Raising the bar

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

2. Generalized homogeneity. If $latex f(I)$ depends only on information $latex I,$ then $latex
   E_I(f(I)X)=f(I)E_I(X)$ (a function of known information is known and behaves like a constant). A special case is $latex E_I(f(I))=f(I)E_I(1)=f(I).$ With $latex f(I)=E_I(X)$ we get $latex E_I(E_I(X))=E_I(X).$ 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. $latex E_I(X+Y)=E_IX+E_IY.$

4. Law of iterated expectations (LIE). If we know about two information sets that $latex I_1\subset I_2,$ then $latex E_{I_1}E_{I_2}X=E_{I_1}X.$ 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 $latex X=EX+(X-EX)$ and denoted $latex \mu=EX,~\varepsilon=X-EX$ to obtain $latex X=\mu+\varepsilon$ with the property $latex E\varepsilon=0.$

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

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

Properties. 1) $latex E_I\varepsilon=E_IX-E_IX=0$ (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,

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

For the components in (1) we have

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

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