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Questions tagged [filtrations]

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13 views

Can a stochastic process be neither adapted to filtration nor previsible?

The idea behind the question arises from my intuition about the concepts of 'adapted to filtration' and 'previsbility'. If a process is adapted, it essentially means that the evolution of the ...
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1answer
52 views

Showing $GL_n^1(\mathbb Z_p)$ with usual $p$-valuation is $p$-saturated.

First a few definitions: $GL_n^1(\mathbb Z_p) = \ker(GL_n(\mathbb Z_p) \rightarrow GL_n(\mathbb Z/p\mathbb Z))$ $\omega: G \rightarrow \mathbb R^{>0} \cup \{\infty\}$ is a $p$-valuation (or ...
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1answer
16 views

Filtration in Markov Chains and stopping times

I am trying to get a better understand of filtrations, and I can't seem to find any simple, concrete examples, so I will try to make one here. Consider a discrete-time Markov chain with states $\{A,B\...
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1answer
52 views

Mistake on Wikipedia: Completeness of Filtration and Completeness of Individual Probability Spaces

A probability space $(\Omega, \mathcal{A}, \mathbb{P})$ is called complete iff every subset $\tilde{N}$ of a set $N \in \mathcal{A}$ of measure $0$ is measurable, i.e. if $\mathbb{P}(N) = 0$ and $\...
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17 views

generator of sigma algebra associated with stopping time

I am studying Martingales and Stopping Times from the 3rd edition of the book "Probability and Measure" by Patrick Billingsley. The following arose while I was reading page 465. Let $(\varOmega,\...
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1answer
23 views

Find values of $a$ and $\lambda$ for which $Z_{0}e^{at+bW_{t}}-\lambda t$ is a martingale

Find values of $a$ and $\lambda$ for which $Z(t)=Z_{0}e^{at+bW_{t}}-\lambda t$ is a martingale. In here $W_{t}$ is a Brownian motion and $a,b\in\mathbb{R}$ can be positive as well as negative, since $...
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13 views

Importance of filtrations that are NOT natural filtrations.

I know the natural filtration intuitively represents the history of the process as the process evolves over time, and hence can be used to talk about conditional probabilities and conditional ...
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1answer
22 views

Is every filtration the natural filtration of some stochastic process?

We have a notion of natural filtrations, which intuitively represents the history of the process as the process evolves over time. We also have a notion of filtrations in general, which are ...
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21 views

associated graded ring is PI implies original ring is PI

This seems like it should be a known result but I didn't find it in a couple of standard references on noncommutative Noetherian rings. Recall that a unital associative ring $R$ is a polynomial ...
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1answer
59 views

What is meant by a filtration “contains the information” until time $t$?

I have problems understanding the concept of a filtration in stochastic calculus. I understand that for example the natural filtration $F_t$ contains only outcomes up to time $t$, but since it is a ...
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25 views

Clarification on a proof about $p$-valued groups.

First we have a few definitions: $(R,v)$ is a filtered ring if $v : R \rightarrow \mathbb R^{\geq0} \cup \{\infty\}$ satisfies: $v(r-s) \geq \min\{v(r), v(s)\}$ $v(rs) \geq v(r) + v(...
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1answer
31 views

How to picture a filtration generated by a Brownian motion?

I know what filtrations generated by discrete random variables are, but I can't apply it to Brownian motions. Please help.
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30 views

Filtration of a set Really dont get it

Let $a\in \mathbb R$. Consider the filtrations $\varphi,\varphi':\mathbb N\to {\rm Sub}(\mathbb R)$ defined by $\varphi(n)=\left(a-\frac{1}{n},a+\frac{1}{n}\right)$ and $\varphi'(n)=\left[a-\frac{1}{n}...
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1answer
30 views

Do we have that $\mathcal{F}_{\infty} = \sigma(X_{t} \colon t \geq 0)$? [closed]

