# Questions tagged [filtrations]

This tag is for questions relating to "Filtration". It has many application in abstract algebra, homological algebra and in measure theory and probability theory for nested sequences of σ-algebras.

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### What is the correct filtration?

Let $B\overset{\circ}{=}\left(B_{t}\right)_{t\geq0}$ denote a Brownian motion in a filtration $\mathcal{F}$. Are $X_{t}=\frac{1}{\sqrt{a}}B_{at}$ ($a>0$ constant) and/or $Y_{t}=tB_{\frac{1}{t}}$ ...
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### Meaning of measurableness [duplicate]

I've done courses on measure theory, advanced stochastic processes, etc.--years and years of using the notion of 'measurableness'. I've now come to the conclusion that, in fact, I do not understand ...
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### Problem with intuition regarding sigma algebras and information

After thinking about it a few years, I still don't quite understand the precise link between sigma algebras and information. (Yes--I know that there's already dozens of questions on MSE by other ...
1 vote
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### With stopping time $T$ and positive constant $t$, show that $T \land t$ is $\mathscr{F}_t$ measurable.

Since $T$ is a stopping time, by definition, for any $t \ge 0$, $\{T \le t\} \in \mathscr{F}_t$. $T \land t = \min(T, t)$ We have to show that $T \land t$ is $\mathscr{F}_t$ measurable, which means ...
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### Karatzas & Shreve 2.6: Show that the hitting time $H_{\Gamma}$ is an optional time.

Consider a stochastic process $X$ with right-continuous paths, which is adapted to a filtration $\{ \mathscr{F}_t \}$. Consider subset $\Gamma \in \mathscr{B}(\mathbb{R}^d)$ of the state space of the ...
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### Karatzas and Shreve Exercise 1.8: If paths of process $X$ are RCLL almost surely, show that $A$ can fail to be in $\mathscr{F}_{t_0}^X$

Karatzas+Shreve Textbook Exercise 1.8: Let $X$ be a [stochastic] process whose sample paths are RCLL almost surely, and let $A$ be the event that $X$ is continuous on $[0,t_0)$. Show that $A$ can ...
1 vote
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### What is joint filtration?

Consider a probability space $(\Omega,\mathcal{F},\mathbb{P})$ with two filtrations $(\mathcal{F}_t)_{t\geq0}$ and $(\mathcal{G}_t)_{t\geq0}$. The book Credit Risk: Modeling, Valuation, and Hedging by ...
1 vote
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### Characterizing strict morphisms in the category of bifiltered vector spaces

Let $k$ be a field, and let $C$ be the category whose objects are finite dimensional $k$-vector spaces endowed with two finite filtrations $W$ and $F$, the former being ascending and the latter ...
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Let $X=(X_t)_{t\ge0}$ be a stochastic process on a probability space $(\Omega,\mathcal{F},P)$ with $X_0=0$ a.s., and let $(\mathcal{F}_t)_{t\ge0}$ denote the filtration generated by $X$, i.e., $\... 2 votes 0 answers 51 views ### How to show$\mathbb{E}(Y_{n+1}^{2} | \mathcal{F}_{n}) = \mu^{2} Y_{n}^{2} + \sigma^{2} Y_{n}$? (Unanswered repost) I'm basically asking an extension of the first part from this post a while back. To parse, it asked: Given a branching process$Y = \{Y_{n} : n \geq 0\}$and the corresponding family of random ... 2 votes 1 answer 45 views ### Solution of stochastic differnetial equation adapted? Consider the following stochastic differential equation: $$dY_t = Z_t dW_t$$ and terminal condition$Y_T = b,$for which holds:$E[|b|^2] < \infty$Furthermore b is adapted to the filtration ... 4 votes 0 answers 67 views ###$A(t)=\int_{0}^{t} I_{(X\geq u)} \,dΛ(u)$is a predictable process I am currently reading the book "Counting Processes and Survival Analysis" by Fleming and Harrington and I am stuck in the proof of Proposition 1.4.2 p.34: Proposition 1.4.2: Let$(T,U)$be ... 4 votes 0 answers 362 views ### How do we naturally extend the average from the Hausdorff Measure for functions with a domain without a gauge function? Motivation: According to this question Some sets have a Hausdorff Dimension$\alpha$but have a zero-dimensional Hausdorff Measure. These sets may have another dimension function, i.e. a function$h:[... 24 views

### If $0 < S < T$ are stopping times, and $A \in \mathcal{F}_S$ show that $T = S1_A + T1_{A^c}$ is a stopping time.

Let $0 < S < T$ two stopping times with respect to the discrete-time filtration $(\mathcal{F}_n)_{n \ge 0}$. Let $A \in \mathcal{F}_S$. Show that the random time $T'$ defined by: \begin{align*} ...
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### What is the spectral sequence associated to this filtration on the de Rham complex?

