Questions tagged [filtrations]
This tag is for questions relating to "Filtration". It has many application in abstract algebra, homological algebra and in measure theory and probability theory for nested sequences of σ-algebras.
185
questions
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2answers
22 views
Are all stochastic processes $\mathcal{F}_t$-measurable?
I am currently reading about martingales, and the notion of a $\mathcal{F}_t$-measurable process has been introduced. It is stated that:
The filtration [of a process] $\mathcal{F}_t$ represents ...
1
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0answers
17 views
The canonical filtration of a Brownian motion is not a subset of the canonical filtration of its absolute process
I was reading the following example from René Schilling's Brownian Motion. However, I cannot understand the final argument. Given that $X_t$ is a Brownian motion and $\mathscr{F}_t^X$ is its completed ...
1
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0answers
24 views
Filtration Generated by Stochastic Process: Exercise 3.4.10 in Cohen-Elliott Stochastic Calculus
This is my first question here at the community, so I thank in advance anyone who is reading and, possibly, answering it.
Recently, I have been doing some exercises from the book of Elliott and Cohen &...
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0answers
58 views
What metric captures the progress of these two filtrations?
Let $X=\Bbb Z[\frac16]^+$ be the non-negative dyadic and ternary rationals, which is a monoid two ways - both mutliplicatively and additively, give or take zero.
Let $X/\langle2,3\rangle$ be the ...
0
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0answers
24 views
Is the filtration of a process “evaluated” at a random time a sigma-algebra?
Consider the stochastic process $(X_n)_{n\geq 1}$, defined by
\begin{align}
X_n = \sum_{k=1}^nY_k, \qquad Y_k= \left\{\begin{array}{ll} \mathcal{N}(0,1), &\; \text{w.p.} \; p \\
0, &\; \text{w....
0
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1answer
26 views
Questions about proving the statement: $\tau(\omega)$ is a stopping time, iff $\{ \tau(\omega) < t\}$ $\in \mathcal{F}_t$, for all $t \geq 0$.
Assume the filtration is right-continuous ($\mathcal{F}_{t+0} := \cap_{s>t}\mathcal{F_s} = \mathcal{F}_t$) and complete, then we have that $\tau(\omega)$ is a stopping time, if and only if $\{ \tau(...
3
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1answer
62 views
Clarification and Proof of 'equivalence' asserted in Matsumura
Let $A$ be a commutative ring with unity, $M$ an $A$-module and $I$ an ideal of $A$.
Now it is fairly obvious that we invoke the usual definition of a Cauchy sequence of $M$ in the $I$-adic topology ...
0
votes
1answer
19 views
A counterxample of a non-adapted process?
I know that, given $\mathbb{I}\subset[0,\infty]$, we call a stochastic process $(X_t)_{t\in\mathbb{I}}$ adapted to the filtration $\left(\mathcal{F}_t\right)_{t\in\mathbb{I}}$ if $X_t$ is for each $t\...
3
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1answer
48 views
Shifting balls in urns that are already occupied by balls
At the time $n=0$ we place $N$ balls in $k$ urns and change this in each step as follows: We choose one of the balls evenly distributed at random (meaning: each ball is chosen with a probability of $\...
1
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0answers
66 views
Find $a_n$ and $b_n$, such that $M_n=a_nY_n+b_n$ defines a Martingale, when $E[Y_{n+1}|F_n]=u_nY_n+v_n$.
Let $(\Omega,\Sigma,P)$ be a propability space and $(\mathcal{F}_n)_{n\in\mathbb{N}_0}=:\mathbb{F}$ a filtration. Let $Y_0,Y_1,\dots$ be a adapted process of integrable random variables. Further let $(...
0
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1answer
57 views
$R$ be a Noetherian semi local ring such that $R/N(R)$ is complete $\mathrm{Jac}(R)$-adically, then $R$ is complete $\mathrm{Jac}(R)$-adically.
Let $R$ be a Noetherian semi local ring, and $I=\text{N}(R)$, $J=\text{Jac}(R)$. If $R/I$ is complete w.r.t. $J$-adic filtration then we have to show that $R$ is complete in $J$-adic filtration.
Thus ...
1
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1answer
51 views
Describe all martingales $(X_n)_{n\in\mathbb{N}}$, such that $X_n\in\{-1,0,1\}$ for all $n\in\mathbb{N}$ with an arbitrary sample space $\Omega$.
