42 votes

Example of filtration in probability theory

Another simple example. The natural filtration when we model tossing a die twice in a row. Here is how it works. Let $X_1$ be the outcome of the first toss. So the values of $X_1$ are in the set $\...
  • 99.3k
23 votes
Accepted

Example of filtration in probability theory

Let me first state an interpretation for the meaning of a filtration: A filtration $\mathcal F_t$ contains any information that could be possibly asked and answered for the considered random process ...
  • 358
18 votes

Example of filtration in probability theory

Take the following simple model: a stochastic process $X$ that starts at some value $0$. From that value, it can jump at time $1$ to either the value $a$, either the different value $b$. And at time $...
17 votes
Accepted

What exactly is a 'predictable process'?

The classical interpretation of $\sigma$-algebras is information. A nice example comes from quantitative finance. Suppose we have a certain amount of money $V_n$ at a certain (discrete) time $t_n$. ...
  • 3,985
17 votes

What is meant by a filtration "contains the information" until time $t$?

In order to understand the intuition behind filtrations, it's a good idea to start with a very particular case: the $\sigma$-algebra generated by a single random variable $X:\Omega \to \mathbb{R}$, i....
  • 114k
11 votes

What is the intuition behind right-continuous filtration?

The idea is that you gain no additional information by taking an infinitesimal step forward in time. Remember that an $\mathit{intersection}$ means that we are taking only the elements contained in ...
8 votes
Accepted

Stochastic processes - Why do we need filtration?

You know the probability of every event in every $\mathcal{F}_t$, but the idea is that at time $t$ you know specifically which event you are in. For an easy example, you can think of flipping two ...
  • 9,427
7 votes
Accepted

Intersection multiplicity does not go down after restriction to closed subvariety: proof using filtrations

I believe the desired result is not true. Let $S=\mathbb{Q}[x,y,z]\big/(xy-z^n)$, with $n\geqslant 2$. For convenience we refer to the coset $p+(xy-z^n)\in S$ as $\bar{p}$, for each $p\in\mathbb{Q}[x,...
6 votes
Accepted

adapted process, translation between measurable and information?

It's not that $\mathcal{F}_t$ itself "contains information about the process up to time $t$" in the way you seem to be assuming. Rather, elements of $\mathcal{F}_t$ are allowed to depend on ...
5 votes
Accepted

Show that the hitting time of an open set for a right-continuous process is a stopping time

For fixed $\omega \in \{\tau<t\}$ there exists $\tau(\omega) \leq s < t$ such that $Y_s(\omega) \in H$. As $H$ is open, there exists some $\epsilon>0$ such that $B(Y_s(\omega),\epsilon) \...
  • 114k
5 votes
Accepted

Simple question regarding stopping times.

Since $S \leq T$, we have $$\{T \leq t\} = \{S \leq t\} \cap \{T \leq t\}.$$ Hence, $$F \cap \{T \leq t\} = \underbrace{(F \cap \{S \leq t\})}_{\in \mathcal{F}_t} \cap \underbrace{\{T \leq t\}}_{\...
  • 114k
5 votes
Accepted

Stopping Times, the $\inf$ is not a stopping time

Your way of writing the event is wrong. Note that $\inf \limits_n \frac{1}{n} = 0$, while $\frac{1}{n} \le 0$ never holds.
  • 19.3k
5 votes
Accepted

Meaning of measurableness

Here is an intuitive answer which may address your concern about $\sigma$-algebras and information. Assume that only $3$ mutually exclusive events may happen at time $T$. Let these be denoted by $\...
  • 1,335
5 votes
Accepted

What is the correct filtration?

$X_t$ and $Y_t$ will be Brownian motions under their natural filtrations; i.e. under $$\mathcal{F}_t^{X} = \sigma(X_s\, : \, s \leq t) \qquad \text{ and } \qquad \mathcal{F}_t^{Y} = \sigma(Y_s\, : \, ...
  • 10.3k
4 votes
Accepted

Exercise on algebra filtrations

I think that it is a matter of definition: I have looked at both M. Sweedler's book on Hopf algebras (p.230, 1969 edition) and E.Abe's book on Hopf algebras (p.20, 1977 edition) and they both seem to ...
  • 7,035
4 votes
Accepted

union of natural filtration vs union of right continuous filtration

$\mathcal{F}_t\subset\tilde{\mathcal{F}}_{t+\delta}$ for any $\delta>0$, so $ \bigcup_{t\geq 0}\mathcal{F}_t\subset \bigcup_{t\geq 0}\tilde{\mathcal{F}}_t$, and conversely the right-continuous ...
  • 1,590
4 votes

