889

Proof: It is clear the definition is well defined because if { fn} and {gn} are twosequences 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 31.4.3, partof which is listed here for convenience.

Theorem 33.0.6 Let {X (t)} be a right continuous nonnegative submartingale ad-apted to the 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 (33.13)

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

L2 (Ω;L2 ([0,T ] ,ν (·))

).

Then letting

Xτ ln,m (t) =

∥∥∥∥∫ t

0( fn− fm)dMτ l

∥∥∥∥U,

Xn,m (t) =∥∥∥∥∫ t

0( fn− fm)dM

∥∥∥∥U=

∥∥∥∥∫ t

0fndM−

∫ t

0fmdM

∥∥∥∥U

It follows Xτ ln,m is a continuous nonnegative submartingale and from Theorem 31.4.3 just

listed,

P([

Xτ l∗n,m (T )> λ

])≤ 1

λ2

∫Ω

Xτ ln,m (T )2 dP

≤ 1

λ2

∫Ω

∫ T

0| fn− fm|2 d [Mτ l ]dP

≤ 1

λ2

∫Ω

∫ T

0| fn− fm|2 d [M]dP

Letting l→ ∞,

P([

X∗n,m (T )> λ])≤ 1

λ2

∫Ω

∫ T

0| fn− fm|2 d [M]dP

Therefore, there exists a subsequence, still denoted by { fn} such that

P([

X∗n,n+1 (T )> 2−n])< 2−n