# 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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### 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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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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### 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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### 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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### 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 ...
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1 vote
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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 ...
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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 ...
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I have this statement in my stochastic processes script: Suppose $X$ is a stochastic process indexed by $I$, then $\mathcal{F}_{n} := \sigma (X_{s} :s \leq t) = \lbrace X_{k}^{-1}(B), \: k \leq n , \: ... • 13 0 votes 1 answer 61 views ### Stopping Time: Diffrences in discrete and continuous time I have a question regarding the definition of stopping times in continuous and discrete time. For continuous time we have the definition that a random variable on a filtered probability space$(\Omega ...
Can anyone please give me an example of a process $X$ whose natural filtration satisfies $(\mathscr{F}_{t}^{X_{+}})_{t \ge 0} \neq (\mathscr{F}_{t}^{X})_{t \ge0}$?