Stochastic integration starts with a Q Wiener process having values in a separable
Hilbert space U. Thus it satisfies the following definition.
Definition 64.1.1Let W
(t)
be a stochastic process with values in U, a real separableHilbert space which has the properties that t → W
(t,ω )
is continuous. Whenevert_{1}< t_{2}<
⋅⋅⋅
< t_{m}, the increments
{W (ti) − W (ti−1)}
are independent, W
(0)
= 0, andwhenever s < t,
ℒ (W (t) − W (s)) = N (0,(t− s)Q)
which means that whenever h ∈ H,
ℒ ((h,W (t)− W (s))) = N (0,(t− s)(Qh,h ))
Also
E ((h1,W (t)− W (s))(h2,W (t)− W (s))) = (Qh1,h2)(t− s).
Here Q is a nonnegative trace class operator. Recall this means
∑∞
Q = λiei ⊗ ei
i=1
where
{ei}
is a complete orthonormal basis, λ_{i}≥ 0, and
∞∑
λi < ∞
i=1
Such a stochastic process is called a Q Wiener process.
Recall that such Wiener processes are always of the form
∞∑
ψk(t)Jgk
k=1
where J is a Hilbert Schmidt operator from a suitable space U_{0} to U and the
ψ_{k} are real independent Wiener processes described earlier. This follows from
Theorem 63.5.4 where you let U_{0}⊆ U be such that for J the inclusion map,
Je_{k} =
√ ---
λk
e_{k} for Q = ∑_{k}λ_{k}e_{k}⊗ e_{k}, the e_{k} an orthonormal set in U. Thus
and that Ψ and Ψ^{∗} are Hilbert Schmidt together.
The filtration will continue to be denoted by ℱ_{t}. It will be defined as the following
normal filtration in which
----------------------------
σ (W (s) − W (r) : 0 ≤ r < s ≤ u)
is the completion of σ
(W (s)− W (r) : 0 ≤ r < s ≤ u)
.
----------------------------
ℱt ≡ ∩u>tσ (W (s)− W (r) : 0 ≤ r < s ≤ u). (64.1.2)
(64.1.2)
and σ
(W (s)− W (r) : 0 ≤ r < s ≤ u)
denotes the σ algebra of all sets of the
form
(W (s)− W (r))−1(Borel)
where 0 ≤ r < s ≤ u.
Definition 64.1.2Let Φ
(t)
∈ℒ
(U, H)
be constanton each interval, (t_{m},t_{m+1}]
determined by a partition of
[a,T]
, 0 ≤ a = t_{0}< t_{1}
⋅⋅⋅
< t_{n} = T. Then Φ
(t)
is said to beelementaryif also Φ
(tm)
is ℱ_{tm}measurable and Φ
(tm)
equals a sum of theform
∑m
Φ(tm)(ω) = Φj XAj
j=1
where Φ_{j}∈ℒ
(U, H)
, A_{j}∈ℱ_{tm}. What does the measurability assertion mean? It meansthat if O is an open (Borel) set in the topological space ℒ
(U,H )
, Φ
(tm)
^{−1}
(O )
∈ℱ_{tm}.Thus an elementary function is of the form
n∑−1
Φ(t) = Φ (tk)X (tk,tk+1](t).
k=0
Then for Φ elementary, the stochastic integralis defined by
∫ t n∑−1
a Φ (s)dW (s) ≡ Φ (tk)(W (t∧ tk+1)− W (t∧ tk)).
k=0
It is also sometimes denoted by Φ ⋅ W
(t)
.
The above definition is the same as saying that for t ∈ (t_{m},t_{m+1}],
∫ t m∑−1
Φ (s)dW (s) = Φ (tk)(W (tk+1)− W (tk))
a k=0
+ Φ (tm )(W (t)− W (tm)). (64.1.3)
The following lemma will be useful.
