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2
votes
0answers
48 views

Condition for a map to carry over to Leray spectral sequences

I am trying to understand the conditions for two Leray spectral sequences to be related by a map. Let $f_1 : X_1 \to Y_1$ and $f_2 : X_2 \to Y_2$ be two continuous maps of topological spaces (with ...
11
votes
0answers
264 views

Mysterious “raison d'être” of filtrations of congruence subgroups

I wonder for long why congruence subgroups seem to arise so naturally in certain filtrations. Everything below is on a local field $F_p$. Filtration for $GL_n$. Casselman and later Jacquet, Piatetski-...
1
vote
1answer
68 views

Whether the first moment two stochastic processes differ can be formulated in terms of filtrations?

I am, struggling to see whether the first moment when two processes are different (in terms of their finite dimensional distributions) can be defined in terms of their filtrations and would ...
1
vote
1answer
48 views

Writing $\sigma$-algebra in terms of predictble processes?

If $X_t$ is a semi-martingale, $\mathfrak{F}_t$ is the $\sigma$-field generated by $X_t$ and $L^2(Pred)$ is the set of all $\mathfrak{F}_t$-predictible processes. Then is it true that: $$ \mathfrak{G}...
3
votes
0answers
83 views

In which sense does the quadratic variation depend on the considered filtration?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a complete right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$ $X$ be an almost surely ...
2
votes
0answers
34 views

Is there a well-developed theory on filtering on sub $\sigma$-fields?

Suppose we are given an observation process $$ dY_t = \mu(Y_t)dt + \Sigma(Y_t)dW_t $$ with valued in $\mathbb{R}^d$ (or potentially in an arbitrary separable Hilbert space). The classical filtering ...
2
votes
0answers
142 views

Stopping time sigma-fields

Let $(F_n)$ be a discrete Filtration and $S_n,S$ (not necessarily finite) stopping times with $S_n\uparrow S$ (increasing convergence). Is it true that the associated sigma-fields satisfy $F_{S_n}\...
2
votes
1answer
78 views

Is the class of acyclic complexes deconstructible?

Let $\mathcal{C}$ be a category, then a class $\mathcal{A}\subseteq \mathcal{C}$ is deconstructible if there is a set $\mathcal{S}\subseteq\mathcal{C}$ such that $\mathcal{A}$ consists of $\mathcal{S}$...
2
votes
1answer
841 views

When is the hitting time of an open set a stopping time?

Let $(\Omega, \mathcal{F},P)$ be a probability space and $(\mathcal{F}_t)_{t \in [0,T]}$ a filtration. Consider an adapted, right-continuous process $X$ taking values in $\mathcal{X}$ and let $B$ be ...
0
votes
0answers
94 views

What is the sigma field of the derivative of a process?

When $t\to X_t$ is an absolutely continuous process ($X_t= X_0+ \int_0^t Y_s dt$ for some measurable process $Y_t$) we have for all $t$ $$\sigma(Y_t) \subset \cap_{\epsilon >0}\sigma(X_{s}, s\in [t,...
1
vote
0answers
82 views

When the completed filtration of a process increases slowly

If $\mathcal{F}_t$ is the filtration of the evaluation process on $C_T$ (continuous function on $[0,T]$). Can we find some law of continuous process $\mathbb{P}$ so that for $t\leq T$ $$\mathcal{F}^{\...
1
vote
0answers
119 views

Is semistability of smooth Weil sheaf preserved under tensor product?

Let $X_0$ be a smooth, geometrically connected scheme over $\mathbb{F}_q$. As usual, let $\tau : \bar{\mathbb{Q}}_{\ell} \simeq \mathbb{C}$ be a fixed isomorphism. Let $\mathcal{C}$ be the category of ...
3
votes
1answer
184 views

Does martingale convergence hold for arbitrary time?

Let $\{\mathcal B_i:i\in I\}$ be a family of $\sigma$-algebras (over the same set $\Omega$) which are totally ordered by inclusion, in the sense that for any $i,j\in I$ either $\mathcal B_i\subset\...
11
votes
5answers
2k views

Properties preserved under passage to augmented filtration

Dear all, generally speaking, my question is about which properties of a stochastic process are preserved when I skip from the original to the augmented filtration. Recall that if $(\mathcal{F}_t)_{...
4
votes
1answer
3k views

Right-continuity of natural filtrations

My question: Is the natural filtration of a right-continuous process also right-continuous? I would say yes, but don't know where to start proving it. Thanks for your help/ideas!
6
votes
2answers
497 views

Does there exist a functorial splitting for the weight filtration (of singular cohomology)?

There are plenty of examples of varieties whose singular cohomology with rational coefficients considered as a mixed Hodge structure does not decompose as the direct sum of its pure (weight) factors. ...
4
votes
1answer
755 views

Do Deligne's decalage and two filtrations for mixed Hodge complexes have a conceptual explanation?

As far as I understood, there are two way to assign weights to a mixed Hodge complex (a way that does not depend on shifts, and another one that does). There is a clever way to relate these to ways ...
3
votes
2answers
316 views

Is the first filtration Hausdorff?

Maybe this is too technical and elementary, but I cannot make up my mind, nor find a reference. The situation is the following: let $X$ be a double cochain (right half-plane) complex of abelian ...
15
votes
1answer
623 views

Semistable filtered vector spaces, a Tannakian category.

Let $k$ be a field (char = 0, perhaps). Let $(V,F)$ be a pair, where $V$ is a finite-dimensional $k$-vector space, and $F$ is a filtration of $V$, indexed by rational numbers, satisfying: $F^i V \...
6
votes
2answers
536 views

If associated-graded of a filtered bialgebra is Hopf, does it follow that the original bialgebra was Hopf?

Warning: older texts use the word "Hopf algebra" for what's now commonly called "bialgebra", whereas now "Hopf" is an extra condition. So as to avoid any confusion, I'll give my definitions before ...
25
votes
4answers
2k views

What is the universal property of associated graded?

Given a filtered vector space (or module over a ring) $0=V_{0}\subseteq V_{1}\subseteq\cdots\subseteq V$, you can construct the associated graded vector space $\mathrm{gr}\left(V\right)=\oplus_{i}V_{i+...