2690 CHAPTER 79. INCLUDING STOCHASTIC INTEGRALS
Now the right side converges to 0 as m,n→ ∞ and so there is a subsequence, denoted withthe index k such that whenever m > k,
E
(sup
t∈[0,T ]⟨Buk−Bum,uk−um⟩(t)
)≤ 1
2k
Note how this implies ∫Ω
∫ T
0⟨Buk−Bum,uk−um⟩dtdP≤ T
2k (79.1.17)
Then consider the martingales Mk (t) considered earlier. One of these is of the form
Mk =∫ t
0
(Φk ◦ J−1)∗Buk ◦ JdW
Then by the Burkholder Davis Gundy inequality and modifying constants as appropriate,
E((Mk−Mk+1)
∗)≤C
∫Ω
(∫ T
0
∥∥∥(Φk ◦ J−1)∗Buk−(Φk+1 ◦ J−1)∗Buk+1
∥∥∥2dt)1/2
dP
≤C∫
Ω
( ∫ T0 ∥Φk−Φk+1∥2 ⟨Buk,uk⟩
+∥Φk+1∥2 ⟨Buk−Buk+1,uk−uk+1⟩dt
)1/2
dP
≤ C∫
Ω
(∫ T
0∥Φk−Φk+1∥2 ⟨Buk,uk⟩dt
)1/2
+C∫
Ω
(∫ T
0∥Φk+1∥2 ⟨Buk−Buk+1,uk−uk+1⟩dt
)1/2
dP
≤C∫
Ω
supt⟨Buk,uk⟩1/2
(∫ T
0∥Φk−Φk+1∥2 dt
)1/2
dP
+C∫
Ω
supt⟨Buk−Buk+1,uk−uk+1⟩1/2
(∫ T
0∥Φk+1∥2 dt
)1/2
dP
≤C(∫
Ω
supt⟨Buk,uk⟩dP
)1/2(∫Ω
∫ T
0∥Φk−Φk+1∥2 dtdP
)1/2
+C(∫
Ω
supt⟨Buk−Buk+1,uk−uk+1⟩dP
)1/2(∫Ω
∫ T
0∥Φk+1∥2 dtdP
)1/2
From the above inequality, 79.1.15 and after adjusting the constants, the above is nolarger than an expression of the form C
( 12
)k/2which is a summable sequence. Then
∑k
∫Ω
supt∈[0,T ]
|Mk (t)−Mk+1 (t)|dP < ∞