Some inverse problems involving conditional expectations

Abstract

Let (Ω, F, P) be a probability space, let H be a sub-σ-algebra of F, and let Y be positive and H-measurable with E[ Y] = 1. We discuss the structure of the convex set CE( Y; H) = { X ∈ p F: Y = E[ X| H]} of random variables whose conditional expectation given H is the prescribed Y. Several characterizations of extreme points of CE( Y; H) are obtained. A necessary and sufficient condition is given in order that CE( Y; H) be the closed, convex hull of its extreme points. For the case of finite F we explicitly calculate the extreme points of CE( Y; H), identify pairs of adjacent extreme points, and characterize extreme points of CE( Y; H) ⌢ CE( Z; G), where G is a second sub-σ-algebra of F and Z ∈ p G. When H = σ( Y) and appropriate topological hypotheses hold, extreme points of CE( Y; H) are shown to be in explicit one-to-one correspondence with certain left inverses of Y. Finally, it is shown how the same approach can be applied to the problem of extremal random measures on R + with a prescribed compensator, to deduce that the number of extreme points is zero or one.