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Questions tagged [filtrations]

This tag is for questions relating to "Filtrations". It has many application in abstract algebra, homological algebra, topology, measure theory and probability theory for nested sequences of $\sigma$-algebras.

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Meaning of $\mathcal{F}_{\min\{n,\tau\}}$

Suppose $\mathcal{F}_n$ is a filtration and $\tau$ is a stopping time. What does $\mathcal{F}_{\min\{n,\tau\}}$ mean in this context? I am struggling to grasp what that should mean since $\tau$ ...
MathMaestro's user avatar
0 votes
1 answer
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Is any (integrable) centered, adapted stochastic process with independent increments a martingale?

I came across the following question: Assume $(X(t), t\geq 0)$ is an integrable, adapted stochastic process on a filtered probability space $(\Omega, \mathcal{F},(\mathcal{F}_t, t\geq 0), \mathbb{P})$ ...
Frank's user avatar
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2 votes
0 answers
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What is the cohomology of a polynomial ring of a cochain complex?

I am struggling to compute the cohomology of a polynomial ring of a fixed cochain complex. Starting with the simplest case, consider a two-step cochain complex of vector spaces $$ V := V_{0} \...
Stańczyk's user avatar
  • 131
3 votes
0 answers
47 views

On stopping time and filtration containments

This is an attempt to clarify an issue from Section 6 in the proof of Theorem 2.2. Bass, R. F., Uniqueness in law for pure jump Markov processes, Probab. Theory Relat. Fields 79, No. 2, 271-287 (1988)....
Sarvesh Ravichandran Iyer's user avatar
-1 votes
1 answer
52 views

Filtration of wedge of spheres

I'm taking an algebraic topology class and playing around definitions, and I can't answer the following question. Let $K$ be a simplicial complex with the homotopy type of a wedge of spheres, $K \...
mat95's user avatar
  • 339
3 votes
1 answer
57 views

Is $\inf\left\{t\in\left[0,1\right]\vert t+B^2_t=1\right\}$ a stopping time?

Problem Let $\left(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P}\right)$ be a filtered probability space such that $(\mathcal{F}_t)_{t\geq 0}$ is a complete and right-continuous filtration ...
Wilfred Montoya's user avatar
2 votes
1 answer
28 views

Two definitions for $\mathcal F_{\infty}$

I have just started learning about Brownian motion $(B_t)_{t \geq 0}$. The book I'm following defines a filtration $(\mathcal F_t)\_{t \geq 0}$ by $\mathcal F_t := \sigma(B_r, 0 \leq r \leq t)$. I ...
Francesco Squillari's user avatar
1 vote
0 answers
18 views

Weibel 5.4.4: How are the mapping cone and spectral sequences related

Let $f\colon B\rightarrow C$ be a map of filtered chain complexes. For each $r\geq 0$, define a filtration on the mapping cone $cone(f)$ by $F_p cone(f)=F_{p-r}B_{n-1}\oplus F_pC_n$. Show that $E_{p}^...
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2 votes
0 answers
21 views

Weibel 5.4.3: How to prove to prove shifting a filtration gives an isomorphic spectral sequence

Given a filtration $F$ on a chain complex $C$, define two new filtrations $\tilde{F}$ and $Dec(F)$ on $C$ by $\tilde{F}_pC_n=F_{p-n}C_n$ and $(Dec F)_pC_n=\{x\in F_{p+n}C_n\colon d(x)\in F_{p+n-1}C_{n-...
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4 votes
0 answers
48 views

intersection homology and naturality

I'm learning about intersection homology, and I'm trying to write a proof of the following statement: Take X,Y two filtered spaces with perversities $p$ and $q$ resp. , a continuous function $f:X\to ...
bml64's user avatar
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4 votes
0 answers
110 views

When is $\text{gr}(V)\cong V$?

Let $V$ be a vector space with an increasing filtration $V_j, j\in \mathbb Z$, we assume the filtration is Hausdorff $\bigcap_j V_j=0$ and exhaustive $\bigcup_j V_j=V$. Consider the associated graded ...
Eric Ley's user avatar
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1 vote
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Why is $\{(\Delta_1,...,\Delta_k) \in B_1 : k\geq 1, B\in \mathcal{B}(\mathbb{R}^k)\}$ a $\cap$-stable generator of $\mathcal{F}_u$?

