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Oct 17

## Conditional-mean-plus-remainder representation

Conditional-mean-plus-remainder representation: we separate the main part from the remainder and find out the remainder properties. My post on properties of conditional expectation is an elementary introduction to conditioning. This is my first post in Quantitative Finance.

## A brush-up on conditional expectations

1. Notation. Let $X$ be a random variable and let $I$ be an information set. Instead of the usual notation $E(X|I)$ for conditional expectation, in large expressions it's better to use the notation with $I$ in the subscript: $E_IX=E(X|I).$

2. Generalized homogeneity. If $f(I)$ depends only on information $I,$ then $E_I(f(I)X)=f(I)E_I(X)$ (a function of known information is known and behaves like a constant). A special case is $E_I(f(I))=f(I)E_I(1)=f(I).$ With $f(I)=E_I(X)$ we get $E_I(E_I(X))=E_I(X).$ This shows that conditioning is a projector: if you project a point in a 3D space onto a 2D plane and then project the image of the point onto the same plane, the result will be the same image as from single projecting.

3. Additivity. $E_I(X+Y)=E_IX+E_IY.$

4. Law of iterated expectations (LIE). If we know about two information sets that $I_1\subset I_2,$ then $E_{I_1}E_{I_2}X=E_{I_1}X.$ I like the geometric explanation in terms of projectors. Projecting a point onto a plane and then projecting the result onto a straight line is the same as projecting the point directly onto the straight line.

## Conditional-mean-plus-remainder representation

This is a direct generalization of the mean-plus-deviation-from-mean decomposition. There we wrote $X=EX+(X-EX)$ and denoted $\mu=EX,~\varepsilon=X-EX$ to obtain $X=\mu+\varepsilon$ with the property $E\varepsilon=0.$

Here we write $X=E_IX+(X-E_IX)$ and denote $\varepsilon=X-E_IX$ the remainder. Then the representation is

(1) $X=E_IX+\varepsilon.$

Properties. 1) $E_I\varepsilon=E_IX-E_IX=0$ (remember, this is a random variable identically equal to zero, not a number zero).

2) Conditional covariance is obtained from the usual covariance by replacing all usual expectations by conditional. Thus, by definition,

$Cov_I(X,Y)=E_I(X-E_IX)(Y-E_IY).$

For the components in (1) we have

$Cov_I(E_IX,\varepsilon)=E_I(E_IX-E_IE_IX)(\varepsilon-E_I\varepsilon)=E_I(E_IX-E_IX)\varepsilon=0.$

3) $Var_I(\varepsilon)=E_I(\varepsilon-E_I\varepsilon)^{2}=E_I(X-E_IX)^2=Var_I(X).$