Kenneth Kuttler

27.3. CONDITIONAL EXPECTATION, SUB-MARTINGALES 741

Proposition 27.3.2 Suppose∫Fp1×Fp2 |x|dλ (X,Y ) (x) < ∞. Then E (X|y) exists for

λY a.e. y and ∫Fp2

E (X|y)dλY =∫Fp1xdλX (x) = E (X).

Proof: ∞ >∫Fp1×Fp2 |x|dλ (X,Y ) =

∫Fp2

∫Fp1 |x|dλX|y (x)dλY (y) and so∫

Fp1|x|dλX|y (x)< ∞,

λY a.e. Now∫Fp2 E (X|y)dλY =

=∫Fp2

∫Fp1xdλX|y (x)dλY (y) =

∫Fp1×Fp2

xdλ (X,Y )

=∫Fp1

∫Fp2xdλY |x (y)dλX (x) =

∫Fp2xdλX (x) = E (X) ■

Definition 27.3.3 Let {Xn} be any sequence, finite or infinite, of random variableswith values in R which are defined on some probability space, (Ω,S ,P). We say {Xn} is amartingale if

E (Xn|xn−1, · · · ,x1) = xn−1

and we say {Xn} is a sub-martingale if

E (Xn|xn−1, · · · ,x1)≥ xn−1.

Recall Lemma 10.15.1, Jensen’s inequality. It is stated next for convenience.

Lemma 27.3.4 If φ : R → R is convex, then φ is continuous. Also, if φ is convex,µ(Ω) = 1, and f ,φ ( f ) : Ω→ R are in L1(Ω), then φ(

∫Ω

f du)≤∫

Ωφ( f )dµ .

Next is the notion of an upcrossing.

Definition 27.3.5 Let {xi}Ii=1 be any sequence of real numbers, I ≤ ∞. Define an

increasing sequence of integers {mk} as follows. m1 is the first integer ≥ 1 such thatxm1 ≤ a, m2 is the first integer larger than m1 such that xm2 ≥ b, m3 is the first integerlarger than m2 such that xm3 ≤ a, etc. Then each sequence,

{xm2k−1 , · · · ,xm2k

}, is called an

upcrossing of [a,b].

Here is a picture of an upcrossing.

Proposition 27.3.6 Let {Xi}ni=1 be a finite sequence of real random variables defined

on Ω where (Ω,S ,P) is a probability space. Let U[a,b] (ω) denote the number of upcross-ings of Xi (ω) of the interval [a,b]. Then U[a,b] is a random variable, in other words, anonnegative measurable function.

27.3. CONDITIONAL EXPECTATION, SUB-MARTINGALES 741Proposition 27.3.2 Suppose fi; gr |w|dAx,y) (x) < ee. Then E(X|y) exists forAy ae. y andE(X|y)dly -[ adhx (x) =E(X).FP2 FPIProof: 0 > fg) grr |@|dA(x,¥) = Sper Spr |@|dA x \y (x) day (y) and so[,, |elddxiy @) <=,Ay a.e. Now frp, E(X|y) day =[ . [ |, Bh xy (x) dy (y) = [ vy cam CAR XY)| | rdhyje(y)dax (x) = | adh x (x) = E(X)FP1 JRP2 FP2Definition 27.3.3 Le {X,,} be any sequence, finite or infinite, of random variableswith values in R which are defined on some probability space, (Q,.7,P). We say {X,} isamartingale ifE (Xn|Xn—1, aa x1) =Xn-1and we say {X,} is a sub-martingale ifE (Xn\Xn-1, aa x1) > Xn-1-Recall Lemma 10.15.1, Jensen’s inequality. It is stated next for convenience.Lemma 27.3.4 if @ : R > R is convex, then @ is continuous. Also, if @ is convex,u(Q) = 1, and f,o(f): Q— Rare in L'(Q), then 0(Jof du) < Jo O(f)du.Next is the notion of an upcrossing.Definition 27.3.5 Le: {xi}_, be any sequence of real numbers, I < 0. Define anincreasing sequence of integers {mx} as follows. my, is the first integer > 1 such thatXm, <a, my is the first integer larger than m, such that xm, > b, m3 is the first integerlarger than mz such that xm, <a, etc. Then each sequence, {Xm oe Xn, } , ls called anupcrossing of [a,b].Here is a picture of an upcrossing.Proposition 27.3.6 Let {X;}"_, be a finite sequence of real random variables definedon Q where (Q,./,P) is a probability space. Let Uig.y|(@) denote the number of upcross-ings of X;(@) of the interval [a,b]. Then Ujgy) is a random variable, in other words, anonnegative measurable function.