If $(\mathcal{F}_{t})_{t \geq 0}$ is the filtration generated by a process $(X_{t})_{t \geq 0}$, one typically sets $$ \mathcal{F}_{\infty} : = \sigma \Big( \bigcup_{t \geq 0} \mathcal{F}_{t} \Big). $$...
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1answer
51 views

Prove Symmetric Random Walk is a Martingale

I'm given a probability space of ($\Omega$, $\mathcal{F}$, $\mathbb{P}$) and am asked to look into a symmetric random walk with its n-step defined as $$ X_n = \Bigg\{ \begin{matrix} 1 & \text{...
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1answer
19 views

Prove conditional expectation of standard normal random variables

Let $X_{i},i=1,2,...$ be a sequence of independent identically standard normally distributed random variables. Let $\{\mathcal{F}_{n},n\in\mathbb{N}\}$ be the natural filtration and $S_{n}=\sum^{n}_{i=...
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1answer
25 views

Where is an error in my deduction? (question about martingales)

Suppose we have a filtration $\{\mathcal{F_{t}},t\geq 0\}$ and a stochastic process $\{ X_{t},t\geq 0\}$ which is adapted to this filtration and also integrable. All we need for this process to be a ...
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48 views

Natural filtration and coin tossing

I have a problem understanding a probably easy task from a book I'm reading at the moment: "Show, that the natural filtration $\ F_2 $, generated by observing the events $\ A_1 $ = {(H,H),(H,T)} and ...
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0answers
33 views

Is Conditional Expectation of stochastic process a martingale?

I have the following: Let $Y$ be an integrable and $F_T$ -measurable random variable. Define $\{X_t\}_{t \in [0,T]}$ by $X_t =E(Y|F_t), \ \forall t \in T$. The $\{F_t\}_{t \in [0,T]}$ should be the ...
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1answer
44 views

$\leq_{\frak K_\lambda}$-increasing continuous

Here (in the context of Abstract Elementary Classes) on the page 43 at the bottom,-6th line, what does it technically mean $$\leq_{\frak K_\lambda}-\text{increasing continuous}$$ ? I think that this ...
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1answer
45 views

Filtration in crystalline Poincaré Lemma

I am trying to understand section 20 in https://stacks.math.columbia.edu/download/crystalline.pdf, especially the proof of Lemma 20.2. If $A\rightarrow B$ is a map of rings and $P=B[x_i]$ is some ...
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0answers
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An example of the fact that from measurability of a random process does not follow measurability of its integral

Let {$ \xi _t(\omega), t\in[0,\infty)$} be a random process and $ \xi _t(\omega)\in \{\mathfrak F_t\}$ (some filtration). If $ \xi _t(\omega) $ is $ \mathfrak F_t $ measurable then $\int_0^t\xi _s(\...
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Alternative formulation of a markov process

I'm wondering how the markov property can be specified as follows, if anyone can provide more details (this looks awfully like the definition for a martingale): $$E[f(X_t)|\mathcal{F}_s]=E[f(X_t)|\...
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1answer
45 views

Prove that $\mathbf{E}(X_{\tau_2}|\mathcal{F}_{\tau_1})=X_{\tau_1}$

Let $(X_n)$ be a martingale with respect to $(\mathcal{F}_n)$ and suppose $\tau_1$ and $\tau_2$ are bounded stopping times such that $\tau_1\leq \tau_2<B<\infty.$ Then $$\mathbf{E}(X_{\tau_2}|\...
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1answer
45 views

How to prove an isomorphism related to a filtration.

Let $A$ be a filtered algebra with a filtration $F_0(A) \subset F_1(A) \subset \cdots \subset A$. Let $I$ be a two sided ideal of $A$. The algebra $A/I$ has a filtration $F_i(A/I)=F_i(A)/(F_i(A)\cap I)...
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1answer
77 views

Proof that augmented filtration is right continuous

Consider a filtered probability space $(\Omega, \mathcal{F}, \mathbb{F}, P)$, where $\mathbb{F}= \left\{\mathcal{F}_t:t\ge 0\right\}$ is a filtration on $(\Omega,\mathcal{F})$. Let $$\mathcal{N} = \...
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1answer
35 views

