# Tag Info

• 16.1k
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 $t_n$. ...
• 4,253

### 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 ...
• 181
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 ...
• 13.5k
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• 120k
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### 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 ...
• 331k
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.
• 20k
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• 13k
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### Is $Z_n = X_n Y_n$ a martingale?

I was somewhat surprised that the answer is negative. (However, see the answer by John Dawkins, who notes that $Z_n$ is a Martingale with respect to some filtration.) Let $\{\xi_n\}$ and $\{\eta_n\}$...
• 22k
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### 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 ...
• 1,610
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 ...
• 7,344

• 120k
Accepted

### Stopping time clarification

The idea is that $F_t$ consists of all events that depend on your stochastic process only up to time $t$. So to say that $\{\omega\in\Omega:\tau(\omega)\leq t\}\in F_t$ means that given any ...
• 331k
Accepted

### Tensor product commutes with associated graded

One can prove that $\phi_k$ is an isomorphism by noting that it maps a basis of the left hand side to a basis of the right hand side. The following bookkeeping device will be useful to find suitable ...
• 4,454

### 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 ...
• 20.8k
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 ...
• 120k
Consider a real-valued Brownian motion $(B_t)_{t \geq 0}$ in its natural filtration, $\left(\mathcal{F}_t\right)_{t \geq0}$. Then, if we define a new process $(X_t)_{t \geq 0} = (B_{t+1})_{t \geq 0}$, ...