Questions tagged [filtrations]
The filtrations tag has no usage guidance.
1
vote
0answers
34 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 ...
2
votes
0answers
53 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
266 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:
$$
\mathfrak{G}...
3
votes
0answers
84 views
In which sense does the quadratic variation depend on the considered filtration?
Let
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(\mathcal F_t)_{t\ge 0}$ be a complete right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$
$X$ be an almost surely ...
2
votes
0answers
34 views
Is there a well-developed theory on filtering on sub $\sigma$-fields?
Suppose we are given an observation process
$$
dY_t = \mu(Y_t)dt + \Sigma(Y_t)dW_t
$$
with valued in $\mathbb{R}^d$ (or potentially in an arbitrary separable Hilbert space). The classical filtering ...
2
votes
0answers
142 views
Stopping time sigma-fields
Let $(F_n)$ be a discrete Filtration and $S_n,S$ (not necessarily finite) stopping times with $S_n\uparrow S$ (increasing convergence).
Is it true that the associated sigma-fields satisfy $F_{S_n}\...
2
votes
1answer
79 views
Is the class of acyclic complexes deconstructible?
Let $\mathcal{C}$ be a category, then a class $\mathcal{A}\subseteq \mathcal{C}$ is deconstructible if there is a set $\mathcal{S}\subseteq\mathcal{C}$ such that $\mathcal{A}$ consists of $\mathcal{S}$...
2
votes
1answer
878 views
When is the hitting time of an open set a stopping time?
Let $(\Omega, \mathcal{F},P)$ be a probability space and $(\mathcal{F}_t)_{t \in [0,T]}$ a filtration. Consider an adapted, right-continuous process $X$ taking values in $\mathcal{X}$ and let $B$ be ...
0
votes
0answers
94 views
What is the sigma field of the derivative of a process?
When $t\to X_t$ is an absolutely continuous process ($X_t= X_0+ \int_0^t Y_s dt$ for some measurable process $Y_t$) we have for all $t$ $$\sigma(Y_t) \subset \cap_{\epsilon >0}\sigma(X_{s}, s\in [t,...
1
vote
0answers
84 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
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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
186 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
509 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
757 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
318 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
625 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
545 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+...
