingale
I am working on this exercise:
Let $(X_{n},\mathcal{F}_{n})$ be a martingale and $\tau$ a $\mathcal{F}_{n}$ stopping time that is almost surely finite. Further assume that $\mathbb{E}|X_{\tau}|<\...
- 5,930
1
vote
0
answers
75
views
Describe the $\sigma$-algebra generated by $T_n = \min(T, n+1)$
For $k \in N$, let $X_k$ be the outcome of the $k$-th toss and let $T$ be the toss in which the first "Heads" appears and define $T_n = T$ if $T \leq n$ and $T_n = n+1$ otherwise. That is,
$$T = \inf\{...
- 77
1
vote
1
answer
463
views
A stopped canonical Markov process is a Markov process
I came across the following question while studying for a Stochastic Processes exam:
Consider the space $(\Omega,\mathcal F)$, where $\Omega$ is the space of real valued continuous functions, and $\...
- 6,669
4
votes
1
answer
959
views
Hitting time of Brownian motion against a square root curve
$B$ denotes Brownian motion and the hitting time I am interested in is
$$\tau = \inf\{t \geq 0: B_t = b\sqrt{a+t}\}$$
where $a,b >0$. I first want to show that $\tau < \infty$ almost surely. I ...
- 5,775
1
vote
0
answers
202
views
Prove that $\inf\left\{t>\sigma:X_t=\varepsilon\right\}$ is a stopping time
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge0}$ be a filtration of $\mathcal A$
$X$ be a real-valued continuous $\mathcal F$-adapted stochastic process on $(\...
- 13.9k
2
votes
2
answers
1k
views
Stopping times, Filtration, Martingales,
I am new here and I have a question.
Definition: Let $ \tau$ be a stopping time, then $\mathcal F_{\tau}=\left\{F\subset \Omega: \forall n \in N \cup \{\infty\} , F\cap(\tau\leq n)\in \mathcal F_{n}\...
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