Why is $\{(\Delta_1,...,\Delta_k) \in B_1 : k\geq 1, B\in \mathcal{B}(\mathbb{R}^k)\}$ a $\cap$-stable generator of $\mathcal{F}_u$? Suppose $(W_t)_t$ is a standard Brownian motion and $0=s_0<s_1&...
Analysis's user avatar
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0 answers
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Continuous process is measurable

I am revising the lecture notes and i stumbled across this proposition: Given $(\Omega, \mathcal{A}, P)$ a probability space, the following holds: If a process $(X_t)_{t\geq 0}$ has continuous ...
Davide's user avatar
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1 vote
1 answer
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morphism on complete filtered group

I am studying complete local fields. For the third time i read proofs which seem similar. There is an application defined on a complete group with a filtration $F_n$ (i'm not sure if it's the right ...
noradan's user avatar
  • 189
0 votes
2 answers
71 views

Definition of predictable stopping time

I am looking for a counter-example to $$ \tau \text{ is a predictable stopping time} \iff \forall t ~~~ \lbrace \tau \leq t \rbrace \in \bigcap_{l<t} \mathcal{F}_l $$ I have already shown that the ...
Justin Ruelland's user avatar
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0 answers
31 views

Are function spaces over a shrinking set vector bundles?

I have function spaces $F(S_t) \subseteq \{f : S_t \to \mathbb{R}\}$ defined over shrinking sets $S_t\subset S_s$ for $t> s$. I have a trivial fiber bundle $[0,\infty) \times F(S_0)$ where the ...
lightxbulb's user avatar
  • 2,057
0 votes
1 answer
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Can an IID sequence be a martingale?

I was looking into Doob's upcrossing inequality, which says that for a supermartingale $X$ and real numbers $b > a$, $$(b-a)\mathbb{E}[N_n([a,b], X)] \leq \mathbb{E}[(X_n -a)^-],$$ where $N_n([a,b],...
Harry Partridge's user avatar
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0 answers
42 views

Super-martingale at stopping time is a super-martingale

I have a question regarding right-continuous super martingales. Let $(X_t, \mathcal{F}_t)_{t\geq0}$ be a right continuous super-martingale, and $\tau$ a $(\mathcal{F}_t)_{t\geq0}$-stopping time. I ...
mathematico's user avatar
1 vote
1 answer
79 views

Stability of an associated graded ring

I'm studying Eisenbud's Commutative Algebra Book, but I need help with the topic of Filtrations and the Artin- Rees Lemma. I'm stuck with the proposition 5.2, I don't even get the first '"clearly&...
Antonio's user avatar
  • 11
0 votes
2 answers
86 views

Is the filtration of double complex Hausdorff?

Let $K^{\bullet,\bullet}$ be a double complex in a general abelian category. I'm wondering if the filtration $$F_I^p(\text{Tot}^n(K^{\bullet , \bullet })) = \bigoplus_{i + j = n, i \geq p} K^{i, j}$$ ...
Z. He's user avatar
  • 502
1 vote
1 answer
33 views

On showing the equality between a filtration $(\mathcal F_t)_{t\ge 0}$ and a twice time-changed filtration $(\tilde{\mathcal F_t)}_{t\ge 0}$

Let $A$ be a strictly increasing continuous process which is adapted to a right-continuous filtration $(\mathcal F_t)_{t\ge0}$. We also suppose that $A_\infty=\infty$. For $t\ge0$, let $C_t=\inf\{s\...
maxjw91's user avatar
  • 486
3 votes
1 answer
150 views

The (AB4) condition in Weibel's chapter on spectral sequences.

I've been revisiting the chapter on spectral sequences in Weibel's An Introduction to Homological Algebra and trying to pay attention to how everything works out in arbitrary abelian categories as ...
Thorgott's user avatar
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4 votes
0 answers
75 views

Karatzas & Shreve, Problem 2.22, Positivity of Stopping Time

Problem 2.22 in Karatzas' & Shreve's Brownian Motion and Stochastic Calculus asks to Prove that if $S$ is an optional time and $T$ is a positive stopping time with $S \le T$ and $S < T$ on $\{...
jro's user avatar
  • 715
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0 answers
30 views

Graphical representation of a filtration

I am searching for a good illustration of the concept of a filtration. So far this https://books.google.de/books?id=OBOGDwAAQBAJ&pg=PA218#v=onepage&q&f=false book had a nice illustration ...
Ggjj11's user avatar
  • 101
1 vote
0 answers
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How to construct filtrations in modal logic that preserve a specific properties to prove finite model properties.

