Regular Martingales and Closable Martingales 📂Probability Theory

Regular Martingales and Closable Martingales

Definition

Let us assume a probability space $( \Omega , \mathcal{F} , P)$ and a martingale $\left\{ ( X_{n} , \mathcal{F}_{n} ) \right\}$ are given.

If for some integrable random variable $\eta$, $X_{n} = E ( \eta | \mathcal{F}_{n} )$ holds, then $\left\{ ( X_{n} , \mathcal{F}_{n} ) \right\}$ is called a regular martingale.
If there exists some integrable random variable $X_{\infty}$ that makes $\left\{ ( X_{n} , \mathcal{F}_{n} ): n = 1 , \cdots , \infty \right\}$ a martingale and if it is $\mathcal{F}_{\infty}$-measurable, then $\left\{ ( X_{n} , \mathcal{F}_{n} ) \right\}$ is called a closable martingale.

$\displaystyle \mathcal{F}_{\infty} = \bigotimes_{n=1}^{\infty} \mathcal{F}_{n}$ does not mean a tensor product but the smallest sigma field that includes all elements of all filtrations $\mathcal{F}_{n}$. It’s not particularly new; in fact, the smallest sigma field that contains all the open sets of a topological space $\Omega$ has been called a Borel sigma field. However, if that’s too difficult, it can simply be accepted as a sigma field that satisfies the condition of the filtration.

Explanation

Considering $E \left( E ( \eta | \mathcal{F}_{m} ) | \mathcal{F}_{n} \right)$ $\eta$ for $m > n$, since $\eta$ is $\mathcal{F}_{n}$-measurable and $\mathcal{F}_{n} \subset \mathcal{F}_{m}$, due to the smoothing property, it becomes $E \left( \eta E ( 1 | \mathcal{F}_{m} ) | \mathcal{F}_{n} \right) = E \left( \eta | \mathcal{F}_{n} \right) = X_{n}$, hence a regular martingale $\left\{ ( X_{n} , \mathcal{F}_{n} ) \right\}$ remains a martingale. Note here that $m$ is not $n+1$ but any integer greater than $n$.
The notion of $n = 1 , \cdots , \infty$, as always with infinity, looks simple but is not that easy. To verify that some $\mathcal{F}_{\infty}$-measurable random variable $X_{\infty}$ exists such that $\left\{ ( X_{n} , \mathcal{F}_{n} ) \right\}_{n \in \overline{\mathbb{N}}}$ becomes a martingale is tantamount to confirming for all natural numbers $n \in \mathbb{N}$ that $E \left( X_{\infty} | \mathcal{F}_{n} \right) = X_{n}$ holds.

Meanwhile, being a regular or a closable martingale is a necessary and sufficient condition. Closable martingales seem to have many useful properties, and regular martingales are easy to construct by proposing some $\eta$, and their equivalence is quite fortunate as one might easily guess.

Theorem

[1]: If it is a regular martingale, then it is a uniformly integrable martingale.
[2]: If it is a uniformly integrable martingale, then it is an L1 convergent martingale.
[3]: If it is an L1 convergent martingale, then it is a closable martingale.
[4]: If it is a closable martingale, then it is a regular martingale.

Proof

[1]

[2]

[3]

[4]

Assuming $\eta:= X_{\infty}$, since $X_{n} = E ( X_{\infty} | \mathcal{F}_{n} )$ holds, it is a closable martingale.

■