# 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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### 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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### 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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### 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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### 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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### $R$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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, $\... 1answer 33 views ### Conditional expectation of exp{xy} given filtration$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}|...
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&...