2094 CHAPTER 62. STOCHASTIC PROCESSES

Lemma 62.8.3 Let X (t) be a submartingale adapted to a filtration Ft . Let {rk} ⊆ [s, t) bea decreasing sequence converging to s. Then

{X (r j)

}∞

j=1 is uniformly integrable.

Proof: First I will show the sequence is equiintegrable. I need to show that for all ε > 0there exists λ large enough that for all n∫

[|X(rn)|≥λ ]|X (rn)|dP < ε.

Let ε > 0 be given. Since {X (r)}r≥0 is a submartingale, E (X (rn)) is a decreasing sequencebounded below by E (X (s)). This is because for rn < rk,

E (X (rn))≤ E (E (X (rk) |Fn)) = E (X (rk))

Pick k such that

E (X (rk))− limn→∞

E (X (rn))

=∣∣∣E (X (rk))− lim

n→∞E (X (rn))

∣∣∣< ε/2.

Then for n > k,∫[|X(rn)|≥λ ]

|X (rn)|dP =∫[X(rn)≥λ ]

X (rn)dP+∫[X(rn)≤−λ ]

−X (rn)dP

=∫[X(rn)≥λ ]

X (rn)dP+∫[X(rn)>−λ ]

X (rn)dP−∫

Ω

X (rn)dP

≤∫[X(rn)≥λ ]

X (rn)dP+∫[X(rn)>−λ ]

E (X (rk) |Fn)dP−∫

Ω

X (rn)dP

≤∫[X(rn)≥λ ]

X (rk)dP+∫[X(rn)>−λ ]

X (rk)dP−∫

Ω

X (rk)dP+ ε/2

=∫[X(rn)≥λ ]

X (rk)dP+∫[X(rn)≤−λ ]

(−X (rk))dP+ ε/2

=∫[|X(rn)|≥λ ]

|X (rk)|dP+ ε/2

≤∫[

sup{|X(r)|≥λ :r∈{r j}∞

j=1

}] |X (rk)|dP+ ε/2 (62.8.29)

From maximal inequalities of Theorem 60.6.4

P

([sup

r∈{rn,rn−1,··· ,r1}|X (r)| ≥ λ

])≤ 2E (|X (t)|+ |X (0)|)

λ≡ C

λ

and so, letting n→ ∞,

P

([sup

r∈{rn}∞n=1

|X (r)| ≥ λ

])≤ C

λ.