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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19 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 ...
1 vote
0 answers
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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 ...
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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 $\...
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0 answers
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Conditional expectation w.r.t. two filtrations [closed]

How can I evaluate this conditional expecation $\mathbb{E}[X|\mathcal{F}_1 \vee \mathcal{F}_2]$? I tried to look for some literature to solve this but I could not find anything relevant. Thanks
3 votes
2 answers
61 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}...
2 votes
1 answer
45 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\...
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1 vote
1 answer
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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 $\overline{\mathcal{F}}=\sigma(\mathcal{F}\cup \mathcal{N})$ ...
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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, ...
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1 vote
1 answer
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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=...
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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: $$ dx(t) = Ax(t)dt + Bdw(t) $$ ...
1 vote
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35 views

Does Empirical Distribution Function Independent of Filtration?

I encountered a problem when reading a paper: We have i.i.d double exponential random variables $X_1,\dots,X_n$. Then consider the empirical distribution function $\hat{F_n}(x) = \frac{1}{n} \sum\...
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25 views

Problem 2.7.1 in Karatzas and Shreve

Let's consider this problem in the book by Karatzas and Shreve. Problem. Let $\{X_t\}$ be a stochastic process and $\{\mathcal{F}^X_t\}$ its natural filtration. Show that the filtration $\mathcal{F}^...
2 votes
2 answers
101 views

Natural filtration of the Brownian motion is not right-continuous

Let $B$ be a Brownian motion on a filtered probability space. Let $\mathcal{F}_t^B$ be the natural filtration associated to $B$. The fact that the natural filtration of the Brownian motion is not ...
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30 views

Isomorphic submodules of a filtration

Is it true that $t^{-n}\mathbb C[[t]]$, $n>0$, is isomorphic to $\mathbb C[[t]]$ as an $\mathbb C[[t]]$-module? I think yes, since $t^{-n}\mathbb C[[t]]$ is a free $\mathbb C[[t]]$-module of rank $...
1 vote
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30 views

Filtration to define a martingale for the difference of two Brownian motions

Suppose we want to find a filtration, say $(\mathcal{F}_t)$, to ensure that $Z_t := B_{8t}-6C_t$ is a martingale, where $B$ and $C$ are two independent Brownian motions. For this, we want to ensure ...
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1 vote
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38 views

Showing that the stopped filtration $\mathcal{F}_{s \vee t} = \mathcal{F}_s \cup \mathcal{F}_t$

Show that $\mathcal{F}_{s \vee t} = \mathcal{F}_s \cup \mathcal{F}_t$ where $\mathcal{F}_{s}, \mathcal{F}_{t}$ and $\mathcal{F}_{s \vee t}$ are stopping time sigma fields. Let's suppose that S > T ...
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-1 votes
2 answers
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$A\in F_\tau\land\tau(\omega)<\infty\Rightarrow A\in F_{\tau(\omega)}$

Let $(\Omega,\Sigma)$ be a measurable space and $(F_t)_{t\in I}$ a filtration. We postulate that $F_t$ is the information available at time $t$. Furthermore, if $\tau$ is a stopping time, then $F_\tau$...
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1 answer
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Showing that the stopped filtration $\mathcal{F}_{s \wedge t} = \mathcal{F}_s \cap \mathcal{F}_t$

Show that $\mathcal{F}_{s \wedge t} = \mathcal{F}_s \cap \mathcal{F}_t$ where $\mathcal{F}_{s}, \mathcal{F}_{t}$ and $\mathcal{F}_{s \wedge t}$ are stopping time sigma fields. Let's suppose that S <...
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1 vote
2 answers
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$\sigma$-algebra of $\tau$-past: Definition vs interpretation

Let $(\Omega,\Sigma)$ be a measurable space and $(F_t)_{t\in I}$ a filtration. We postulate that $F_t$ is the information available at time $t$. Lastly, let $\tau$ be a stopping time. According to ...
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1 vote
0 answers
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Harder-Narasimhan filtration of tensor product of coherent sheaves

Let $\mathcal{F}_1,\mathcal{F}_2$ torsion-free coherent sheaves on a smooth, projective, irreducible, reduced scheme $X$ (id est, $X$ is a smooth projective variety) of dimension $n\geq1$ over an ...
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17 views

Attaching a random sequence to another

In probability it is often necessary, given two sequences of random variables $\mathbf{X}$ and $\mathbf{Y}$, to attach one to another in the following way: $(X_1,Y_1,X_2,Y_2,\ldots)$. This is useful ...
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2 votes
0 answers
54 views

