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Characterizing filtrations generated by a stopping time

Setup Let $\Omega$ be the set of càdlàg functions $f : [0,\infty) \to \mathbb R^d$ equipped with the Skorokhod topology for any $d \geq 1$, and let $X_t(\omega) = \omega(t)$ for any $\omega \in \Omega,...
Sarvesh Ravichandran Iyer's user avatar
2 votes
1 answer
101 views

Normal bundle of veronese as iteration extension of symmetric powers

In this post, an answer claims that the normal bundle of the Veronese decomposes into a filtration, such that the associated graded is $$\bigoplus_{i=2}^d S^i T,$$ where $T$ is the tangent bundle. But ...
maxo's user avatar
  • 33
1 vote
0 answers
69 views

Does weak convergence of filtrations preserve progressive measurability?

Suppose on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ I have a sequence of filtrations $\mathbb{F}^n =(\mathcal{F}^n_t)_{t \geq 0}$ generated by Brownian motions $W^n$ for each $n$, ...
PeterGoGo's user avatar
2 votes
1 answer
276 views

Maps from $\mathbb A^1/ \mathbb G_m$ to Coherent sheaves

I am reading this paper https://arxiv.org/abs/1608.04797 Let $\Theta$ be the stack $\mathbb A^1/{\mathbb G_m}$. Let $X$ be a smooth projective curve of genus $g$ over a field $k$. Let $Coh_P$ be the ...
angry_math_person's user avatar
3 votes
1 answer
131 views

How to compute the integer corresponding to a class in $G_0(B_{\mathrm{red}})$ for a commutative noetherian ring $B$?

$\newcommand{\red}{\mathrm{red}}$Let $k$ be an algebraically closed field of characteristic zero and $m$ be a positive integer. Let $R$ be the subring $k[x,xy,xy^2,…,xy^m]$ of the polynomial ring $k[x,...
Boris's user avatar
  • 549
2 votes
1 answer
215 views

Is every filtration on an abelian category strict?

It's well-known that the category of filtered objects in an abelian category is generally not an abelian category. One must generally add extra structure to ensure that all morphisms are strict for ...
David Corwin's user avatar
  • 15.1k
7 votes
0 answers
171 views

Has anyone written about filtered Tannakian categories?

tl;dr Is there any source that discusses the concept of a filtered Tannakian category? I'm writing a paper with this notion and want to know if it's ever been discussed. The original book by Saavedra-...
David Corwin's user avatar
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2 votes
0 answers
72 views

Filtration of norm-one elements of quaternion algebra over local field with respect to an involution

Let $K$ be a local non-archimedean field, with ring of integers $\mathcal{O}_K$, uniformizing element $\varpi_K$, and residue field $\mathcal{O}_K/\varpi_K\mathcal{O}_K \cong \mathbb{F}_q$. For ...
pbarron's user avatar
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1 vote
0 answers
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When enlarging a filtration makes a stochastic processes into a solution to an SDE

Let $n$ be a positive integer and let $(Y_t)_{t\in [0,1]}$ on $\mathbb{R}^n$ be a stochastic process defined on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in [0,1]},\mathbb{P}...
ABIM's user avatar
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2 votes
2 answers
366 views

Dimension of the associated graded module at an ideal

Let $I$ be an ideal of a Noetherian local ring $(R, \mathfrak m)$. Define the associated graded ring $G_I(R):=\bigoplus_{n=0}^\infty I^n/I^{n+1}$. Then $G_I(R)$ is a Noetherian ring of the same ...
Louis 's user avatar
  • 279
1 vote
1 answer
105 views

Properties of filtrations preserved by a DG-algebra homomorphism

Suppose we have a homomorphism $f : A^{\bullet} \longrightarrow B^{\bullet}$ of differential graded algebras over a field $k$, and consider the filtration \begin{align*} A^{\bullet} \supseteq F^0A^{\...
michiganbiker898's user avatar
4 votes
0 answers
461 views

Filtration over tensor product

Let $$ M \supset M_1 \supset \ldots \supset M_n \supset \ldots \text{ and } N \supset N_1 \supset \ldots \supset N_n \supset \ldots$$ be exhaustive decreasing filtrations of modules over a commutative ...
Pierre21's user avatar
  • 385
1 vote
1 answer
221 views

can we take skeletons of covering maps to give new covering maps?

