Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [filtrations]

The tag has no usage guidance.

-1
votes
1answer
27 views

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

If $(\mathcal{F}_{t})_{t \geq 0}$ is the filtration generated by a process $(X_{t})_{t \geq 0}$, one typically sets $$ \mathcal{F}_{\infty} : = \sigma \Big( \bigcup_{t \geq 0} \mathcal{F}_{t} \Big). $$...
1
vote
1answer
20 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{...
0
votes
1answer
15 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=...
1
vote
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 ...
0
votes
0answers
30 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 ...
1
vote
0answers
22 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 ...
0
votes
1answer
43 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 ...
1
vote
1answer
42 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 ...
1
vote
0answers
19 views

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(\...
0
votes
0answers
12 views

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)|\...
1
vote
1answer
42 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}|\...
1
vote
1answer
38 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)...
1
vote
1answer
53 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} = \...
1
vote
1answer
26 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 $...
1
vote
1answer
41 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\...
1
vote
1answer
34 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\...
1
vote
0answers
54 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. ...
0
votes
0answers
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 ...
2
votes
2answers
36 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 ...
0
votes
1answer
37 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 $(\Omega, \mathcal{F}, \mathbb{P}, \mathbf{F})$ ...
0
votes
0answers
27 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/...
1
vote
1answer
56 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 ...
1
vote
1answer
43 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 $\{\...
1
vote
0answers
47 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, ...
3
votes
2answers
140 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 ...
0
votes
0answers
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 ...
2
votes
0answers
30 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 ...
3
votes
2answers
80 views

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 $(F_n)_{n\in\mathbb{N}}$...
0
votes
3answers
63 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 ...
0
votes
0answers
37 views

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 ...
2
votes
1answer
118 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 $X,Y:\mathbb{R}_{+}\times\Omega\to\mathbb{R}$ ...
1
vote
0answers
15 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 ...
3
votes
1answer
63 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 ...
0
votes
1answer
21 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{...
5
votes
0answers
72 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 $[0,1]$. $\mathcal{F_n}=\sigma([\frac{k-1}{2^n},\frac{k}{2^n}))$ ...
1
vote
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 ...
1
vote
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 ...
1
vote
0answers
74 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}...
0
votes
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 ...
1
vote
0answers
45 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 ...
0
votes
1answer
31 views

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 ...
2
votes
0answers
70 views

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\...
0
votes
1answer
59 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 $\...
1
vote
0answers
26 views

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-$\sigma-$...
1
vote
1answer
85 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\...
3
votes
0answers
43 views

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\{...
0
votes
2answers
40 views

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 ...
4
votes
0answers
115 views

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{...
0
votes
0answers
25 views

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-...
1
vote
0answers
46 views

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 ...