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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What is the correct filtration?

Let $B\overset{\circ}{=}\left(B_{t}\right)_{t\geq0}$ denote a Brownian motion in a filtration $\mathcal{F}$. Are $X_{t}=\frac{1}{\sqrt{a}}B_{at}$ ($a>0$ constant) and/or $Y_{t}=tB_{\frac{1}{t}}$ ...
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3 votes
2 answers
93 views

Meaning of measurableness [duplicate]

I've done courses on measure theory, advanced stochastic processes, etc.--years and years of using the notion of 'measurableness'. I've now come to the conclusion that, in fact, I do not understand ...
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2 votes
2 answers
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Problem with intuition regarding sigma algebras and information

After thinking about it a few years, I still don't quite understand the precise link between sigma algebras and information. (Yes--I know that there's already dozens of questions on MSE by other ...
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1 vote
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For every $n \geq 0$ and $A_{n} \in \Omega_{n}$, $\mathbb{E} \left [X_{n+1} \mathit{1}_{A_{n}} \right ]=0$. Explain.

I have the following lemma in one of my probabilistic courses. For $(X_{i})_{i}$ be a sequence of i.i.d random variables such that $\mathbb{P}(X=1)=\mathbb{P}(X=1)=\frac{1}{2}$ Let $\Omega_{n}=\sigma(...
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1 vote
1 answer
59 views

Is the difference of random walk a martingale

Suppose $X_i \sim i.i.d. N(0,1), i=1,...$, $S_n=X_1+...+X_n$ let's $ Y_{i}^{\left( v \right)} := X_i \textbf{1}_{ \left\{ \left| X_i \right| \leq v \right\} }, X_{i}^{\left( v \right)} := \left( Y_{i}...
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5 votes
1 answer
36 views

Filtrations of stopping times.

My textbook introduces this deffinition. Let $(\Omega, \mathcal{F} , \mathcal{F}_t \mathbb{P}) $ be a filtered probability space. Let $\tau$ be a stopping time, Then define $\mathcal{F}_{\tau} : = \{A ...
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2 votes
1 answer
91 views

Does $\sigma(\bigcap_{n=1}^\infty \mathcal{F}_n)=\bigcap_{n=1}^\infty \sigma(\mathcal{F}_n)$ hold?

Let $(\Omega,\mathcal{F})$ be a measurable space and $\mathcal{F}_1 \supseteq \mathcal{F}_2 \supseteq...$ be a sequence of decreasing subsets of $\mathcal{F}$. For $\mathcal{A} \subseteq \mathcal{F}$ ...
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  • 417
-1 votes
1 answer
28 views

Show that $f$ is a linear function if $(f(Xn))$ is martingale with respect to the filtration $(Fn)$. [closed]

$f$ is a real continuous function, $(f(X_{n}))$ $n=0,1,2,..$ is martingale with respect to the filtration $(F_{n})$, where $F_{n}=\sigma(X_{0},..,X_{n})$, and $(X_{n})$ is martingale with respect to ...
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0 votes
1 answer
29 views

Composition factors of intersection of modules

For simplicity we assume that $R$ is an Artinian ring, such that any finitely generated $R$-modul has a composition series. To such an $R$-modul $M$ we associate the set of composition factors $C(M)$. ...
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With optional time $T$ on $\mathscr{F}_t$, and $T_n = \frac{k}{2^n}$ on $\{ \frac{k-1}{2^n} \le T < \frac{k}{2^n} \}$, show $T_n$ is a stopping time

Let $T$ be an optional time for the filtration $\{ \mathscr{F}_t \}_{t \ge 0}$, and define the sequence of random times $T_n$ by: \begin{align*} T_n = T = \infty \; \text{on} \; \{T = \infty\}, \; \;...
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1 vote
1 answer
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Given sequence of optional times $T_n$ and optional time $T = \inf_n T_n$, show $\mathscr{F}_{T+} = \cap_n \mathscr{F}_{T_n+}$

Show that if $\{ T_n \}_n$ is a sequence of optional times and $T = \inf_n T_n$, then $T$ is also an optional time and $\mathscr{F}_{T+} = \cap_n \mathscr{F}_{T_n+}$. The first part is easy. We can ...
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  • 2,334
1 vote
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Why is the natural filtration of Brownian motion not $\mathcal{B}(\mathbb{R})$ for all $t$?