I am trying to calculate some relative de Rham cohomology, but I am not too skilled with hypercohomology or spectral sequences, and the situation becomes more complicated because (1) the base is not ...
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### Example/meaning of filtration on a group $(\mathbb{R},+)$

Serre, in his Lie algebras and Lie groups, gives the definition of a filtration $\omega$ on a group $G$ as a map $\omega:G\to R\cup\{+\infty\}$ such that $\omega(e)=+\infty$ where $e$ is the identity ...
1 vote
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### Is possible to construct a "relevant information theory" from the notion of filtration?

Information theory and Shannon's formula provide a measure of uncertainty in specific situations where the outcomes and their probabilities are known. But in many situations, this information theory ...
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### Martingale with respect to different filtration

I consider two stochastic processes $X_t = B_t$ and $Y_t = B_{t-1}$ with their natural filtrations $\mathcal{F}^X$ and $\mathcal{F}^Y$, where $B_t$ is a Brownian motion. It is well known that $X_t$ is ...
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### Natural filtration for a transformation of a Brownian motion

Let $W$ be a standard Brownian motion on the filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in T},\mathbb{P})$, where $(\mathcal{F}_t)_{t\in T}$ is the natural filtration of $W$, i....
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### Relation between filtrations of stochastic processes

We consider a stochastic process $X_t$ that takes values in $\mathbb{R}$. Now we consider the following processes: $U_t = X_t + t$, $W_t = X_t^2$, $Y_t = \sin(X_t^2)$, $Z_t = e^{X_t}$. Their natural ...
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### Filtration with respect to which reverse brownian motion is martingale

I was wondering with respect to what filtration the process $X_t=\begin{cases}\frac{1}{t}B_{1/t}\text{ if }t>0 \\ 0\text{ if }t=0 \end{cases}$ $\;\;\;$($B_t$ is a Brownian motion) is a martingale. ...
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Let $G$ be a finite group and $V$ a finite-dimensional representation of $G$ (or $G$-module) over $\mathbb{C}$. Furthermore, suppose we have a filtration of $G$-modules $$0=V_0\subset V_1\subset\cdots ... 1 vote 1 answer 61 views ### filtration generated by function of brownian motion Consider a filtration F^{B_1,B_2} generated by two independent Brownian motions B_1,B_2. Now I define W:= a B_1 + \sqrt{1-a^2} B_2 and consider the filtration F^W generated by W. Are F^{B_1,... 1 vote 1 answer 77 views ### martingale representation of two independent Brownian motion Let W^1_t and W^2_t be two independent standard Brownian motions. Then W^2_t is a martingale with respect to the its own filtration but not adapted to the filtration generated by W^1_t. I want ... 0 votes 1 answer 52 views ### Can an ascending filtration be used to define completion of a vector space/ring? Let V: = V_0 \supseteq V_1 \supseteq V_2 \supseteq V_3 \supseteq \ldots be a descending filtration of a vector space V. Then one can define as basis of open sets given by \{ v + V_k, v \in V, k \... 2 votes 1 answer 87 views ### Short exact sequences of filtered vector spaces are split Let k be a field. By a filtered vector space over k, I mean a pair (V,F) where V is a finite dimensional k-vector space and F=(F^pV)_{p\in \mathbb{Z}} is an increasing filtration of V by ... 0 votes 0 answers 28 views ### Filtration of Leray spectral sequence for open embedding Let X be a smooth complex projective variety/compact Kähler manifold and D a normal crossing divisor. Let j: U:=X-Z\hookrightarrow X be the open embedding. We have two spectral sequences for ... 2 votes 0 answers 47 views ### Let \sigma,\tau be stopping times. \{\sigma<\tau\}\in\mathcal{F}_{\sigma}? Consider a filtered prob space (\Omega,\mathcal{F},(\mathcal{F}_s)_{s\in [0,T]},\mathbb{P}) satisfying the usual conditions. Let \sigma,\tau be stopping times. Is it possible to show that$$\{\...
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Given a filtration $(\mathcal{F}_n)_{n\geq 0}$ and a stopping time $T$, show that there exists an adapted process $(X_n)_{n\geq 0}$ such that $$T=\inf \{n \geq 0 : X_n >0\}$$ I'm really lost where ...