I am following a course on modal logic and I have issues with a specific area, namely filtrations that preserve frame properties. A definition for a filtration is given here: https://en.wikipedia.org/...
DueledGalois's user avatar
2 votes
1 answer
80 views

Why $\limsup_{t\rightarrow s} \frac{B_t-B_s}{f(t-s)}$ is not $\mathcal F_s$ measurable?

Let $(B_t)_{t\geq 0}$ be a Brownian motion equipped with its natural filtraton $\mathcal F_t = \sigma(B_s, 0 \leq s\leq t)$ and let $f$ be a measurable function such that $f(x) >0$ for all $x >0$...
ProbabilityLearner's user avatar
5 votes
0 answers
74 views

What is the origin of the word "filtration" in measure theory.

What is the origin of the word filtration in measure theory? What is the mental image that motivated this wording? What inspired this wording outside of the mathematical world?
user3188216's user avatar
1 vote
0 answers
25 views

Hodge polygon of tensor filtration

Let $V$ and $W$ be finite-dimensional vector spaces over a field $k$ with (exhaustive, separated, finite, descending) filtrations $F^\bullet$ and $G^\bullet$, respectively. On $V \otimes_k W$, we can ...
gimothytowers's user avatar
1 vote
1 answer
90 views

A question on the "usual" conditions of a probability space?

I am reading a paper where it states the following: "Let $(\Omega, \mathcal{F},\{{\mathcal{F}_t}\}_{t≥0}, \mathbb{P})$ be a complete probability space with a filtration $\{{\mathcal{F}_t}\}_{t≥0}$...
Leo's user avatar
  • 193
0 votes
0 answers
26 views

Filtration generated by two-dimensional processes

The following question seems intuitively true to me but I would appreciate if someone could provide a through/rigorous explanation: Let f(t) and g(t) be continuous-time processes defined on the same ...
David Chan's user avatar
0 votes
0 answers
59 views

Modification of a stochastic process and complete filtration

I consider $B_t$ $t\in[0,T]$ a (real valued) stochastic process adapted for the filtration $\mathcal{F}_t$ and $\bar{B}_t$ à modification of $B_t$. I would like to show that $\bar{B}_t$ is adapted ...
G2MWF's user avatar
  • 1,339
3 votes
0 answers
73 views

Filtration dependence of martingales

Let $Y_1,...,Y_T$ be iid. random variables with $E_P(Y_1)=0$ and $P(Y_1\neq 0)>0$. Consider the filtration generated by $Y$, i. e. $\mathcal{F}_0=\{\emptyset, \Omega\}$ and $\mathcal{F}_t=\sigma(...
Analysis's user avatar
  • 2,450
0 votes
0 answers
62 views

Short exact sequence of persistence modules

I am currently trying to work out a elementary proof of the following statement: Let $X$ be a simplicial complex with a filtration $\mathbb{X}: X=\bigcup_{n\in\mathbb{N}} X_n$, let $k\in\mathbb{N}$ ...
littleD's user avatar
0 votes
0 answers
52 views

Joint filtration vs join of filtrations for counting processes

Let $X,Y$ be counting processes without common jumps, i.e. $\Delta X \Delta Y = 0$ $P$-a.s. Denote by $\mathcal{F}_t^X$ and $\mathcal{F}^Y_t$ the filtrations generated by $X$ and $Y$, respectively. Is ...
northwiz's user avatar
  • 325
3 votes
1 answer
107 views

On the structure of a probability space for the extension of a measure via a martingale

This is a question about theorem 3.2.2 of James Norris's notes on advanced probability (http://www.statslab.cam.ac.uk/~james/Lectures/ap.pdf). Let $(\Omega,\mathcal{F}, (\mathcal{F}_n)_{n∈ \mathbb{N}},...
Huggy's user avatar
  • 61
2 votes
0 answers
150 views

Prove Bernstein’s inequality for martingales

I am new to machine learning theory and recently read Lamma A.8 in P363 of the book Prediction, Learning, and Games: Bernstein’s inequality for martingales: Let $X_1, X_2, \cdots, X_n$ be a bounded ...
Jimmy Zhao's user avatar
1 vote
0 answers
48 views

Measurability with respect to complete filtration

I don't know if this is a difficult question, but I have absolutely no intuition about it. Let us consider a probability space $(\Omega, \mathcal F, \mathbb P)$ supporting a $\mathbb R-$valued ...
GasJuRo's user avatar
  • 31
0 votes
0 answers
25 views

Coradical filtration and socle series $C_n=\mathrm{Soc}^{n+1}(C)$

I am reading the book Hopf Algebras and Their Actions on Rings by Susan Montgomery. In page 64, she said $C_n=\mathrm{Soc}^{n+1}(C)$, where $C$ is a coalgebra with coradical filtration $\{C_n \}$ and ...
Z.B. Zuo's user avatar
  • 510
0 votes
1 answer
50 views