I'm trying to get a sense of what the $\sigma$-algebras in a filtration actually contain. More specifically, suppose $W_t$ is standard $\mathbb{R}$-Brownian motion defined on a filtered probability ...
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1 vote
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Showing natural filtration of counting process is the same as filtration at $n-1$'st jump $T_{n-1}$ given the event $\{T_{n-1}\le t < T_n\}$

Consider a simple counting process $N(t)$ and $(\mathcal{F}_t)_{t\ge 0}$ the natural filtration and let $T_1,T_2,...,$ be the associated jumps, which are all $\mathcal{F}_t$-stopping times. I want to ...
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Karatzas + Shreve 2.22: For optional time $S$, stopping time $T$, prove that $\mathscr{F}_{S+} \subset \mathscr{F}_T$

Prove that if $S$ is an optional time and $T$ is a positive stopping time such that with $S \le T$ and $S < T$ on $\{S < \infty\}$, then $\mathscr{F}_{S+} \subset \mathscr{F}_T$. The following ...
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2 votes
0 answers
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Karatza + Shreve 2.19: Show that $Y_t = \int_0^t f(s, X_s) \, ds; t \ge 0$ is progressively measurable

Let $X = \{X_t, \mathscr{F}_t; 0 \le t < \infty \}$ be a progressively measurable process, and let $T$ be a stopping time of $\mathscr{F}_t$. With $f(t,x): [0, \infty) \times \mathbb{R}^d \to \...
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  • 2,334
0 votes
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With stopping time $T$ and positive constant $t$, show that $T \land t$ is $\mathscr{F}_t$ measurable.

Since $T$ is a stopping time, by definition, for any $t \ge 0$, $\{T \le t\} \in \mathscr{F}_t$. $T \land t = \min(T, t)$ We have to show that $T \land t$ is $\mathscr{F}_t$ measurable, which means ...
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  • 2,334
3 votes
1 answer
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Karatzas & Shreve 2.6: Show that the hitting time $H_{\Gamma}$ is an optional time.

Consider a stochastic process $X$ with right-continuous paths, which is adapted to a filtration $\{ \mathscr{F}_t \}$. Consider subset $\Gamma \in \mathscr{B}(\mathbb{R}^d)$ of the state space of the ...
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3 votes
1 answer
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Karatzas and Shreve Exercise 1.8: If paths of process $X$ are RCLL almost surely, show that $A$ can fail to be in $\mathscr{F}_{t_0}^X$

Karatzas+Shreve Textbook Exercise 1.8: Let $X$ be a [stochastic] process whose sample paths are RCLL almost surely, and let $A$ be the event that $X$ is continuous on $[0,t_0)$. Show that $A$ can ...
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1 vote
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What is joint filtration?

Consider a probability space $(\Omega,\mathcal{F},\mathbb{P})$ with two filtrations $(\mathcal{F}_t)_{t\geq0}$ and $(\mathcal{G}_t)_{t\geq0}$. The book Credit Risk: Modeling, Valuation, and Hedging by ...
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1 vote
0 answers
61 views

Characterizing strict morphisms in the category of bifiltered vector spaces

Let $k$ be a field, and let $C$ be the category whose objects are finite dimensional $k$-vector spaces endowed with two finite filtrations $W$ and $F$, the former being ascending and the latter ...
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1 vote
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26 views

$\sigma$-algebra generated by stochastic process [closed]

Let $X=(X_t)_{t\ge0}$ be a stochastic process on a probability space $(\Omega,\mathcal{F},P)$ with $X_0=0$ a.s., and let $(\mathcal{F}_t)_{t\ge0}$ denote the filtration generated by $X$, i.e., $\...
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2 votes
0 answers
51 views

How to show $\mathbb{E}(Y_{n+1}^{2} | \mathcal{F}_{n}) = \mu^{2} Y_{n}^{2} + \sigma^{2} Y_{n}$? (Unanswered repost)

I'm basically asking an extension of the first part from this post a while back. To parse, it asked: Given a branching process $Y = \{Y_{n} : n \geq 0\}$ and the corresponding family of random ...
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2 votes
1 answer
45 views

Solution of stochastic differnetial equation adapted?

Consider the following stochastic differential equation: $$dY_t = Z_t dW_t$$ and terminal condition $Y_T = b,$ for which holds: $E[|b|^2] < \infty$ Furthermore b is adapted to the filtration ...
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4 votes
0 answers
67 views

$A(t)=\int_{0}^{t} I_{(X\geq u)} \,dΛ(u)$ is a predictable process

I am currently reading the book "Counting Processes and Survival Analysis" by Fleming and Harrington and I am stuck in the proof of Proposition 1.4.2 p.34: Proposition 1.4.2: Let $(T,U)$ be ...
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4 votes
0 answers
362 views

How do we naturally extend the average from the Hausdorff Measure for functions with a domain without a gauge function?