Example of a progressively measurable process not optional

Let $X$ be a stochastic process on a filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\geq 0},\mathbb{P})$ with value in $\mathbb{R}$ equipped with the Borel sigma algebra and the ...
1 vote
1 answer
121 views

A stochastic process with right continuous paths and usual conditions to the adapted filtration is progressively measurable

I'm reading Stochastic Processes by Richard F. Bass and got stuck by Exercise 1.3 for a while. Let $\{\mathcal{F}_t\}$ be a filtration satisfying the usual conditions (complete and right-continuous) ...
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1 vote
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Definition of $\mathcal{F}_{t-}$

Let $(\Omega, \mathcal{F},\mathbb{F}, P)$ be a filtered probability space, where $\mathbb{F}=\left(\mathcal{F}_t,\,t\ge 0\right)$ is a filtration. I am curious about the rigorous definition of $\...
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1 vote
1 answer
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Stochastic processes: understanding a complete filtration

My understanding is that the filtration for a stochastic process represents the information known, meaning that at time $t$ I know which sets in $\mathscr{F}_t$ are true or false. A filtration is ...
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1 vote
1 answer
104 views

Prove that $T$ is a stopping time

Let $(X, \mathcal A, \mathcal A_n, \mu)$ be a $\sigma$-finite filtered measure space and let $X_n$, $n\in\mathbb N$ be a sequence of $\mathcal A_n$ measurable real valued functions. For a fixed Borel ...
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0 votes
1 answer
32 views

Equivalent Definitions of Hodge Structure

I have read some materials on Hodge structures and all of them state the equivalence of definitions from the following perspectives: (suppose that we are considering a Hodge structure of weight $n\in \...
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1 vote
1 answer
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Calculate $E[X_t|\mathcal{F}_s^X]$ for each $s < t$

I need help understanding a solution, I have two thirds figured out but I cannot understand one part, I explain: Problem: Let $X$, $Y$ be two random variables in a probability space ($\Omega, \mathcal{...
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72 views

Rees algebra is finitely generated if associated graded algebra is

$ \newcommand{\cn}{\colon} \newcommand{\<}{\leqslant} \newcommand{\>}{\geqslant} \newcommand{\ss}{\subset} \newcommand{\k}{\mathrm{k}} \newcommand{\gr}{\mathrm{gr}} \newcommand{\R}{\mathrm{R}} \...
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2 votes
0 answers
80 views

Algebra is noetherian if associated graded algebra is noetherian

$ \newcommand{\cn}{\colon} \newcommand{\<}{\leqslant} \newcommand{\>}{\geqslant} \newcommand{\ss}{\subset} \newcommand{\gr}{\mathrm{gr}} $ Let $A$ be associative commutative algebra with unity ...
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0 answers
17 views

Writing intersection of events with respect to a filtration

I am a novice and hence come of my understanding maybe wrong so please look at the scenario accordingly. Suppose we have a filtration $\{ \mathcal F_k \}_{k \geq 0}$. Suppose I have events $E_1,E_2,...
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1 answer
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Is $E[X_sX_t^2]=0 \,\,s<t$ for a MDS?

Let $(X_t,\mathcal F_t)$ be a stationary martingale difference sequence (MDS). Can we say that $$E[X_sX_t^2]=0 \quad s<t \quad ?$$ For $s>t$ we can use the the law of iterated expectations and ...
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2 votes
2 answers
137 views

Is $Z_n = X_n Y_n$ a martingale?

Suppose $X_n$ and $Y_n$ are independent martingales with respect to filtration $\mathcal{F}_n$. Is $Z_n$ a martingale with respect to the same filtration, where $Z_n = X_n Y_n$ and we know that $\...
0 votes
0 answers
26 views

Expectation of Brownian Motion adapted for Filtration

I just started learning about Brownian Motions and martin gales and have the following issue. If $X_t$ is a brownian motion, I cannot understand how the results below are different when adapting for ...
0 votes
1 answer
22 views

Filtration with same adaptable and predictable processes

Let us have a toss of two coins and our $\Omega=(hh,tt,ht,th)$. 1) give a filtration so that all stochastical processes are also adaptable and also predictable. 2)given the filtration $F_{0}= trivial,...
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1 vote
2 answers
39 views

Interpretation of adapted process?

