Proof of Grouping Independence and the Fresh-Start Sigma-Algebra of an Independent-Increment Process

Step 0 (a Dynkin factorization principle). Call a collection of events a $\pi$-system if it contains $\Omega$ and is closed under pairwise intersection. We first record: if $\mathcal{A}$ and $\mathcal{B}$ are $\pi$-systems of events with $P(A\cap B)=P(A)\,P(B)$ for all $A\in\mathcal{A}$ and $B\in\mathcal{B}$, then the same identity holds for all $A\in\sigma(\mathcal{A})$ and $B\in\sigma(\mathcal{B})$, where $\sigma(\cdot)$ denotes the generated $\sigma$-algebra. Indeed, fix $B\in\mathcal{B}$ and let $\mathcal{L}=\{A\in\mathcal{F}:P(A\cap B)=P(A)P(B)\}$. Then $\Omega\in\mathcal{L}$; if $A,A'\in\mathcal{L}$ with $A\subseteq A'$ then $P((A'\setminus A)\cap B)=P(A'\cap B)-P(A\cap B)=(P(A')-P(A))P(B)$, so $A'\setminus A\in\mathcal{L}$; and if events $A_n\in\mathcal{L}$ increase to $A$, then writing $A=A_1\sqcup\bigsqcup_{n\ge1}(A_{n+1}\setminus A_n)$ and intersecting with $B$, countable additivity of $P$ (probability space) gives $P(A\cap B)=\lim_nP(A_n\cap B)=\lim_nP(A_n)P(B)=P(A)P(B)$, so $A\in\mathcal{L}$. Hence $\mathcal{L}$ is a $\lambda$-system containing $\mathcal{A}$, and Dynkin's $\pi$-$\lambda$ theorem gives $\sigma(\mathcal{A})\subseteq\mathcal{L}$. Now fix $A\in\sigma(\mathcal{A})$ and repeat the argument with $\mathcal{L}'=\{B\in\mathcal{F}:P(A\cap B)=P(A)P(B)\}\supseteq\mathcal{B}$ to obtain $\sigma(\mathcal{B})\subseteq\mathcal{L}'$.

Part (a). If $I$ or $J$ is empty the claim is trivial, every event being independent of $\varnothing$ and $\Omega$; so assume both are nonempty. Let $\mathcal{A}_I$ consist of $\Omega$ together with all finite intersections $\bigcap_{h\in I'}\{\xi_h\in B_h\}$ with $I'\subseteq I$ finite nonempty and each $B_h$ a Borel set, and define $\mathcal{A}_J$ likewise. Each is a $\pi$-system: the intersection of two such events is again one, merging the index sets and replacing $B_h$ by the intersection of the two Borel sets where an index occurs in both. Moreover $\sigma(\mathcal{A}_I)=\sigma(\xi_h:h\in I)$, since the events $\{\xi_h\in B\}$ with $h\in I$ and $B$ Borel generate the latter and lie in $\mathcal{A}_I$. For $A=\bigcap_{h\in I'}\{\xi_h\in B_h\}\in\mathcal{A}_I$ and $B=\bigcap_{h\in J'}\{\xi_h\in B'_h\}\in\mathcal{A}_J$, the independence of $\xi_1,\dots,\xi_q$ - applied with the Borel sets $B_h$ for $h\in I'$, $B'_h$ for $h\in J'$ ($I'$ and $J'$ being disjoint), and $\mathbb{R}$ for every other index, and then again with $\mathbb{R}$ outside $I'$ alone and outside $J'$ alone - gives

The cases $A=\Omega$ or $B=\Omega$ are trivial. Step 0 now yields the independence of $\sigma(\xi_h:h\in I)$ and $\sigma(\xi_h:h\in J)$, which is part (a).

Part (b). Let $\mathcal{C}_{\mathrm{past}}$ consist of $\Omega$ and all finite intersections $\bigcap_{i=1}^{p}\{X_{s_i}\in B_i\}$ with $0\le s_1<\dots<s_p\le r$ and $B_i$ Borel, and let $\mathcal{C}_{\mathrm{fut}}$ consist of $\Omega$ and all finite intersections $\bigcap_{j=1}^{m}\{X_{r+u_j}-X_r\in B'_j\}$ with $0<u_1<\dots<u_m$ and $B'_j$ Borel. Both are $\pi$-systems, exactly as in part (a). Also $\sigma(\mathcal{C}_{\mathrm{past}})=\mathcal{F}^X_r$, the natural filtration at $r$ being generated by the variables $X_s$ with $s\le r$; and $\sigma(\mathcal{C}_{\mathrm{fut}})=\sigma(X_{r+u}-X_r:u\ge0)$, since the increment for $u=0$ is the constant $0$, whose level events are $\varnothing$ or $\Omega$, so omitting $u=0$ loses nothing. By Step 0 it suffices to prove $P(A\cap B)=P(A)P(B)$ for $A\in\mathcal{C}_{\mathrm{past}}$ and $B\in\mathcal{C}_{\mathrm{fut}}$, the cases $A=\Omega$ or $B=\Omega$ being trivial; so fix $A=\bigcap_{i}\{X_{s_i}\in B_i\}$ and $B=\bigcap_{j}\{X_{r+u_j}-X_r\in B'_j\}$ as displayed.

Let $0=v_0<v_1<\dots<v_q$ enumerate, in increasing order, the distinct elements of $\{0\}\cup\{s_1,\dots,s_p\}\cup\{r\}\cup\{r+u_1,\dots,r+u_m\}$, and set $D_h=X_{v_h}-X_{v_{h-1}}$ for $h\in\{1,\dots,q\}$ (differences of random variables are random variables, as noted in the definition of independent increments). By the independent-increments property applied to the partition $v_0<v_1<\dots<v_q$, the family $D_1,\dots,D_q$ is independent. Put $I=\{h:v_h\le r\}$ and $J=\{h:v_{h-1}\ge r\}$; these are disjoint, since $v_h\le r\le v_{h-1}$ is impossible for $v_{h-1}<v_h$.

For each $j$: since $r$ and $r+u_j$ are partition points, telescoping gives $X_{r+u_j}-X_r=\sum_{h\in J:\,v_h\le r+u_j}D_h$, a finite sum of the variables $D_h$ with $h\in J$; hence $B\in\sigma(D_h:h\in J)$, finite sums of the generating variables being measurable with respect to the generated $\sigma$-algebra.

For each $i$: on the event $\{X_0=c\}$, telescoping from $v_0=0$ gives $X_{s_i}=c+\sum_{h\in I:\,v_h\le s_i}D_h$. Define

The events $A$ and $\tilde{A}$ agree on $\{X_0=c\}$, whose complement is a null event by hypothesis, so $P(A)=P(\tilde{A})$ and $P(A\cap B)=P(\tilde{A}\cap B)$. By part (a) applied to the family $D_1,\dots,D_q$ and the disjoint index sets $I$ and $J$,

which completes part (b).