Describe all martingales $(X_n)_{n\in\mathbb{N}}$, such that $X_n\in\{-1,0,1\}$ for all $n\in\mathbb{N}$ with an arbitrary sample space $\Omega$.
This Question evolved out of this Question where $\...
0
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1answer
24 views
Does the Martingale Representation theorem hold both ways?
Can the Martingale Representation theorem be used to assume that the integral with respect to Brownian motion, $B(t,\omega)$,
$$X=\int^{T}_{0}B^{4}(t,\omega)dB(t,\omega)$$
is a square integrable ...
2
votes
1answer
52 views
Stochastic processes - Why do we need filtration?
In the theory of stochastic process, besides the $\sigma$-algebra $\mathcal {F}$, we have an increasing sequence of $\sigma$-algebras $\{{\mathcal {F}}_{{t}}\}_{{t\geq 0}} $ called filtration.
...
0
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1answer
24 views
Relation between Stopping times
I am having a hard time trying to understand the following relation:
Consider that a stopping time is defined by $\{\tau \leq n \} \in \mathcal{F}_{n}$.
Now take two stopping times $\phi$ and $\tau$ ...
2
votes
0answers
10 views
Is the product of a constant and a stopping time also a stopping time?
If $\tau$ is a stopping time wrt $\mathcal{F}=\{\mathcal{F}_t : t \geq 0\}$ and $\alpha \in \mathbb{R}$, then is $\alpha \tau$ also a stopping time?
I know that if $\alpha \geq 1$, then $\mathcal{F_{...
0
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0answers
16 views
Show that filtrations are equal
Hey how to show that natural filtrations generated by processes $W(t)$ and $W_Q(t)=\frac{\mu-r}{\sigma}t+W(t)$ are the same where $\mu\in\mathbb{R},r,\sigma>0$? So that $F_t=\sigma\{W(s):0\le s\le ...
1
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0answers
49 views
Is it true that $\mathbb{E}\{X_n\}=\mathbb{E}\{X_m\}\Rightarrow\mathbb{E}\{X_n|\mathcal{F}_m\}=\mathbb{E}\{X_m|\mathcal{F}_m\}=X_m$?
Given a filtration $(\mathcal{F}_{n})_{n}$ and a sequence of random variables $(X_n)_{n}$, is it true that, for $m\leq n$,
$$\mathbb{E}\{X_n\}=\mathbb{E}\{X_m\}\Rightarrow\mathbb{E}\{X_n|\mathcal{F}_m\...
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0answers
3 views
Independence relation for joint filtrations
Let $X$ be a random variable with first moment.
Let $\mathcal A$ and $\mathcal B$ be sub-$\sigma$-algebras.
Let $\mathcal A$ be independent of $\sigma(X) \vee \mathcal B$.
Does it hold that?
$$
E( X ...
3
votes
0answers
51 views
What does it mean by saying 'a random variable $\mathit X$ is $\mathcal G$-measurable'?
This is the definition of "measurability":
Let $\mathit X$ be a random variable defined on $(\Omega,\mathcal F,\mathbb P)$. Let $\mathcal G$ be a $\sigma$-algebra of subsets of $\Omega$. If $\sigma(\...
0
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0answers
13 views
I am trying to construct a process $Y$ in the martingale representation of the following two terms
(i) $M_t = \mathbb{E}[B_t^2|F_t]$
(ii) $M_t = \mathbb{E}[\text{max}\{B_t,0\}|F_t]$
Where $B$ is a one-dim. Brownian Motion and $F$ its $P$-completed canonical filtration.
I think in (ii) we can ...
1
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1answer
20 views
In a filtered module $\sum x_n$ converges if $x_n$ tends to 0 in $M$
Let $M$ be a filtered module which is Hausdorff and complete with respect to the topology defined by the filtration. I want to show that if the sequence $\{x_n\}$ tends to $0$ the series $\sum x_n$ ...
1
vote
1answer
109 views
Show that $X$ is a submartingale, given some assumptions. Is the following solution correct?
Let $X=(X_n)_{n>0}$ be an increasing sequence of integrable r.v.'s, each $X_n$ being $\mathcal{F}_n$-measurable. Show that $X$ is a submartingale.
MY SOLUTION
What I have to show is that, given ...
