While reading the textbook "Martingale Methods in Financial Modelling" by Musiela and Rutkowski I am puzzled with a new definition (i.e. "the family of decompositions") that I never encountered and that I could not find within the book. Consider the space
$$ \Omega=\left\{\omega=\left(a_1,...,a_T\right)|\text{ for all }t=1,...,T,\text{ either }a_t=1\text{ or } a_t=0\right\} $$
and, for each $p\in\left(0,1\right)$, the probability measure
$$ \mathbb{P}\left\{\omega\right\}=p^{\sum_{t=1}^Ta_t}\,\left(1-p\right)^{T-\sum_{t=1}^Ta_t}. $$ Given $0<u<1<d$ consider the sequence of iid andom variables
$$ \xi_t\left(\omega\right) = u\,a_t+d\,(1-a_t), $$
and the sequence $S_t$ of random variables with $S_0=c>0$ and
$$ S_{t}\left(\omega\right) = \xi_t\left(\omega\right)\,S_{t-1}\left(\omega\right) ,~~t=1,...T. $$
The problem comes now: the authors say to consider the "family of decompositions of $\Omega$" $\mathcal{D}_t^{S}$ generated by the random variable $S_{\tau}$ with $\tau<t$, that is
$$ \mathcal{D}_t^{S}=\mathcal{D}\left(S_0,S_1,...,S_t\right). $$ Any guess on how the $\mathcal{D}\left(S_0,S_1,...,S_t\right)$ is formally defined?
It would be enough also to understand how the $\sigma-$algebra
$$ \mathcal{F}_t^{S}=\sigma\left(S_0,S_1,...,S_t\right) $$
is described. Which is, for example,
$$ \sigma\left(S_0,S_1\right)=\left\{...~~~~....\right\} ? $$
Usually when you have a question about a concept you meet in the book, check out Index. In this book Index states that decompositions can be found on p. 611. There you see that decomposition is just partition: collection of non-overlapping subsets whose union is the whole set. At the same time, that disagrees with the statement on p. 47 that family of decompositions is an increasing family of $\sigma$-fields, since these two objects are different. I guess that's a typo.
Now, to the filtration. So your sample space is $\Omega = \{0,1\}^n$ since $n$ for me is easier to deal with than $T^*$. Now, once you know all the prices $S_0,S_1,\dots, S_n$ you know the whole path $\omega\in \Omega$. If you just know a part of the price, you can only say to which set this path belong. For example, $D(S_0)$ consists of the following two sets: $\{0\}\times \{0,1\}^{n-1}$ and $\{1\}\times \{0,1\}^{n-1}$, and the corresponding $\sigma$-algebra is all sets obtained from the latter two by intersections, unions etc. Can you now answer your own question?