Terminology Filtered probability space

Question

What is the difference between a "filtered probability space" and a "Stochastic base"? I see both terms used but it is not clear to be me if there is a difference. Maybe one is complete but the other need not be?

Answer 1

In this paper [A Basic Course on General Stochastic Integration] from 1977 by M. Métivier and J. Pellaumail on page 1, a stochastic basis is defined as a family $(\Omega, \mathscr{F}, (\mathscr{F}_t)_{t\in T})$, where $(\Omega,\mathscr{F})$ is a measurable space and $(\mathscr{F}_t)_{t\in T}$ is an filtration on $\mathscr{F}$, for a given $T\subset \mathbb{R}$.

A filtered probability space e.g. $(\Omega, \mathscr{F}, \mathrm{P}, (\mathscr{F}_t)_{t\in T})$, where $(\Omega, \mathscr{F}, \mathrm{P})$ is a probability space, is denoted as "probabilized" stochastic basis (or only as stochastic basis).

They call a probabilized stochastic basis complete if $(\Omega, \mathscr{F}, \mathrm{P})$ is a complete probability space and if $A\in \mathcal{F}_t$ for all $t\in T$ when ever $A\in \mathscr{F}$ with $\mathrm{P}(A)=0$. If $0\in T$ (as mostly) then the latter is equivalent in requiring that $\mathscr{F}_0$ contains all the subsets of all $\mathrm{P}$-nullsets.

On page 2 in the book [Limit Theorems for Stochastic Processes] of J. Jacod and A.N. Shiryaev there is also a definition of a stochastic basis given. J. Jacod and A.N. Shiryaev define a stochastic basis as a filtered probability space, where the underlying filtration is right continuous. If additionally the filtration contains all nullsets and the underlying probability space is complete then they talk about a complete stochastic basis.

If a filtered probability space $(\Omega, \mathscr{F}, \mathrm{P}, (\mathscr{F}_t)_{t\in T})$ is a complete stochastic basis (according to the definition of Jacod and Shiryaev) then it is also refered as a filtered probability space satisfying the usual conditions [cf. Jacod and Shiryaev, p. 2]. Often in the latter case the filtration is also said to be satisfying the usual conditions see for instance in the book "Brownian motion and stochastic calculus" of Karatzas and Shreve.

Conclusion:

I think you can treat the word "stochastic basis" equivalent as a "filtered probability space" according to Métivier and Pellaumail. See also how stochastic basis is defined by the authors here https://www.sciencedirect.com/topics/mathematics/stochastic-basis (as I could see it is mostly defined as filtered probability space). But in my opinion, one should be careful when introducing these terms and not just assume that others know what you are talking about.

Some further remarks:

1. There is also this interesting terminology: Let $(\Omega, \mathscr{F}, \mathrm{P}, (\mathscr{F}_i)_{i\in I})$ be a filtered probability space, where $I=[0,+\infty)$ or $I=[0,T]$ $(T>0)$. Then the filtration $(\mathscr{F}_i)_{i\in I}$, is called normal if $\mathscr{F}_0$ contains all $\mathrm{P}$-nullsets and is right continuous, cf. [Stochastic Equations in Infinite Dimensions] by Da Prato and Zabczyk on p. 75.
2. In this book [Financial Mathematics] by Yuliya Mishura on p. 2 a family of non decreasing $\sigma$-fields $\mathscr{F}_0\subset \mathscr{F}_1\subset \ldots \mathscr{F}_T \subset \mathscr{F}$ on a probability space $(\Omega,\mathscr{F},\mathrm{P})$ is called a flow of $\sigma$-fields, a stochastic basis, or a filtration on $(\Omega,\mathscr{F},\mathrm{P})$. So this author seem to treat the word filtration and stochastic basis as the same, which is actually in line with the definition of Métivier and Pellaumail if you briefly think about it.
3. The terminology stochastic basis seem to appear also in a different context. I could find for instance this on the web: STOCHASTIC BASES IN FRECHÉT SPACES, Wojciech Herer

Answer 2

A filtered probability space is a probability space $(\Omega,\mathcal F,\mathbb P)$ equipped with a filtration $(\mathcal F_t)_{t\in\mathbb R_+}$.

Some authors define a stochastic basis as a filtered probability space whose filtration is right-continuous, that is $$ \forall t\in\mathbb R_+,\quad F_t=\cap_{s>t}\mathcal F_s. $$ (see for instance Jacod-Shiryaev:Limit theorems for stochastic processes, for a definition)

Other authors might define a stochastic basis as a filtered probability space whose filtration is right-continuous and complete, that is for all $t\in\mathbb R_+$, $F_t=\cap_{s>t}\mathcal F_s$ and $\mathcal F_t$ contains all the $\mathbb P$-null sets of $\mathcal F$.

As far as I am concerned, I never use the term "stochastic basis". I rather say "probability space equipped with a right-continuous filtration" for instance. It's longer, yes, but at least it is unequivocal.