1
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0answers
27 views
What does $X \in \mathbb{F}$ mean for $\mathbb{F}$ being a filtration
I am reading lecture notes which say
Let $\mathbb{F}$ be a history, the process $X$ satisfies $X \in \mathbb{F}$ and ...
Now, I know what a history/filtration is, X is, in my understanding, a ...
1
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0answers
150 views
Computing expectation of random variable with respect to filtration
Consider a counting process $N$ and a binary random variable $V \in \{ 0,1 \}$. $N$ has intensity process $\lambda$ with respect to $\mathcal F^{N,V}$, the filtration generated by $N$ and $V$. In ...
0
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0answers
8 views
An example of the filtration of sigma algebras where F0 is not the trivial sigma algebra
Does anybody have any examples when F0 is not the trivial sigma algebra? I know it is normal to presume that it always is but for an assignment I have to find an example where F0 is not trivial and ...
2
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0answers
40 views
Computation of Associated Graded Module
I am trying to compute $\mathrm{gr}_m(P)$ where $m=\langle X,Y\rangle $ and $P=\langle X^2-Y^3\rangle$ in the power series ring $R=\mathbb C[[X,Y]]$ with the $m$-adic filtration and show that it is ...
0
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0answers
22 views
Grillet, Proposition VI.9.2 (minor detail)
Having a quick problem with a line in Grillet's Abstract Algebra, Prop. VI.9.2 on p. 267-8 on the topic of Filtrations and Completions. He says that the ideal
$$\widehat{\mathfrak{a}}_j = \{ (x_1 + \...
1
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1answer
31 views
Relation between two versions of the Second Borel Cantelli lemma
In Durett's book, the second Borel Cantelli lemma is as follows:
Let $\{F_n\}$ be a filtration, and $A_n\in F_n$ be a sequence of events. Then, $\{A_n \text{ i.o.}\}=\{\omega:\sum_{n=1}^\infty P(A_n|...
1
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0answers
40 views
How to prove that a version of adapted process X is also adapted wrt Filtration?
We have a process $X_1$ and a modification of that process $X_2$.
Criteria to be a modification is as follows: P($X_1t$ = $X_2t$) = 1 for all t in interval I.
A filtration is just an increasing ...
0
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0answers
14 views
Can the addition of a stochastic drift result in smaller generated filtration?
For some $T>0$, let $(B_t)_{t\in[0;T]}$ be a standard Brownian motion in $(\Omega, \mathcal{F}, \mathbb{P})$. Let $\mathbb{F}$ be the filtration generated by $B$. Is it possible to find an $\mathbb{...
0
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0answers
9 views
Filtrations for multiple different stochastic processes
In an answer to this post concerning filtrations and stochastic processes, the following statement is made:
Another point which you may be missing here is that you should really think of the ...
1
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0answers
33 views
Motivation for Grading and Filtration
I have started reading about graded rings and modules and filtered rings and modules.
For grading at least,I can see the polynomials as a prototype,graded by usual degree. But I can't seem to find ...
0
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0answers
22 views
Filtration at infinity and sigma-algebra of events strictly prior to infinity.
I'm a worker and I'm studying Stochasstic processes from the book Stochastic Calculus and Application by Cohen and Elliott.
In particular, I am reading the Chapter 4 on Discrete Time Stochastic ...
1
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1answer
25 views
On $\kappa$-filtrations and clubs.
Let $A$ be a set of cardinality $\kappa$. A $\kappa$-filtration of $A$ is an indexed sequence $\{A_{\nu}:\nu<\kappa\}$ such that for all $\mu,\nu<\kappa$:
the cardinality of $A_{\nu}$ is $<\...
2
votes
1answer
38 views
Uniformly integrable martingale problem: typo (wrong filtration)?
I am working on the following problem: Let $(X_n,{\cal F}_n)_{n\in\mathbb N}$ be a uniformly integrable martingale, let $M\in{\mathbb R}_{>0}$ and let $\tau=\inf\{n\in\mathbb N:|X_n|\ge M\}.$ (Here,...
0
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1answer
20 views
Understanding the join of sigma-algebras (filtrations)
Let $(\mathcal{F}_n)_{n\geq0}$ be a filtration, i.e. $\mathcal{F}_0 \subset \mathcal{F}_1 \subset \dots $ and define the join of the sigma-algebras (filtration) $\mathcal{F}_\infty := \bigvee_{n\geq 0}...