Motivation: According to this question Some sets have a Hausdorff Dimension $\alpha$ but have a zero-dimensional Hausdorff Measure. These sets may have another dimension function, i.e. a function $h:[...
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0 votes
1 answer
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If $0 < S < T$ are stopping times, and $A \in \mathcal{F}_S$ show that $T = S1_A + T1_{A^c}$ is a stopping time.

Let $0 < S < T$ two stopping times with respect to the discrete-time filtration $(\mathcal{F}_n)_{n \ge 0}$. Let $A \in \mathcal{F}_S$. Show that the random time $T'$ defined by: \begin{align*} ...
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4 votes
0 answers
47 views

What is the spectral sequence associated to this filtration on the de Rham complex?

I am trying to calculate some relative de Rham cohomology, but I am not too skilled with hypercohomology or spectral sequences, and the situation becomes more complicated because (1) the base is not ...
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2 votes
0 answers
46 views

When $\mathbb{P}(C\cup D) = 0$? (Martingale)

Theorem: Let $(X_n,\mathcal{F}_n)_{n\geq 0}$ be a martingale with $|X_n - X_{n-1}| \leq M<\infty$. Let $C := \{{\lim X_n \ \mbox{exists and} < \infty}\} $ and $D:=\{\limsup X_n = +\infty \mbox{...
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1 vote
1 answer
40 views

Does $Y$ $\mathcal{F_t}$-adapted and $X$ adapted to $Y$ imply that $X$ is $\mathcal{F_t}$-adapted?

Suppose $(Y_t)_{t \geq0}$ is stochastic process which is adapted to a filtration $(\mathcal{F_t})_{t\geq0}$ (meaning $Y_t$ is $\mathcal{F_t}$-measurable for every $t\geq0$). Suppose another stochastic ...
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Existence of a filtration with respect to which a sequence becomes a martingale difference.

Suppose $(X_t,\mathcal F_t)_{t \in \mathbb Z}$ is such that $\mathbb E(X_t\mid \mathcal F_{t-1}) = 0$ for every $t$, where $(\mathcal F_t)$ is a filtration on a probability space $(\Omega,\mathcal F,\...
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  • 449
5 votes
1 answer
84 views

Example/meaning of filtration on a group $(\mathbb{R},+)$

Serre, in his Lie algebras and Lie groups, gives the definition of a filtration $\omega$ on a group $G$ as a map $\omega:G\to R\cup\{+\infty\}$ such that $\omega(e)=+\infty$ where $e$ is the identity ...
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1 vote
0 answers
37 views

Is possible to construct a "relevant information theory" from the notion of filtration?

Information theory and Shannon's formula provide a measure of uncertainty in specific situations where the outcomes and their probabilities are known. But in many situations, this information theory ...
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30 views

Martingale with respect to different filtration

I consider two stochastic processes $X_t = B_t$ and $Y_t = B_{t-1}$ with their natural filtrations $\mathcal{F}^X$ and $\mathcal{F}^Y$, where $B_t$ is a Brownian motion. It is well known that $X_t$ is ...
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  • 611
0 votes
0 answers
24 views

Natural filtration for a transformation of a Brownian motion

Let $W$ be a standard Brownian motion on the filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in T},\mathbb{P})$, where $(\mathcal{F}_t)_{t\in T}$ is the natural filtration of $W$, i....
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0 votes
0 answers
17 views

Relation between filtrations of stochastic processes

We consider a stochastic process $X_t$ that takes values in $\mathbb{R}$. Now we consider the following processes: $U_t = X_t + t$, $W_t = X_t^2$, $Y_t = \sin(X_t^2)$, $Z_t = e^{X_t}$. Their natural ...
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  • 611
2 votes
1 answer
74 views

Filtration with respect to which reverse brownian motion is martingale

I was wondering with respect to what filtration the process $X_t=\begin{cases}\frac{1}{t}B_{1/t}\text{ if }t>0 \\ 0\text{ if }t=0 \end{cases}$ $\;\;\;$($B_t$ is a Brownian motion) is a martingale. ...
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2 votes
1 answer
55 views

Prove that sigma algebra of random variables ordered negatively is a filtration

I have a problem with a tricky exercise in stochastic process that take negativ integers for the time. Let be $(U_{t})_{t \in T}$ sequence of iid random variables in $L^2(\Omega,\mathcal{A},p)$ with $...
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0 votes
1 answer
33 views

"unnatural" filtration?