Simple example of an adapted process that is a martingale for one filtration but not for another

I have seen this question. I haven't had Brownian motion yet, is there a simple, more basic example?
Analysis's user avatar
  • 2,450
0 votes
1 answer
139 views

$L^1$ Convergence of expectation of conditional expectation

The underlying space is a probability space. Suppose we have a filtration $\{\mathscr{F}_n\}_{n\in \mathbb N}$ and denote $\cup_{n\in \mathbb N} \mathscr{F}_n$ by $\mathscr{F}_\infty$. Suppose we have ...
Sam Wong's user avatar
  • 2,287
2 votes
1 answer
57 views

Conditional expectation of i.i.d variables at random times

Let $\mathbb{F}=(\mathcal{F_n})_{n \in \mathbb{N}}$ be a filtration on some probability space, $(X_n)$ a series of $\mathbb{F}$-adapated i.i.d RVs with mean $\mu$, and $(\tau_n)$ a series of stopping ...
Tanakak's user avatar
  • 47
1 vote
1 answer
35 views

Induced filtration on polynomial ring with coefficients in a filtered associative algebra

Let $k$ be a commutative ring with $\deg(t)=0$, and let $k[t]$ be the ungraded polynomial ring in the variable $t$, centred in degree zero. Let $A$ be an associative $k$-algebra with increasing ...
Flavius Aetius's user avatar
1 vote
0 answers
54 views

associated graded of ungraded polynomial ring is isomorphic to graded polynomial ring.

I would like to discuss a fine observation that I made which is not discussed in the literarure. Maybe because it is easy. Nevertheless, I think that it is quite important. Let $k$ be a commutative ...
Flavius Aetius's user avatar
1 vote
0 answers
53 views

Filtration degree vs Grading degree

Consider the polynomial ring $\mathbb C[t]$ with $\deg t=0$, that is, we view $\mathbb C[t]$ as ungraded ring concentrated in degree zero. On the other hand there is a decreasing filtration of $\...
Flavius Aetius's user avatar
3 votes
2 answers
195 views

How can a Filtration be interpreted as Information / Filtration for repeated Coin toss

Say we throw a coin three times. The sample space is then given by $\Omega=\{HHH, HHT, HTH, ..., THT, TTT\}$. The Filtration is given by: $\mathbb{F}_0=\{\emptyset, \Omega\}$ $\mathbb{F}_1=\mathbb{F}...
ben.kaltinger's user avatar
2 votes
1 answer
84 views

Proof check: Filtration $\mathbb F$ is not right-continuous

Question Let $\Omega=C([0,2])$ (set of continious funtions) and $X$ a stochastic process on $\mathbb R$ such that $$X_t(\omega):=\omega(t)$$ with the natural filtration $\mathbb F:=(\mathcal F_t)_{t\...
scholar's user avatar
  • 113
1 vote
1 answer
53 views

How to show the equivalence between two definitions of completion of sigma algebra

Given a Probability space $(\Omega,\mathcal{A},P)$ , I knew the definiton of the completion of a sub-sigma algebra $\mathcal{F}$ to be $\overline{\mathcal{F}}=\sigma(\mathcal{F}\cup \mathcal{N})$ ...
Dovahkiin's user avatar
  • 1,285
0 votes
0 answers
45 views

Action on associated graded algebra inducing action on filtered algebra

Suppose $Q$ is a filtered algebra, with associated graded algebra $\text{gr}(Q)$. If we have an action of a ring $R$ on $\text{gr}(Q)$ (i.e. $\text{gr}(Q)$ is an $R$-module) then it seems clear that, ...
Ted Jh's user avatar
  • 479
1 vote
1 answer
33 views

How to construct a base compatible with two filtrations

Suppose $V$ is a $m$-dimensional vector space over some field $k$. We define filtration to be a ascending chain of subspaces (for some $n$): $$Fil:Fil_0=0\subset Fil_1\subsetneq \cdots\subsetneq Fil_n=...
Richard's user avatar
  • 1,412
1 vote
0 answers
27 views

What is the filtered probability space used to study linear SDEs with constant coeficients?

Context: I am currently working with Kalman-like filters. As a result I deal with linear Stochastic Differential Equations (SDEs) with constant coefficients such as: $$ dx(t) = Ax(t)dt + Bdw(t) $$ ...
FeedbackLooper's user avatar

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