2
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0answers
11 views
Classification of stopping times in a Poisson Process setting
Due to the general theory of stochastic processes by Dellacherie and Meyer we know that split any stopping time into an accesible and totally inaccessible part, that is
$T=T_I\wedge T_A$ on $\{T<\...
1
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1answer
32 views
inclusion of filtration
Let $(\Omega,F,P)$ be probability space and $T>0$, let $\lbrace W(t), 0\leqslant t\leqslant T\rbrace $ be SBM and $F^W_{s,t}=\sigma \lbrace W(r)-W(s); s\leqslant r\leqslant t\rbrace \vee N$
with $...
1
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1answer
23 views
Problem comprehending an equation about Stopping Times
I am given the following definition:
T is a stopping time if $\{T⩽t\}∈F_t$ for all t. $\{S∧T⩽t\}=\{S⩽t\}∪\{T⩽t\}$ both of which are in $F_t$.
I do not understand the $\cup$ sign.
In my logic (...
1
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0answers
18 views
$E\left[Y|F^X_T\right]=E[Y|X_T]$? Reducing the conditional stopped sigma algebra in a natrual filtration setting.
I have the following problem. A random variable $Y$ depends on $X_T$, where $X$ is a strong Markov process generating the filtration of the space, but not on what has happend earlier than $T$. ($Y$ is ...
1
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1answer
65 views
Why is this Brownian motion event not measurable w.r.t. the natural filtration?
In my lecture on stochastic processes it is stated that the natural filtration $\mathcal{F_t}^0=\sigma(\forall s\leq t: \omega\mapsto \omega(s)$ is measurable$)$ is not a good choice for Brownian ...
1
vote
3answers
138 views
Help understanding the definition of a “filtration” in probability theory
I am having trouble understanding wikipedia's definition of filtration in probability theory:
Definition "filtration"
Let $(\Omega ,\mathcal {A}, P)$ be a probability space
Let $I$ be a totally ...
1
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0answers
50 views
If we have $(\Omega, \mathscr{F}, P)$ and a filtration $\{\mathscr{F}_t^W; 0 \leq t \leq T\},$ how can we justify that $\mathscr{F} = \mathscr{F}_T$?
Let $(\Omega, \mathscr{F}, P)$ be a probability space and $T>0.$ Also, let $\{B_t; 0 \leq t \leq T\}$ be a Brownian motion that generates the filtration $\{\mathscr{F}_t^W; 0 \leq t \leq T\}.$ I ...
4
votes
2answers
83 views
Why does a filtration of a group consist of normal subgroups, and not any subgroups?
See here:
In algebra, filtrations are ordinarily indexed by $\mathbb {N}$ , the set of natural numbers. A filtration of a group $G$, is then a nested sequence $G_{n}$ of normal subgroups of $G$ (...
0
votes
0answers
28 views
Why is it necessary to have a topology associated to a filtration of algebraic objects?
Why is it necessary that when we are defining a filtration of algebraic objects there must be a topology associated to the filtration?
For example, a descending filtration of group has the topology ...
0
votes
2answers
338 views
Proving a Wiener Process is a Martingale
Let $W_t$ be a Wiener process. Let $X_t = W_t^2 − t$. Show that {${X_t
;t ≥ 0}$} is a martingale with respect to the natural filtration.
To prove that it is in fact a Martingale I must prove 2 ...
1
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0answers
32 views
Examples of when filtrations matter for martingales
I am trying to hone my understanding of martingales with respect to their filtrations.
The definition of a martingale is the following:
Let $(\Omega,\mathscr{F},P)$ be a probability space, $\...
0
votes
1answer
33 views
Conditional expectation of exp{xy} given filtration $F_1$
Let $Y_1, Y_2$ be independent en standard normal distributed and $F_1$ a natural filtration. For $0<\lambda <1$, I need to compute the following conditional expectation:
$E[e^{\lambda Y_1Y_2}|...
0
votes
0answers
17 views
Mesurability of Wiener Process
Let $(W_u)_{u\geq0}$ be a Wiener process defined on a probability space $(\Omega,\mathbb P, \mathscr{F}, \sigma(W_u)_{u\geq0})$ where $\sigma(W_t)$ is the $\sigma$-algebra generated by $W_t$. Let $s&...
