I have a problem with a tricky exercise in stochastic process that take negativ integers for the time.
Let be $(U_{t})_{t \in T}$ sequence of iid random variables in $L^2(\Omega,\mathcal{A},p)$ with $T=\{-1,-2,...\}$ non negativ set of integers and let be $S_t := U_t+U_{t+1}+...+U_{-1}$.
If we define $\mathcal{H}_t$:= $\sigma(S_t,U_{t-1},U_{t-2},...)$, prove that $(\mathcal{H}_t)_{t \in T}$ is a filtration.
The definition of filtration is that for every time $t1 <= t2$, $\mathcal{F_{t_{1}}} \subset \mathcal{F_{t_{2}}}$ so I can't find a way to prove it because we work with negativ integers in time and it changes the inclusion.
Well, $$ S_{-1}=U_{-1}, S_{-2}=U_{-1}+U_{-2}=S_{-1}+U_{-2},\ldots, $$ and in general, $S_{t}=S_{t+1}+U_{t}$. Thus, \begin{align} \mathcal{H}_{-1}&=\sigma(U_{-1},U_{-2},U_{-3},\ldots),\\ \mathcal{H}_{-2}&=\sigma(S_{-2},U_{-3},U_{-4},\ldots)=\sigma(S_{-2})\vee\sigma(U_{-3},U_{-4}\ldots) \\ &\qquad\subset\sigma(U_{-1},U_{-2})\vee \sigma(U_{-3},U_{-4},\ldots) \\ &=\mathcal{H}_{-1}, \\ \cdots \\ \mathcal{H}_{t}&=\sigma(S_{t},U_{t-1},U_{t-2},\ldots)=\sigma(S_{t})\vee\sigma(U_{t-1},U_{t-2}\ldots) \\ &\qquad\subset\sigma(S_{t+1},U_{t})\vee \sigma(U_{t-1},U_{t-2},\ldots) \\ &=\mathcal{H}_{t+1}. \end{align}