Let \((\Omega, \mathscr{A}, P)\) be a probability space and let \(\left(X_{n}ight)_{n \in \mathbb{N}}\) be a sequence of independent identically distributed random variables such that \(P\left(X_{n}=0ight)=P\left(X_{n}=2ight)=\frac{1}{2}\). Set \(M_{k}:=\) \(\prod_{n=1}^{k} X_{n}\). Show that there exists no any filtration \(\left(\mathscr{A}_{n}ight)_{n \in \mathbb{N}}\) and no random variable \(M\) such that \(M_{k}=\mathbb{E}^{\mathscr{A}_{k}} M\).

[ compare this with Example 23.3 (xi).]

Remark. This example shows that not all martingales can be obtained as conditional expectations of a single function.

Data from example 23.3 (xi)

Transcribed Image Text:

(xi) Let (un)neN CL (A) L (A), Un > 0, be independent functions (in the sense of (x)). The products Pn=uu gale w.r.t. the filtration Un, neN, form a submartin- un) if, and only if, fun du >1 :o(u, 2,. for all n. This follows directly from LPm Pn+1 du = [ AP 1 APnn+1 d (23.7) [ 1.4pm d - [ Un+1 d du = [ Pn du [unt: Un+1 A