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.

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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 ...
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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 ...
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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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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 ...
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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....
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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(...
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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 ...
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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\...
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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 $\...
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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 $(...
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$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 ...
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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 $\...
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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 ...
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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. ...
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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 $\tau$ ...
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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_{...
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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 ...
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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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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 ...
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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(\...
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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 ...
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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 $\sum x_n$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 + \...
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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|...
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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 ...
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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{...
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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 ...
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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 ...
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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 ...
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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 $<\...
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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,...
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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}...
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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<\...
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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 $...
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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 (...
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$E\left[Y|F^X_T\right]=E[Y|X_T]$? 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 ...
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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 ...
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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 ...
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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 ...
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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 $G$ (...
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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 ...
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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 ...
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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, $\...
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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}|...
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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&...