2158 CHAPTER 63. THE QUADRATIC VARIATION OF A MARTINGALE

and so by Jensen’s inequality, |M (T )| ≤ E (|M (a)| |FT ) . Therefore,∫[|M(T )|≥λ ]

|M (T )|dP ≤∫[|M(T )|≥λ ]

E (|M (a)| |FT )dP

=∫[|M(T )|≥λ ]

|M (a)|dP. (63.7.34)

Now by Theorem 62.5.3,

P([|M (T )| ≥ λ ])≤ 1λ

E (|M (a)|)

and so since a given L1 function is uniformly integrable, there exists δ such that if P(A)< δ

then ∫A|M (a)|dP < ε.

Now choose λ large enough that

1λ

E (|M (a)|)< δ .

Then for such λ , it follows from 63.7.34 that for any stopping time bounded by a,∫[|M(T )|≥λ ]

|M (T )|dP < ε.

This shows M is DL.

Example 63.7.11 Let {X (t)} be a nonnegative submartingale with t→E (X (t)) right con-tinuous so {X (t)} can be considered right continuous. Then {X (t)} is DL.

To show this, let T be a stopping time bounded by a > 0. Then by the optional samplingtheorem, ∫

[X(T )≥λ ]X (T )dP≤

∫[X(T )≥λ ]

X (a)dP

and now by Theorem 60.6.4 on Page 1969

P([X (T )≥ λ ])≤ 1λ

E(X+

a).

Thus if ε > 0 is given, there exists λ large enough that for any stopping time, T ≤ a,∫[X(T )≥λ ]

X (T )dP≤ ε

Thus the submartingale is DL.Now with this preparation, here is the Doob Meyer decomposition.