Compensation Martingale

Let $Y_1, Y_2, . . .$ be an adapted sequence, and let $c_n\in \mathcal{R}$, $n \ge 1$. (a) Suppose that $E(Y_{n+1} | F_n) = Y_n + c_n$. Compensate suitably to exhibit a martingale. (b) Suppose that $...
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1answer
42 views

show $\sup\{t \in \mathbb{N}_0 : S_t =1\}$ is a stopping time

Let $(X_n)_{n \in \mathbb{N}_0}$ be a sequence of independent and identically distributed random variables with $$\mathbb{P}(X_1 = 1) = \mathbb{P}(X_1 = -1) = \frac{1}{2}.$$ Define $S_t = \sum\...
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1answer
36 views

Why $S\in \mathcal F_T$ but not in $\sigma (T)$?

Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space. I'm trying to understand better $\sigma -$algebra in probability, in particular the $\sigma -$algebra $$\mathcal F_T=\{A\in \mathcal F\mid A\...
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0answers
61 views

Right Continuous Adapted Processes without right-continuous Filtration.

Let $\mathscr{F}$ be a filtration and $\mathscr{F}^+$ be the right continuous version. ($\mathscr{F} = \{\mathscr{F}_t\}_{t\in [0,T]})$ Suppose $X$ is right-continuous $\mathscr{F}$-adapted process. ...
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6 views

Ascending Filtrations of an Almost Free Non-Free Group

Let $\kappa$ be a cardinal. Say that an abelian group is $\kappa$-free if every subgroup generated by a set of cardinality less than $\kappa$ is free. Now let $G$ be a $\kappa$-free group of infinite ...
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2answers
41 views

How to understand a probability space in dicrete time

It is common in probability to define a prob. space as : $$(\Omega,\mathscr{F},P)$$ This can be understood as sample space, events, and probabilities for each event. However I don't know how to to ...
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1answer
38 views

Predictable graph of a random set

Assume that we work on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ equipped with a filtration $\mathcal{F}_t, t\ge 0$ and stochastic basis $(\Omega, \mathcal{F}, \mathbb{P}, \mathbf{F})$ ...
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28 views

Why do we need to include filtrations in the definition of probability spaces when talking about stochastic processes.

In the first line of these notes the author defines his stochastic process. Using a filtration. What is the importance of filtrations? https://warwick.ac.uk/fac/sci/maths/people/staff/...
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1answer
81 views

adapted process, translation between measurable and information?

Although there are plenty of questions and answers on understanding the intuition for adapted process like this post and this post I am still unclear on how an adapted filtration 'captures the ...
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1answer
53 views

Stopping time clarification

May I please seek someone's help to clear my understanding about stopping time. According to the Wikipedia definition: random variable $\tau:\Omega \rightarrow I$ is called stopping time if $\{\...
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0answers
56 views

Largest filtration for a martingale

Let $(X_n)_{n \in \mathbb{N}}$ be independent random variables where $P(X_n = 2^{n}) = P(X_n = -2^{n}) = \frac{1}{2}$ and $M_n = \sum_{k=1}^n X_k$ for $n \in \mathbb{N}$. Find the largest filtration, ...
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2answers
165 views

Convergence in probability of conditional expectation

Suppose I have a sequence of random variables $X_n$ and $\sigma$-fields $\mathcal{F_n} \subseteq \mathcal{F}_n'$. Suppose that $\mathbb{E}[X_n \mid \mathcal{F}_n']$ converges to a constant $c$ in ...
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33 views

Methods for filtering position signal

I'm working on a location detection algorithm. This is my outcome - postion: Here's first derivative after time - velocity: [ My goal is to apply some filtration to the signal for smoothing the ...
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0answers
33 views

Are local martingales semimartingales in an enlarged filtration (under particular assumption)?

