2142 CHAPTER 63. THE QUADRATIC VARIATION OF A MARTINGALE

ThusνM+N ([s, t])≤ 2(νM ([s, t])+νN ([s, t]))

By regularity of the measures, this continues to hold with any Borel set F in place of [s, t].

Theorem 63.5.5 The integral is well defined and has a continuous version which is a localmartingale. Furthermore it satisfies the Ito isometry,

E

(∣∣∣∣∣∣∣∣∫ t

0f dM

∣∣∣∣∣∣∣∣2U

)=∫

Ω

∫ t

0f (s)2 d [M] (s)dP

Let the norm on GN ∩GM be the maximum of the norms on GN and GM and denote by ENand EM the elementary functions corresponding to the martingales N and M respectively.Define GNM as the closure in GN ∩GM of EN ∩EM . Then for f ,g ∈ GNM,

E((∫ t

0f dM,

∫ t

0gdN

))=∫

Ω

∫ t

0f gd [M,N] (63.5.20)

Proof: It is clear the definition is well defined because if { fn} and {gn} are two se-quences of elementary functions converging to f in L2

(Ω;L2 ([0,T ] ,ν (·))

)and if

∫ 1t0 f dM

is the integral which comes from {gn} ,

∫Ω

∣∣∣∣∣∣∣∣∫ 1t

0f dM−

∫ t

0f dM

∣∣∣∣∣∣∣∣2 dP

= limn→∞

∫Ω

∣∣∣∣∣∣∣∣∫ t

0gndM−

∫ t

0fndM

∣∣∣∣∣∣∣∣2 dP

≤ limn→∞

∫Ω

∫ T

0||gn− fn||2 dνdP = 0.

Consider the claim the integral has a continuous version. Recall Theorem 62.9.4, partof which is listed here for convenience.

Theorem 63.5.6 Let {X (t)} be a right continuous nonnegative submartingale adapted tothe normal filtration Ft for t ∈ [0,T ]. Let p≥ 1. Define

X∗ (t)≡ sup{X (s) : 0 < s < t} , X∗ (0)≡ 0.

Then for λ > 0

P([X∗ (T )> λ ])≤ 1λ

p

∫Ω

X (T )p dP (63.5.21)

Let { fn} be a sequence of elementary functions converging to f in

L2 (Ω;L2 ([0,T ] ,ν (·))

).