Lemma 64.1.3Let f,g ∈ L^{2}
(Ω;H )
and suppose g is G measurable and f is ℱmeasurable whereℱ⊇G. Then
E ((f,g) |G) = (E (f|G ),g) a.e.
H H
Similarly if Φ is G measurable as a map into ℒ
(U,H)
with
∫
||Φ||2dP < ∞
Ω
and f is ℱ measurable as a map into U such that f ∈ L^{2}
(Ω;H )
, then
E (Φf|G) = ΦE (f|G).
Proof: Let A ∈G. Let
{gn}
be a sequence of simple functions, measurable with
respect to G,
m∑n
gn (ω ) ≡ ankXEnk (ω)
k=1
which converges in L^{2}
(Ω;H )
and pointwise to g.Then
∫ ∫
(E (f|G),g)H dP = lim (E (f|G),gn)H dP
A n→∞ A
∫ m∑n ( n ) ∫ m∑n n
= nli→m∞ A E (f|G),akXEnk H dP = lnim→∞ A E((f,ak)H |G)XEnkdP
∫ km=1 ∫ k=1( ( m ) )
∑ n (( n n) ) ∑ n n n
= nli→m∞ A E f,akXEk H |G dP = nli→m∞ AE f, akXE k |G dP
∫ k=1 ∫ ∫ k=1 H
= lim E ((f,gn)H |G)dP = lim (f,gn)H dP = (f,g)H dP
n→ ∞ A n→ ∞ A A
which shows
(E (f|G),g)H = E ((f,g)H |G)
as claimed.
Consider the other claim. Let
∑mn
Φn (ω) = ΦnkXEn (ω),Enk ∈ G
k=1 k
where Φ_{k}^{n}∈ℒ
(U,H )
be such that Φ_{n} converges to Φ pointwise in ℒ
(U,H )
and
also
∫
||Φ − Φ||2dP → 0.
Ω n
Then letting A ∈G and using Corollary 19.2.6 as needed,
∫
ΦE (f|G )dP
A
∫ ∫ m∑n
= nli→m∞ ΦnE (f|G)dP = nli→m∞ ΦnkE (f|G)XEnk dP
A A k=1
∑mn n∫ ∑mn n∫ ( )
= nli→m∞ Φ k A E (f |G)XEnkdP = nli→m∞ Φ k A E XEnkf|G dP
k=m1 ∫ ∫ m k=1
= lim ∑ nΦn X nf dP = lim ∑ n ΦnX nfdP
n→ ∞k=1 k A Ek n→∞ Ak=1 k Ek
∫ ∫ ∫
= lim Φnf dP = lim Φf dP ≡ E(Φf |G )dP
n→ ∞ A n→∞ A A
Since A ∈G is arbitrary, this proves the lemma. ■
Lemma 64.1.4Let J : U_{0}→ U be a Hilbert Schmidt operator and let W
(t)
be theresulting Wiener process
∞
W (t) = ∑ ψ (t)Jg
k=1 k k
where
{gk}
is an orthonormal basis for U_{0}. Let f ∈ H. Then considering one of theterms of the integral defined above,
( )
E (Φ (tk)(W (t∧ tk+1)− W (t ∧tk)),f)2
(( ))
= E (W (t∧ tk+1)− W (t ∧tk)),Φ(tk)∗ f 2
= (t∧ t − t∧ t)E (||||J∗Φ(t )∗f||||2 ).
k+1 k k U0
Proof: For simplicity, write ΔW_{k}
(t)
for W
(t∧ t )
k+1
− W
(t∧ t)
k
and
Δ_{k}
(t)
=
(t∧ t )
k+1
−
(t∧ t )
k
. If Φ
(t )
k
were a constant, then the result would follow right
away from the fact that W
(t)
is a Wiener process. Therefore, suppose for disjoint
E_{i},
m∑
Φ (tk)(ω) = ΦiXEi (ω)
i=1
where Φ_{i}∈ℒ
(U, H)
and E_{i}∈ℱ_{tk}. Then, since the E_{i} are disjoint,