# Martingales: why bother conditioning on the filtration when we can condition on random variables instead?

In some discussion regarding martingales, suppose we have a random process $$\{X_k\}_{k \geq 0}$$, authors use

$$\mathbb{E}[X_{k+1}|\mathcal{F}_k]$$

to denote the expectation with respect to a filtration, i.e., a increasing $$\sigma$$-algebra of the past random variables, $$\mathcal{F}_k = \sigma(X_0, X_1, \ldots, X_k\}$$.

A rough interpretation is that we are conditioning on increasing amount of information.

But why bother conditioning on a $$\sigma$$-algebra? (which is not unique, and can be very large, i.e., $$\mathcal{F}_k = \text{power set}$$, and contains things like $$\varnothing$$ which makes no sense in terms of information necessary)

Why not simply condition on the past random variables themselves? That is,

$$\mathbb{E}[X_{k+1}|X_k, X_{k-1}, \ldots, X_0]$$

Isn't the latter expression more interpretable and direct?

In what situation is conditioning on the filtration better than conditioning on the random variables themselves? Ultimately, I don't see why we need to condition on the filtration.

• You are probably thinking of discrete random variables. For general random variable the two conditional expectations are one and the same. What is your definition of $E(X_{k+1}|X_k,x_{k-1},...,X_0)$? – Kabo Murphy Jul 11 at 7:34

Suppose that we are studying a martingale differences sequence $$\left(X_k\right)_{k\geqslant 1}$$ with respect to a filtration $$\left(\mathcal F_k\right)_{k\geqslant 0}$$. For a fixed $$R>0$$, let $$X_{k,R}:= X_k\mathbf 1\{\lvert X_k\rvert\leqslant R\}$$. This is still measurable with respect to $$\sigma(X_k)$$ hence to $$\sigma\left(X_1,\dots,X_k\right)$$, but we may lose the martingale difference property (unless we are in the conditionally symmetric case).
Therefore, we define $$D_{k,R}:= X_k\mathbf 1\{\lvert X_k\rvert\leqslant R\}-\mathbb E\left[X_k\mathbf 1\{\lvert X_k\rvert\leqslant R\}\mid \mathcal F_{k-1}\right];$$ $$D'_{k,R}:= X_k\mathbf 1\{\lvert X_k\rvert\gt R\}-\mathbb E\left[X_k\mathbf 1\{\lvert X_k\rvert\gt R\}\mid \mathcal F_{k-1}\right]$$ and $$\left(D_{k,R},\mathcal F_k\right)$$, $$\left(D'_{k,R},\mathcal F_k\right)$$ are both martingales differences sequences statisfying $$X_k=D_{k,R}+D'_{k,R}$$.
If we had taken instead the natural filtrations of $$(X_k\mathbf 1\{\lvert X_k\rvert\leqslant R\})_{k\geqslant 1}$$ (respectively $$(X_k\mathbf 1\{\lvert X_k\rvert\gt R\})_{k\geqslant 1}$$), that is, $$D_{k,R}:=X_k\mathbf 1\{\lvert X_k\rvert\leqslant R\}-\mathbb E\left[X_k\mathbf 1\{\lvert X_k\rvert\leqslant R\}\mid \mathcal \sigma(X_{1,R},\dots ,X_{k-1,R})\right]$$ $$D'_{k,R}:=X_k\mathbf 1\{\lvert X_k\rvert\gt R\}-\mathbb E\left[X_k\mathbf 1\{\lvert X_k\rvert\gt R\}\mid \mathcal \sigma(X_1-X_{1,R},\dots ,X_{k-1}-X_{k-1,R})\right]$$ then we would not be sure that the equality $$X_k=D_{k,R}+D'_{k,R}$$ holds.