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.
307
questions
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19
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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 ...
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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 ...
- 745
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0
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9
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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 $\...
- 745
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0
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28
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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
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61
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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
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1
answer
45
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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\...
- 113
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1
answer
24
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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 $\overline{\mathcal{F}}=\sigma(\mathcal{F}\cup \mathcal{N})$ ...
- 904
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0
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31
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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, ...
- 469
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1
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25
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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 $n$):
$$Fil:Fil_0=0\subset Fil_1\subsetneq \cdots\subsetneq Fil_n=...
- 1,190
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14
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What is the filtered probability space used to study linear SDEs with constant coeficients?
Context:
I am currently working with Kalman-like filters. As a result I deal with linear Stochastic Differential Equations (SDEs) with constant coefficients such as:
$$
dx(t) = Ax(t)dt + Bdw(t)
$$
...
- 1,674
1
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0
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35
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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\...
- 51
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25
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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}^...
- 1,098
2
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2
answers
101
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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 ...
- 1,098
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0
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30
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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 $...
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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 ...
- 115
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38
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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 ...
- 573
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2
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39
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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$...
- 2,614
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1
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49
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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 <...
- 573
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2
answers
95
views
$\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 ...
- 2,614
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45
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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
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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 ...
- 604
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54
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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 ...
- 1,098
1
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1
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121
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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) ...
- 129
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0
answers
53
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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 $\...
- 965
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192
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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 ...
- 11
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1
answer
104
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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 ...
- 521
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1
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32
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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 \...
- 11
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1
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26
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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{...
- 655
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0
answers
72
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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}}
\...
- 145
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0
answers
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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 ...
- 145
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0
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17
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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,...
- 251
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1
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41
views
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 ...
- 4,057
2
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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 $\...
- 53
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0
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26
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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 ...
- 55
0
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1
answer
22
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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,...
- 851
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2
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39
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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 ...
- 791
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21
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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_{...
- 411
2
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0
answers
37
views
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 ...
- 604
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0
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19
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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,\...
user1089451
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{...
- 87
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votes
1
answer
77
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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 ...
- 153
0
votes
1
answer
50
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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 ...
- 2,264
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1
answer
33
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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 ...
- 315
1
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2
answers
179
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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 ...
- 315
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0
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30
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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 ...
- 56
1
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1
answer
39
views
$\mathcal{F}_\infty$-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 = \...
- 43
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,...
- 173
0
votes
0
answers
18
views
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 , \: ...
- 13
0
votes
1
answer
61
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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 ...
- 13
1
vote
1
answer
34
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Example of natural filtration which is not right-continuous
Can anyone please give me an example of a process $X$ whose natural filtration satisfies $(\mathscr{F}_{t}^{X_{+}})_{t \ge 0} \neq (\mathscr{F}_{t}^{X})_{t \ge0}$?
