Lecture Notes - Department of Mathematics and Statistics - Queen's ...

32CHAPTER 4. MARTINGALES AND FOSTER-LYAPUNOV CRITERIA FOR STABILIZATION OF MARKOV CHAINS
Thus, the conditional expectation given by the sigma-field generated by A (which is F A = {∅, Ω, A, Ω \ A} is
given by:
E[X|F A ] = X A 1 ω∈A + X A C1 ω/∈A .
We note that conditional probability can be written as:
P(A|G) = E[1 {x∈A} |G],
hence, conditional probability is a special case of conditional expectation.
Let {B i } be a finite (or countable) partition of Ω which generate the σ−field G. It must be that E[X|ω 1 ∈ B 1 ] =
E[X|ω 2 ∈ B 1 ] since otherwise E[X|G] would not be measurable on G. As such for all B ∈ G, we have that
∫
B
∫
X(ω)P(dω) =
B
∫
E[X|B]P(dω) = E[X|B] P(dω) = P(B)E[X|B]
B
The notion of conditional expectation is key for the development of stochastic processes which evolve according
to a transition kernel. This is useful for optimal decision making when a partial information is available with
regard to a random variable.
The following discussion is optional until the next subsection.
Theorem 4.1.1 (Radon-Nikodym) Let µ and ν be two σ−finite positive measures on (Ω, F) such that ν(A) =
0 implies that µ(A) = 0 (that is µ is absolutely continuous with respect to ν). Then, there exists a measurable
function f : Ω → R such that for every A:
∫
µ(A) = f(ω)ν(dω)
A
The representation above is unique, up to points of measure zero. With the above discussion, the conditional
expectation X = E[X|F ′ ] exists for any sub-σ-field, subset of F.
It is a useful exercise to now consider the σ-field generated by an observation variable, and what a conditional
expectation means in this case.
Theorem 4.1.2 Let X be a X valued random variable, where X is a complete, separable, metric space and Y be
another Y−valued random variable, Then, a random variable X is F Y (the σ−field generated by Y ) measurable
if and only if there exists a measurable function f : Y → X such that X = f(Y (ω)).
Proof: We prove the more difficult direction. Suppose X lived in a countable space {a 1 , a 2 , . . . , a n , . . . }. We
can write A n = X −1 (a n ). Since X is measurable on F Y , A n ∈ F Y . Let us now make these sets disjoint: Define
C 1 = A 1 and C n = A n \ ∪ n−1
i=1 A i. These sets are disjoint, but they all live in F Y . Thus, we can construct a
measurable function defined as f(ω) = a n if ω ∈ C n :
The issue is now what happens when X lives in an uncountable, complete, separable, metric space such as R.
We can define countable sets which cover the entire state space (such as intervals with rational endpoints to
cover R). Let us construct a sequence of partitions Γ n = {B i,n , ∪ i B i,n = X} which gets finer and finer (such
as for R: [⌊X(ω)⌋, ⌈X(ω)n⌉)/n). We can define a sequence of measurable functions f n for every such partition.
Furthermore, f n (ω) converges to a function f(ω) pointwise, the limit is measurable.
⋄
With the above, the expectation E[X|Y = y 0 ] can be defined, this expectation is a measurable function of Y .
4.1.2 Some Properties of Conditional Expectation:
One very important property is given by the following.
Iterated expectations: