Proof of a Martingale with respect to Filtration

Question

I'm having a problem with stochastic analysis, needed in my Advanced Mathematical Finance Course. We have:

Let $(\zeta _k)_{k≥1}$ be a sequence of independent random variables with the expected value equal to 1. We are asked to prove that $$(∏_{k=1}^n \zeta_k)_{n≥1} $$

is a martingale with respect to filtration generated by this sequence.

Need any help with this problem, thank you!!

Answer 1

Assuming the random variables $\prod_{k=1}^n\zeta_k$ are integrable for each $n$, the result is a consequence of the two following facts about conditional expectation:

• If $\mathcal G\subset\mathcal F$ is a $\sigma$-algebra and $Y$ is $\mathcal G$-measurable, then for each random variable $X$ integrable such that $XY$ is integrable, we have $E[XY\mid\mathcal G]=YE[X\mid\mathcal G]$.
• If $X$ is independent of $\mathcal G$, then $E[X\mid \mathcal G]=E[X]$.