What would be an example of a simple/useful stochastic process $(X_t)_{t\in T}$ for $T=\mathbb{N}$ or $T=\mathbb{R}$ where it is useful to consider a filtration different from the natural filtration $\...
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0 votes
1 answer
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Cyclic elements in filtrations of representations

Let $G$ be a finite group and $V$ a finite-dimensional representation of $G$ (or $G$-module) over $\mathbb{C}$. Furthermore, suppose we have a filtration of $G$-modules $$0=V_0\subset V_1\subset\cdots ...
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  • 400
1 vote
1 answer
61 views

filtration generated by function of brownian motion

Consider a filtration $F^{B_1,B_2}$ generated by two independent Brownian motions $B_1,B_2$. Now I define $W:= a B_1 + \sqrt{1-a^2} B_2$ and consider the filtration $F^W$ generated by W. Are $F^{B_1,...
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  • 451
1 vote
1 answer
77 views

martingale representation of two independent Brownian motion

Let $W^1_t$ and $W^2_t$ be two independent standard Brownian motions. Then $W^2_t$ is a martingale with respect to the its own filtration but not adapted to the filtration generated by $W^1_t$. I want ...
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  • 451
0 votes
1 answer
52 views

Can an ascending filtration be used to define completion of a vector space/ring?

Let $V: = V_0 \supseteq V_1 \supseteq V_2 \supseteq V_3 \supseteq \ldots$ be a descending filtration of a vector space $V$. Then one can define as basis of open sets given by $\{ v + V_k, v \in V, k \...
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  • 536
2 votes
1 answer
87 views

Short exact sequences of filtered vector spaces are split

Let $k$ be a field. By a filtered vector space over $k$, I mean a pair $(V,F)$ where $V$ is a finite dimensional $k$-vector space and $F=(F^pV)_{p\in \mathbb{Z}}$ is an increasing filtration of $V$ by ...
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  • 1,350
0 votes
0 answers
28 views

Filtration of Leray spectral sequence for open embedding

Let $X$ be a smooth complex projective variety/compact Kähler manifold and $D$ a normal crossing divisor. Let $j: U:=X-Z\hookrightarrow X$ be the open embedding. We have two spectral sequences for ...
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2 votes
0 answers
47 views

Let $\sigma,\tau$ be stopping times. $\{\sigma<\tau\}\in\mathcal{F}_{\sigma}$?

Consider a filtered prob space $(\Omega,\mathcal{F},(\mathcal{F}_s)_{s\in [0,T]},\mathbb{P})$ satisfying the usual conditions. Let $\sigma,\tau$ be stopping times. Is it possible to show that $$\{\...
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  • 141
1 vote
0 answers
58 views

Constructing an adapted process for a given stopping time

Given a filtration $(\mathcal{F}_n)_{n\geq 0}$ and a stopping time $T$, show that there exists an adapted process $(X_n)_{n\geq 0}$ such that $$T=\inf \{n \geq 0 : X_n >0\}$$ I'm really lost where ...
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0 votes
0 answers
17 views

Is there a filtration in discrete time that is not right-continuous?

In discrete time, I've read that a convention for any filtration is to assume that $\mathcal{F}_{t}=\mathcal{F}_{s} \ \forall s \in [t,t+1[$. Wouldn't this make $\mathcal{F}_{t+}=\cap_{s>t} \...
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2 votes
0 answers
43 views

Prove that if $\mathrm{gr}(A)$ is Noetherian without zero-divisors, then so is $A$.

Let $A$ be a filtered commutative algebra and $\mathrm{gr}(A)$ the associated graded algebra. Prove that if $\mathrm{gr}(A)$ is Noetherian without zero-divisors, then so is $A$. Associated graded ...
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1 vote
1 answer
78 views

Equivalent definitions of a Hodge structure

Generally, a Hodge structure of weight $k$ on a finitely generated abelian group $H$ is defined as a decomposition of the complexification: $$ H\otimes \mathbb C = \bigoplus_{p+q=k} H^{p,q}, $$ where ...
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  • 1,602
0 votes
1 answer
24 views

Prove equality of stopped sigma algebras under right continuity

Given the definitions: $\mathcal{F}_\tau := \{A ∈ \mathcal{F} : A ∩ \{τ ≤ t\} ∈ \mathcal{F}_t \quad ∀t ≥ 0 \}$ $\mathcal{F}_{\tau+} := \{A ∈ \mathcal{F} : A ∩ \{τ < t\} ∈ \mathcal{F}_t\}$ How do we ...
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