1948 CHAPTER 60. CONDITIONAL, MARTINGALES

Lemma 60.1.5 If X ≤ Y, then E (X |S )≤ E (Y |S ) a.e. Also

X → E (X |S )

is linear.

Proof: Let A ∈S . ∫A

E (X |S )dP≡∫

AXdP

≤∫

AY dP≡

∫A

E (Y |S )dP.

Hence E (X |S )≤ E (Y |S ) a.e. as claimed. It is obvious X → E (X |S ) is linear.

Theorem 60.1.6 (Jensen’s inequality)Let X (ω) ∈ I and let φ : I→ R be convex. Suppose

E (|X |) ,E (|φ (X)|)< ∞.

Thenφ (E (X |S ))≤ E (φ (X) |S ).

Proof: Let φ (x) = sup{anx+bn}. Letting A ∈S ,

1P(A)

∫A

E (X |S )dP =1

P(A)

∫A

XdP ∈ I a.e.

whenever P(A) ̸= 0. Hence E (X |S )(ω) ∈ I a.e. and so it makes sense to considerφ (E (X |S )). Now

anE (X |S )+bn = E (anX +bn|S )≤ E (φ (X) |S ).

Thussup{anE (X |S )+bn}

= φ (E (X |S ))≤ E (φ (X) |S ) a.e.

60.2 Discrete MartingalesDefinition 60.2.1 Let Sk be an increasing sequence of σ algebras which are subsets of Sand Xk be a sequence of real-valued random variables with E (|Xk|)< ∞ such that Xk is Skmeasurable. Then this sequence is called a martingale if

E (Xk+1|Sk) = Xk,

a submartingale ifE (Xk+1|Sk)≥ Xk,

and a supermartingale ifE (Xk+1|Sk)≤ Xk.

Saying that Xk is Sk measurable is referred to by saying {Xk} is adapted to Sk.