Equivalent characterization of the martingale property

Question

I saw in several books and articles that the following lemma has been used:

Let $M_t$ be a stochastic process such that $\mathbb{E}[|M_t|] < \infty$ for all $t$. Is that true that $M_t$ is a martingale (w.r.t its natural filtration) if and only if $\mathbb{E}[(M_{t_{n+1}} - M_{t_n})\prod_{k=1}^{n}f_k(M_{t_k})] = 0$ for all choices of $t_1 < t_2 < ... < t_{n+1}$ and for all choices of bounded measurable functions $f_1, ...., f_n$.

I wanted to know, is the above statement correct? Is there a reference for it? How does one prove such a statement? I know that it is true if we take some arbitrary $\mathcal{F}_{t_n}$ measurable R.V. instead of $\prod_{i=1}^{n}f_k(M_{t_k})$, but why such a product is sufficient?

Answer 1

Recall the monotone class theorem. Let $\mathscr A$ be a collection of subsets that contains $\Omega$ and is closed under intersection. Let $\mathscr H$ be a vector space of real-valued functions on $\Omega$ satisfying

1. if $A\in \mathscr A, \mathbf 1_A \in \mathscr H$
2. if $f_n \in \mathscr H$ are nonnegative and increase to a bounded function $f$, then $f\in \mathscr H$

Then $\mathscr H$ contains all bounded functions on $\Omega$ that are measurable with respect to $\sigma(\mathscr A)$.

Now, recall that $\mathcal F_t = \sigma(\text{elementary cylinders})$, where an elementary cylinder is a set of the form $B = \{x: x_{t_1}\in B_1,\dots,x_{t_n}\in B_n\}$ for $B_i\in \mathcal B(\mathbb R)$ and $0\leq t_1 <\dots<t_n \leq t$. Let $\mathscr H$ be the set of real valued functions satisfying $E[M_t f] = E[M_s f]$ for $0\leq s < t$. If the mentioned property in your post holds, then clearly $\mathscr H$ satisfies the conditions of the monotone class theorem, and the thesis follows.