Let $X$ be an $n$-dimensional cell complex. We attach an $(n+1)$-cell $e^{n+1}$ to $X$ and obtain a new cell complex $X'$. Take the universal cover (or a general covering space) $\tilde X'$ of $X'$. ...
Shiquan Ren's user avatar
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0 answers
81 views

If $W$ is a Markov chain and $N$ is a Poisson process, then $\left(W_{N_t}\right)_{t\ge0}$ is Markov

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(E,\mathcal E)$ be a measurable space, $(W_n)_{n\in\mathbb N_0}$ be a time-homogeneosu Markov chain on $(\Omega,\mathcal A,\...
0xbadf00d's user avatar
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1 vote
1 answer
122 views

References about transfinite socle series

I would like to know if there is any literature that discusses "transfinite socle series" of a ring module. Below is my attempt at defining the series. Let $R$ be an associative unital ring and $...
Batominovski's user avatar
2 votes
1 answer
236 views

Poset filtrations

Consider a chain complex $C$ and a poset $P$ so that there is a filtration by subcomplexes $C^p$ of $C$ where $p\in P$ in such a way that $p<q$ implies $C^p \leqslant C^q$. As a second option, ...
Pedro's user avatar
  • 1,554
1 vote
0 answers
53 views

Ordered sequence of elements of poset relevant to some filtration -- highest weight category

Let $\mathcal{C}$ be a highest-weight category with $\Lambda$ as a interval-finite poset -- I'm using a definition of the highest-weight category given by Cline, Parshall and Scott and it is presented ...
jpatrick's user avatar
  • 129
8 votes
0 answers
188 views

Standard reference/name for "initial ideals $\Leftrightarrow$ associated graded rings"

Let $R$ be commutative ring with a $\mathbb Z$-grading $\deg$ and let $I\subset R$ be an ideal. On one hand, we may consider the initial ideal $\mathrm{in}_{\deg}(I)$. That is the space spanned by ...
Igor Makhlin's user avatar
  • 3,493
0 votes
0 answers
84 views

If $(Y_n)_{n\in\mathbb N_0}$ and $(N_t)_{t\ge0}$ are stochastic processes, what is the filtration generated by $\left(Y_{N_t}\right)_{t\ge0}$?

Let $(\Omega,\mathcal A)$ and $(E,\mathcal E)$ be measurable spaces $(Y_n)_{n\in\mathbb N_0}$ be a $(E,\mathcal E)$-valued stochastic process on $(\Omega,\mathcal A)$ $(N_t)_{t\ge0}$ be a $\mathbb ...
0xbadf00d's user avatar
  • 181
2 votes
0 answers
69 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 ...
BrianT's user avatar
  • 1,217
11 votes
0 answers
314 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-...
Desiderius Severus's user avatar
1 vote
1 answer
85 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 ...
pzryumov's user avatar
1 vote
1 answer
69 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}...
ABIM's user avatar
  • 4,775
3 votes
0 answers
212 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 ...
0xbadf00d's user avatar
  • 181
2 votes
0 answers
219 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}\...
nullptr's user avatar
  • 93
2 votes
1 answer
103 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 $\mathcal{S}$...
luvemil's user avatar
  • 31
3 votes
1 answer
2k 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 ...
Elias Strehle's user avatar
0 votes
0 answers
132 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,...
Filtrask's user avatar
1 vote
0 answers
118 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}^{\...
Filtrask's user avatar
1 vote
0 answers
141 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 ...
js21's user avatar
  • 7,199
3 votes
1 answer
214 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\...
Joel Moreira's user avatar
  • 1,701
12 votes
5 answers
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)_{...
lpdbw's user avatar
  • 121
9 votes
1 answer
5k 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!
Johannes's user avatar
  • 305
7 votes
2 answers
754 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. ...
Mikhail Bondarko's user avatar
4 votes
1 answer
1k 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 ...
Mikhail Bondarko's user avatar
3 votes
2 answers
478 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 ...
Agustí Roig's user avatar
  • 1,945
15 votes
1 answer
708 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 \...
Marty's user avatar
  • 13.1k
9 votes
2 answers
913 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 ...
Theo Johnson-Freyd's user avatar
33 votes
4 answers
4k 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+...
Anton Geraschenko's user avatar