A process $(X_{t})_{t \in T}$ is $(\mathcal{F}_{t})_{t \in T}$-adpated if for every $t$, $X_{t}$ is $\mathcal{F}_{t}$-measurable. But since the variable $X_{t}$ is interpreted as the state of process ...
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How do we interpret the random variable defined as $X_{\tau} = \sum_{n=0}^{\infty}X_{n}1_{\{\tau = n\}}$?

Given an adapted process $(X_{n})_{n\in\mathbb{N}}$ and a stopping time $\tau$ with respect to the filtration $(\mathcal{F}_{n})_{n\in\mathbb{N}}$, we can define the random variable: \begin{align*} X_{...
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2 votes
0 answers
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Bounded convergence theorem with filtrations

Suppose in a filtered probability space $(\Omega,\mathbb{P},\{\mathcal{F}_n\},\mathcal{F})$ we have a sequence of almost surely vanishing adapted random variables $\varepsilon_n$. By the conditional ...
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0 votes
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Determine the natural filtration of a given stochastic process

Let $\Omega=\{\omega_1,\omega_2,\omega_3,\omega_4\}$ and $X=\{X_t\}_{t=0,1,2}$, where $$X(0)=3, X(1,\omega)=\begin{cases}5, \omega=\omega_1,\omega_2 \\ 2, \omega=\omega_3,\omega_4 \end{cases}, X(2,\...
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2 votes
0 answers
115 views

Is the inverse limit of quotient the quotient of inverse limits?

I am recently studying inverse limits and I have the following issue : Le $V$ be a vector space and $W \subset V$ a subspace of $V$. Assume that $V$ is equipped with a decreasing filtration $(\mathcal{...
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1 answer
77 views

How to prove the Martingale's property if we have a special stopping time?

$(X_n)$ is a sequence of $(F_n)$-adapted integrable random variables, where $(F_n)$ is a Filtration and $X_0=0$. I have to prove that 1)$X_n$ is a martingale with a respect to $F_n$ iff 2)for any ...
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0 votes
1 answer
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Markov Chain wrt filtration Definition?

Let $\{X_t\}$ be a Markov chain on probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with state space $(\chi, \mathbb{B}(\chi))$. Let $\{\mathcal{F_t}\}$ be a filtration and let $\{X_t\}$ be ...
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0 votes
1 answer
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Understanding the relationship between filtration $\mathcal{F}_t$ and an observed trajectory $O_{t}$

Introduction: I understand the filtration $\mathcal{F}_t$ to model all knowledge of a stochastic process $\{X_t:t=0,1,2,\dots,T\}$ up to time $t$ which in this case is discrete-time (due to my basic ...
1 vote
2 answers
179 views

Filtration versus natural filtration: intuition and misunderstandings

I currently have an understanding of what a filtration is and will illustrate this through an example. From this example, I will convey what my idea of natural filtration is. My question is similar to ...
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0 answers
30 views

Right-continuity of an augmented filtration of a strong Markov process

This is a question related to the Proposition 2.7.7 of Karatzas and Shreve's Brownian Motion and Stochastic Calculus. This proposition tells us Proposition 2.7.7: For a d-dimensional strong Markov ...
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1 vote
1 answer
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$\mathcal{F}_\infty$-measurablity of a random variable [closed]

Let $X$ be a r.v. and $Y_1, Y_2, \dots$ be i.i.d. integrable r.v.'s independent of $X$, with $E[Y_i]=0$. Here, we consider $Z_i = X + Y_i$, and let $\mathcal{F}_n$ be filtration s.t. $\mathcal{F}_n = \...
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1 vote
0 answers
23 views

Chaining a conditional likelihood function

Before I proceed, allow me to apologise in advance for potential abuse of notations. Let $\{x_t:\Omega \to \mathbb{R}\}_{t=1,\cdots,n}$ be a stochastic process, defined on a probability space $(\Omega,...
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Natural Filtration Generators

I have this statement in my stochastic processes script: Suppose $X$ is a stochastic process indexed by $I$, then $\mathcal{F}_{n} := \sigma (X_{s} :s \leq t) = \lbrace X_{k}^{-1}(B), \: k \leq n , \: ...
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0 votes
1 answer
61 views

Stopping Time: Diffrences in discrete and continuous time

I have a question regarding the definition of stopping times in continuous and discrete time. For continuous time we have the definition that a random variable on a filtered probability space $(\Omega ...
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1 vote
1 answer
34 views

Example of natural filtration which is not right-continuous

Can anyone please give me an example of a process $X$ whose natural filtration satisfies $(\mathscr{F}_{t}^{X_{+}})_{t \ge 0} \neq (\mathscr{F}_{t}^{X})_{t \ge0}$?

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