2244 CHAPTER 65. STOCHASTIC INTEGRATION
Theorem 65.5.3 The stochastic integral 65.5.7 is well defined. It also is a continuousmartingale and does not depend on the choice of J and U1. Furthermore,
E
(∣∣∣∣∫ t
aΦ(s)dW
∣∣∣∣2H
)=∫ t
aE(||Φ||2
L2(Q1/2U,H)
)ds
Proof: First of all, it is obvious that it is well defined in the sense that the same stochas-tic process is obtained from two different sequences of elementary functions. This followsfrom the isometry of Proposition 65.1.5 with U1 in place of U and Q1/2U in place of U0.Thus if {Ψn} and {Φn} are two sequences of elementary functions converging to Φ ◦ J−1
in L2([a,T ]×Ω;L2
(JQ1/2U,H
)),
E
(∣∣∣∣∫ T
a(Φn (s)−Ψn (s))dW
∣∣∣∣2H
)=∫ T
aE(||(Φn−Ψn)◦ J||2
L2(Q1/2U,H)
)ds (65.5.8)
Now for Φ ∈L2 (U1,H) and {gk} an orthonormal basis for Q1/2U,
||Φ◦ J||2L2(Q1/2U,H) ≡
∞
∑k=1|Φ(J (gk))|2H = ||Φ||2
L2(JQ1/2U,H)
because, by definition, {Jgk} is an orthonormal basis in JQ1/2U. Hence 65.5.8 reduces to∫ T
aE(||(Φn−Ψn)||2L2(JQ1/2U,H)
)ds
which is given to converge to 0. This reasoning also shows that the sequence{∫ t
a ΦndW}
is indeed a Cauchy sequence in L2 (Ω,H).Why is
∫ ta ΦdW a continuous martingale? The integrals
∫ ta ΦndW are martingales and
so, by the maximal estimate of Theorem 62.5.3,
P
([sup
t∈[a,T ]
∣∣∣∣∫ t
aΦndW −
∫ t
aΦmdW
∣∣∣∣H≥ λ
])≤ 1
λ2 E
(∣∣∣∣∫ T
a(Φn−Φm)dW
∣∣∣∣2)
=1
λ2
∫ T
aE(||(Φn−Φm)◦ J||2
L2(Q1/2U,H)
)ds
=1
λ2
∫ T
aE(||(Φn−Φm)||2L2(JQ1/2U,H)
)ds (65.5.9)
which is given to converge to 0 as m,n→ ∞. Therefore, there exists a subsequence {nk}such that
P
([sup
t∈[a,T ]
∣∣∣∣∫ t
aΦnk dW −
∫ t
aΦnk+1dW
∣∣∣∣H≥ 2−k
])≤ 2−k.
Consequently, by the Borel Cantelli lemma, there is a set of measure zero N such that if ω /∈N, then the convergence of
∫ ta Φnk dW to
∫ ta ΦdW is uniform on [a,T ] . Hence t→
∫ ta ΦdW
is continuous as claimed.