I'm studying filtration enlargements and I bumped into the following problem. Assume there are two filtrations $\mathbb{F}=(\mathcal{F}_t)_{t \in T}$ and $\mathbb{H}=(\mathcal{H}_t)_{t \in T}$ on the ...
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Prove $(X_n, F_n) $ Martingale $\iff \int_{F} X_{n+1} = \int_{F} X_{n} \forall F \in F_n$

I have some additional questions to this exercise: Let $(\Omega, F, F_n, P)$ filtered probability space. Let $(X_n)_{n\in\mathbb{N}} \in \mathcal{L}^1(P)$, which is adopted to $(F_n)_{n\in\mathbb{N}}$...
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3answers
71 views

Sub-sigma-algebras: infinite coin flips example on wikipedia

On the wiki there is this section about sub sigma-algebras for the first $n$ coin flips Specifically, they say that after the first $n$n coin flips we can describe the observed information in terms ...
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1answer
128 views

Law of a Markov process uniquely determined by its 2-dimensional distributions

I am stuck with the following problem about Markov processes: Let $(\Omega,\mathcal{F}, \mathbb{P}, \mathcal{F}_t)$ be a filtered probability space. If $X,Y:\mathbb{R}_{+}\times\Omega\to\mathbb{R}$ ...
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0answers
16 views

Characterization of a set in the augmented filtration $\mathcal{F}_t^{\mu}=\sigma(\mathcal{F}_t^X, \mathcal{N}^{\mu})$

Can I write any set in $\mathcal{F}_t^{\mu}=\sigma(\mathcal{F}_t^X, \mathcal{N}^{\mu})$ as $A\cup B$ where $A \in \mathcal{F}_t^X $ and $B \in \mathcal{N}^{\mu}$ ? I am quite convinced that this is ...
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1answer
79 views

Where is the Strong Markov property(SM) being used in the proof that augmented filtration of a Strong Markov process is right continuous?

I was self-reading Section 2.7 of Karatzas and Shreve's Brownian Motion and Stochastic Calculus and Proposition 7.7 claims that the Strong Markov processes augmented filtration is right continuous. In ...
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1answer
22 views

What are the generating sets of $\mathcal{F}_t^X=\sigma(X_s , 0 \leq s \leq t) $?

When can $\mathcal{F}_t^X=\sigma(X_s , 0 \leq s \leq t) $ be generated by sets of the form $F=\{X_{t_0} \in \Gamma_1\, X_{t_1} \in \Gamma_2 ,\dots, X_{t_n} \in \Gamma_n \}$ where $\Gamma \in \mathcal{...
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0answers
74 views

Underlying Random Variable of Conditional Expectation

Consider interval $[0,1]$ with its Borel $\sigma$-algebra and Lebesgue measure on it. It is known that $f$ is an integrable function on $[0,1]$. $\mathcal{F_n}=\sigma([\frac{k-1}{2^n},\frac{k}{2^n}))$ ...
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1answer
23 views

How can I show that $\mathcal{F}_t^X$ is generated by sets of the form $F=\{(X_{t_1},\dots, X_{t_n}) \in \Gamma\}$

How can I show that $\mathcal{F}_t^X$ is generated by sets of the form $F={(X_{t_1},\dots, X_{t_n}) \in \Gamma}$ where $\Gamma \in\mathcal{B}(\mathbb{R}^n)$ and $0=t_1< \dots <t_n=t$. Do I need ...
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1answer
26 views

Why is the following sigma algebra $\mathcal{F}_n=\pi_n^{-1}(\mathcal{B}(\{0,1\}^n))$ finite?

We consider the measurable space $(M,\mathcal{B}(M))$ where $M=\{0,1\}^{\mathbb{N}}=\{\omega_1,\omega_2,\dots\},\omega_i=0$ or $ \omega_i=1$ i.e the space of all binary sequences indexed by the ...
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0answers
78 views

Predictable Projection of a Stopped Process

Given a filtered probability space $( \Omega, \mathcal{F}, (\mathcal{F}_t)_t, \mathbb{P})$ and an $\mathcal{F} \otimes \mathcal{B}(\mathbb{R}_+)$-measurable bounded process $X: \Omega \times \mathbb{R}...