It is time to generalize the integral. The following diagram illustrates the ingredients of
the next lemma.
W (t) ∈ U1 J← Q1∕2U Φ→ H
U
↓ Q1 ∕2
1∕2 J 1∕2
U1 ⊇ JQ U ←1−1 Q U
↓ Φ
Φn ↘
H
Lemma 64.5.1Let Φ ∈ L^{2}
( ( 1∕2 ))
[a,T]× Ω;ℒ2 Q U, H
and suppose also that Φ isprogressively measurable with respect to the usual filtration associated with the Wienerprocess
L
W (t) = ∑ ψ (t)Jg
k=1 k k
which has values in U_{1}for U_{1}a separable real Hilbert space such that J : Q^{1∕2}U → U_{1}isHilbert Schmidt and one to one,
{g }
k
an orthonormal basis in Q^{1∕2}U. Then lettingJ^{−1} : JQ^{1∕2}U → Q^{1∕2}U be the map described in Definition 64.4.1, it followsthat
Φ ∘J−1 ∈ L2([a,T]× Ω;ℒ (JQ1 ∕2U,H )).
2
Also there exists a sequence of elementary functions
{Φn }
having values in ℒ
(U1,H)
_{0}which converges to Φ ∘ J^{−1}in L^{2}
( ( ))
[a,T]× Ω;ℒ2 JQ1 ∕2U,H
.
Proof: First, why is Φ ∘J^{−1}∈ L^{2}
( ( 1∕2 ))
[a,T]× Ω; ℒ2 JQ U,H
? This follows from the
observation that A is Hilbert Schmidt if and only if A^{∗} is Hilbert Schmidt. In fact, the
Hilbert Schmidt norms of A and A^{∗} are the same. Now since Φ is Hilbert Schmidt, it
follows that Φ^{∗} is and since J^{−1} is continuous, it follows
( −1)
J
^{∗}Φ^{∗} =
( −1)
Φ ∘J
^{∗} is
Hilbert Schmidt. Also letting ℒ_{2} be the appropriate space of Hilbert Schmidt
operators,
Since Φ is given to be progressively measurable, so is Φ ∘J^{−1}. Therefore, the existence of
the desired sequence of elementary functions follows from Proposition 64.3.2 and Lemma
64.4.5. ■
Definition 64.5.2Let Φ ∈ L^{2}
([a,T]× Ω; ℒ2(Q1∕2U,H ))
and beprogressivelymeasurable where Q is a self adjoint nonnegative operator defined on U. LetJ : Q^{1∕2}U → U_{1}be Hilbert Schmidt. Then the stochastic integral
∫ t
ΦdW (64.5.7)
a
(64.5.7)
is defined as
∫ t
nli→m∞ ΦndW in L2(Ω;H )
a
where W
(t)
is a Wiener process
∞
∑ ψk(t)Jgk, {gk} orthonormal basis in Q1∕2U,
k=1
and Φ_{n}is an elementary function which has values in ℒ
(U ,H )
1
and converges to Φ ∘J^{−1}in
2( ( 1∕2 ))
L [a,T]× Ω;ℒ2 JQ U,H ,
such a sequence exists by Lemma 64.4.5and Proposition 64.3.2.
U
↓ Q1 ∕2
U1 ⊇ JQ1 ∕2U ←J Q1∕2U
1−1
↓ Φ
Φn ↘
H
It is necessary to show that this is well defined and does not depend on the choice of
U_{1} and J.
Theorem 64.5.3The stochastic integral 64.5.7is well defined. It also is a continuousmartingale and does not depend on the choice of J and U_{1}. Furthermore,
(|∫ | ) ∫
|| t ||2 t ( 2 )
E | a Φ (s)dW |H = a E ||Φ ||ℒ2(Q1∕2U,H) ds
Proof: First of all, it is obvious that it is well defined in the sense that the same
stochastic process is obtained from two different sequences of elementary functions. This
follows from the isometry of Proposition 64.1.5 with U_{1} in place of U and Q^{1∕2}U in place
of U_{0}. Thus if
{Ψ }
n
and
{Φ }
n
are two sequences of elementary functions converging to
Φ ∘ J^{−1} in L^{2}
([a,T ]× Ω;ℒ (JQ1 ∕2U,H ))
2
,
(| |2)
||∫ T || ∫ T ( 2 )
E (|| a (Φn (s)− Ψn (s))dW || ) = a E ||(Φn − Ψn )∘J ||ℒ2(Q1∕2U,H) ds
H
(64.5.8)