Discrete approximation of one step martingale

Question

Definitions:

Two real valued random variables $X_0$ and $X_1$ are called a one step martingale if $E[X_1| X_0] = X_0$.

We say the one step martingale is in $L^2$ if both $X_0$ and $X_1$ are in $L^2(P)$.

Question:

Given an $L^2$ one step martingale $(X_0, X_1)$ does there exist a sequence $(Y_0^n, Y_1^n)$ of $L^2$ one step martingales satisfying the following two conditions?

• $Y_0, Y_1$ are simple random variables, i.e. variables taking only a finite number of values.
• $Y_0^n, Y_1^n$ converge to $X_0, X_1$ respectively in $L^2$.

Answer 1

Let $\mathcal F_i^n$ be a sequence of $\sigma$-algebras increasing to $\mathcal F(X_i)$, the $\sigma$-algebra generated by $X_i$ and let $Y_1^n=\mathbb E(X_1|\mathcal F^n_1\vee \mathcal F^n_0)$ and $Y_0^n=\mathbb E(X_1|\mathcal F^n_0)$.

Then $Y^n_1$ and $Y^n_0$ converge pointwise a.s. to $X_1$ and $X_0$ by Doob's martingale convergence theorem. They therefore converge in $L^2$ to $X_1$ and $X_0$ also.

From the tower law, we see that $\mathbb E(Y^n_1|\mathcal F^n_0)=Y^n_0$.