For a discrete stopping time $\tau$, $\mathcal{F}_\tau^X = \sigma(X(t\wedge\tau):t\ge 0).$

Let $(X_n)_{n \in \mathbb{N}}$ be a stochastic process and $\tau: \Omega \to \mathbb{N}$ an $\mathcal{F}^X$-stopping time. You have already shown that $X(\tau \wedge n)$ is $\mathcal{F}_{\tau}^X$-...
  • 114k
4 votes

Conditional expectation as a random variable new

According to the definition of the conditional expectation we need a random variable $E[X\mid F_n](\omega)$ for which $$\int_A E[X\mid F_n](\omega)\ dP=\int_A X(\omega)\ dP$$ for all $A\in F_n$and $E[...
  • 20k
4 votes
Accepted

Where is the Strong Markov property(SM) being used in the proof that augmented filtration of a Strong Markov process is right continuous?

You are right that the strong Markov property is not used in its full generality; somehow, it's an artifact of the definition of the (strong) Markov property by Karatzas & Shreve. They say that $(...
  • 114k
4 votes

Help understanding the definition of a "filtration" in probability theory

Sigma algebras are often thought of as containing "information". Conditioning on a larger sigma algebra corresponds to "knowing more" about the values of random variables (more things are measurable ...
  • 20.4k
4 votes
Accepted

Uniformly integrable martingale problem: typo (wrong filtration)?

No, it's not a typo; $(\mathcal{F}_{n \wedge \tau})_{n \in \mathbb{N}}$ is a filtration. It holds for any two stopping times $S \leq T$ that $\mathcal{F}_S \subseteq \mathcal{F}_T$ (see e.g. this ...
  • 114k
4 votes
Accepted

Conditional expectations of $X$ with respect to a filtration converges to $X$

Let $X_n=\mathbb{E}\left[X\mid\mathcal{F}_n\right]$. This is a closed martingale so it converges in $L^1$ and almost surely to $\tilde{X}$. For all $n\geq 1$ and $A\in\mathcal{F}_n$, the convergence ...
  • 813
4 votes
Accepted

Example/meaning of filtration on a group $(\mathbb{R},+)$

Very good that you want to try out the definition on one of the standard examples! However, as people have pointed out in comments, it turns out that the group $(\mathbb R, +)$ just does not serve as ...
3 votes
Accepted

Filtered objects and associated graded objects

For any two $R$ modules, $A$ and $B$, there are some extensions $M$ that fit into a long exact sequence of the form $0 \to A \to M \to B \to 0$. Up to equivalence they are classified by the $R$ module ...
  • 19.8k
3 votes
Accepted

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

$\bigvee_n \mathcal{F}_n$ is the $\sigma$-algebra generated by $\bigcup_n \mathcal{F}_n$ (i.e., smallest $\sigma$-algebra containing $\bigcup_n \mathcal{F}_n$). This is needed because in general, $\...
  • 82.7k
3 votes
Accepted

Augmentation generated by Brownian motion are right continuous?

The first equality is indeed a direct consequence of Lévy's backward convergence theorem. For the second one the reasoning goes as follows: Choose $h>0$ sufficiently small such that $t_{k-1} \leq t ...
  • 114k
3 votes

Sigma algebra generated by the stopped process.

Yes, the two $\sigma$-algebras coincide. For any discrete stopping time $\tau: \Omega \to \mathbb{N}_0$ the random variable $X_{\tau}$ is $\mathcal{F}_{\tau}$-measurable, and this implies that $$\...
  • 114k
3 votes

Exercise in measure theory/probability

The text gives you a hint for this: "For any given $\omega$ find an $\alpha \ldots$". To extend this hint a little: Let $\omega\in\Omega$, with $\omega=(\omega_1,\omega_2,\ldots)$ and with infinitely ...
  • 9,988
3 votes

Example of filtration in probability theory

This is a variation on @GEdgard example, that can be applied to real valued random variables. The probability plays no role. Let $X_j$ be the projection of $\mathbb{R}^n$ onto the the $j^{th}$ ...

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