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Questions tagged [filtrations]

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1answer
35 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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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
40 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
35 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
41 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
25 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
28 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
32 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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42 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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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
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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
34 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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26 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
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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
36 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
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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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114 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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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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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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2answers
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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
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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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A product of a martingale and an indicator process it's still a martingale?

Let $\tau_{n}$ be an increasing sequence of stopping times and let $M_{t}^{n}$ be a locale martingale w.r.t $F_{t}$ on $]\tau_{n},\tau_{n+1}]$ for each $n>0$. ($F_{t}$ filtration with usual ...
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1answer
116 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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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
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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
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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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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
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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
25 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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67 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}...
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2answers
23 views

$\sigma$-algebra generated by a subset of a set

Let $\Omega$ = {1,2,3,4} and consider the collection of subsets of $\Omega$, $\mathcal A$={{1,2}}. The problem is to find the $\sigma$-algebra $\mathcal F$ generated by $\mathcal A$. I know this is a ...
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0answers
43 views

How to tell if a Random variable $Y_N$ is a discrete-time martingale?

I understand that the definition of a discrete-time martingale in a filtered probability space $(\Omega, F, \mathbb{P}, (F_n)_{n\geq 0} ) $ is $M = (M_n)_{n\geq 0}$ such that: $M$ is adapted with ...
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1answer
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Can a filtration on a measure space have real-number indexing (giving a continuum of $\sigma$-algebras?

My book "Measures, Integrals and martingales" defines a filtration as a sequence $F_t$ of $\sigma$-algebras. That means that there are at most countable $\sigma$ algebras in the filtration. But in ...
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0answers
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Condition on filtration for convergence of conditional expectations

I'm trying to determine conditions on a filtration $\{ \mathcal{F}_t \}_t$ such that the identity $\lim\limits_{\Delta \longrightarrow 0+} E\big[ f(t,\Delta) | \mathcal{F}_{(t-\Delta)-} \big] = \lim\...
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1answer
56 views

How to prove that the sigma-algebras generated by two processes are equal?

Let $X_t$ and $\xi_t$ be two stochastic processes and let $\mathcal{M}_t$ be a sigma-algebra generated by $X_t$ and $\mathcal{N}_t$ a sigma-algebra generated by $\xi_t$. I'm trying to show that $\...
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Right-continuity of (initially) enlarged filtration

I've asked myself the following simple question: Given a right-continuous filtration $(\mathcal{F}_t)_{t \geq 0}$ on some probability space $(\Omega, \mathcal{A}, \mathbb{P})$ and some sub-$\sigma-$...
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1answer
83 views

Conditional expectation as a random variable new [closed]

We consider the probability space $([0,1), F , \lambda )$ , where $F$ denotes the Borel-$\sigma$-field on $[0,1)$ and $\lambda$ is the Lebesgue-measure. We define for each $n\...
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Trace of the $\sigma$-algebra generated by the predictable rectangles

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $T>0$ $(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration of $\mathcal A$ Let $$\mathcal R_a:=\bigcup_{F\in\mathcal F_a}F\times\left\{...
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2answers
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Filtrations and topological ring

At page $105$ of Introduction to Commutative Algebra of M. Atiyah, there is the following claim: C. Given a ring $A$ and an ideal $I\subset A$, consider the filtration $(I^n)_{n\in\mathbb{N}}$, then ...
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Girsanov theorem and filtrations

Let $\{W_t\}$ be a standard Wiener process on a probability space $(\Omega, \mathcal{F},P)$. Let $\mathcal{F}^W$ be the natural filtration generated by $\{W_t\}$. Let $\{\theta_t\}$ be an $\mathcal{...
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A philosophical question about filtrations and information sets

In continuous-time stochastic calculus when we face an optimal control problem how can we restrict our choices to be "pre" variables rather than "post". To be clear, suppose that we are in a discrete-...
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The associated graded of a filtered coalgebra

Given a coalgebra $C$ with a filtration $F$ such that $\Delta(F^n C)\subset \sum_{i=0}^n F^i C\otimes F^{n-i} C$, how does the coproduct manifest in the associated graded? Do we get something to the ...
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Mapping from natural filtration to sequence of events $X(t)$

Consider a discrete time stochastic process $X: T \times \Omega \mapsto \mathbb{R}$ defined on a probability space $(\Omega, \mathcal{F}, \mu)$. Being defined as the smallest sub-$\sigma$-algebra of $\...
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1answer
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Commutation between σ-algebra and intersection

I wonder the following property is true or not: $$ \sigma(\bigcap_n \mathcal C_n)=\bigcap_n \sigma (\mathcal C_n) $$ where $\mathcal C_n$ are decreased collections of sets in a measurable space. And "...
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1answer
505 views

Conditional expectation on a filtration defined by a stopping time

I am trying to answer this question. The $\sigma$-algebra $\mathcal F_\tau$ defined by a stopping time $\tau$ is such that $A\in \mathcal F_\tau$ iff the event $A \cap \{\tau=t\} \in \mathcal F_\...
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1answer
141 views

How can we show that $\mathcal F_\tau\cap\left\{\tau=t\right\}=\mathcal F_t$? for any $\mathcal F$-stopping time $\tau$?

Let $(\Omega,\mathcal A)$ be a measurable space $I\subseteq\mathbb R$ $(\mathcal F_t)_{t\in I}$ be a filtration of $\mathcal A$ If $\tau$ is an $\mathcal F$-stopping time, then $$\mathcal F_\tau:=\...
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2answers
1k views

Example of filtration in probability theory

I'm studying Martingales and below here filtrations. Given af probability space $(\Omega, F, P)$ I define a filter $(F_n)$ as a increasing sequence of $\sigma$-algebras of $F$, such that $F_t \subset ...
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1answer
207 views

Show that for complete and right-continuous filtration, and $\sigma = \tau$ a.s. with $\tau$ stopping time, then $\sigma$ stopping time

I am trying to solve the following exercise: Let $\mathcal F = \{ F(t) \}_{t\in [0,T]}$ be a complete and right-continuous filtration, $\tau$ a stopping time for $\mathcal F$ and $\sigma$ a random ...
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1answer
85 views

Is $\mathcal{F}_\infty$ a $\sigma$-algebra? (Chung Theorem 9.4.8.)

We let $\mathcal{F}_\infty=\bigvee_n\mathcal{F}_n$. (What does $\bigvee$ usually denote? Is $\bigvee$ different from $\cup$? Can't find it in the text.) The theorem tells us when we have $\lim_n E(Y|\...