3.2 The Weak Stochastic Integral
3.2.3 An Extension of The Class of Integrands
A third step in our construction of the weak stochastic integral is to extend it to a class of families of random variables with only almost sure second moments. More specifically, we want to extend our theory to the following class of integrands:
Definition 3.2.17. Let Λ2,locw (M ; T ) denote the collection (of equivalence classes) of
families X = {X(r, ω, u) : r ∈ [0, T ], ω ∈ Ω, u ∈ U } of Hilbert space-valued maps satisfying the following conditions:
(1) X(r, ω, u) ∈ Φqr,u, for all r ∈ [0, T ] , ω ∈ Ω , u ∈ U ,
(2) X is qr,u-predictable, i.e. for each φ ∈ Φ , the mapping [0, T ] × Ω × U → R+
given by (r, ω, u) 7→ qr,u(X(r, ω, u), φ) is PT ⊗ B(U )-measurable.
(3) P ω ∈ Ω : Z T 0 Z U
qr,u(X(r, ω, u))2µ(du)λ(dr) < ∞
= 1. (3.35)
As before, we will sometimes denote Λ2,locw (M ; T ) by Λ2,locw (T ) when is clear to which
cylindrical martingale-valued measure M we are referring.
One can easily check that the space Λ2,locw (T ) is a linear space. We equip this space
with the vector topology T2,locM generated by the local base of neighbourhoods of zero {Γ,δ : > 0, δ > 0} , where Γ,δ is given by Γ,δ = X ∈ Λ2,locw (T ) : P ω ∈ Ω : Z T 0 Z U
qr,u(X(r, ω, u))2µ(du)λ(dr) >
≤ δ
.
3.2. The Weak Stochastic Integral 57
Hence, under the topology T2,locM , a sequence {X(n)}n∈N converges to X in Λ2,locw (T )
if and only if Z T
0
Z
U
qr,u(Xn(r, u) − X(r, u))2µ(du)λ(dr)→ 0,P as n → ∞. (3.36)
Proposition 3.2.18. The space (Λ2,locw (T ), T2,locM ) is a complete, metrizable topological
vector space.
Proof. On Λ2,locw (T ) , we introduce the translation invariant metric dΛ given by
dΛ(X, Y ) = E G Z T 0 Z U
qr,u(X(r, u) − Y (r, u))2µ(du)λ(dr)
, (3.37)
for all X, Y ∈ Λ2,locw (T ) , where G : R → R is given by G(x) = 1+xx , for each x ∈ R. It
is clear that dΛ is well-defined due to (3.35).
Let X ∈ Λ2,locw (T ) and > 0 . Because G is increasing and from Markov’s inequality
we have P Z T 0 Z U
qr,u(X(r, u))2µ(du)λ(dr) >
≤ 1 + E G Z T 0 Z U
qr,u(X(r, u)))2µ(du)λ(dr)
= 1 +
dΛ(X, 0) (3.38) On the other hand, because the function G is bounded by 1 , we have
dΛ(X, 0) = E G Z T 0 Z U
qr,u(X(r, u))2µ(du)λ(dr)
≤ 1 + P Z T 0 Z U
qr,u(X(r, u))2µ(du)λ(dr) <
+P Z T 0 Z U
qr,u(X(r, u))2µ(du)λ(dr) >
≤ + P Z T 0 Z U
qr,u(X(r, u))2µ(du)λ(dr) >
(3.39) Then, it follows from (3.38) and (3.39) that dΛ generates a vector topology equivalent
to T2,locM . Therefore, (Λ2,locw (T ), T2,locM ) is a metrizable topological vector space. The
proof of the completeness can be carried out by following similar arguments to those used in the proof of Proposition 2.4 of Bojdecki and Jakubowski [13]. Remark 3.2.19. In general the space Λ2,locw (T ) is not locally convex. This fact will
have important consequences for the construction of the strong stochastic integral (see Remark 3.3.35). Indeed, if P is an atomless measure (see Definition 1.12.7 of Bogachev [8], p.55) we can show that every convex neighbourhood of zero is identical to Λ2,locw (T ) ,
and hence Λ2,locw (T ) is not locally convex. To prove this, we will adapt the arguments
used in Remarque 1 of Badrikian [7], p.2.
Assume V is a convex neighbourhood of zero of Λ2,locw (T ) . Then, there exist some
, δ > 0 such that Γ,δ ⊆ V . Let Aδ given by
Aδ = X ∈ Λ2,locw (T ) : P ω ∈ Ω : Z T 0 Z U
qr,u(X(r, ω, u))2µ(du)λ(dr) > 0
≤ δ
.
Chapter 3. Stochastic Integration in Duals of Nuclear Spaces 58
Then, Aδ ⊆ Γ,δ ⊆ V .
As P is atomless, there exist n ∈ N and pairwise disjoint subsets Ω1, · · · , Ωn∈ F such
that P(Ωi) ≤ δ and Ω =Sni=1Ωi (see Theorem 1.12.9 of Bogachev [8], p.55).
Now, for each i = 1, . . . , n , let
Λi= X ∈ Λ2,locw (T ) : Z T 0 Z U
qr,u(X(r, ω, u))2µ(du)λ(dr) = 0, if ω /∈ Ωi
.
As P(Ωi) ≤ δ , then we have Λi ⊆ Aδ. Moreover, note that
Pn
i=1Λi = Λ 2,loc
w (T ) and
that for each a > 0 , aΛi = Λi.
Hence, for any a1, . . . , an such that ai > 0 and Pni=1ai = 1 , then n X i=1 aiΛi = n X i=1 Λi= Λ2,locw (T ),
and because Λi ⊆ Aδ, for all i = 1, . . . , n , then the convex hull of Aδ (see Schaefer
[93], p.39) is equal to Λ2,locw (T ) . This implies that the convex hull of V is equal to
Λ2,locw (T ) , because Aδ⊆ V . Now, since V is convex this implies V = Λ2,locw (T ) .
The extension of the weak stochastic integral to the elements of Λ2,locw (T ) will be
provided by the following result.
Theorem 3.2.20. Let X ∈ Λ2,locw (T ) . Then,
(1) There exists an increasing sequence {τn}n∈N of {Ft}-stopping times satisfying
limn→∞τn= T ( P-a.e.) and such that for each n ∈ N, 1[0,τn]X ∈ Λ
2 w(T ) .
(2) There exists a unique c`adl`ag real-valued locally zero-mean square integrable mar- tingale ˆIw(X) = { ˆItw(X)}t∈[0,T ] such that for any sequence of {Ft}-stopping times
{σn}n∈N satisfying limn→∞σn = T ( P-a.e.) and 1[0,σn]X ∈ Λ
2
w(T ) for each
n ∈ N, the process ˆIw(X) satisfies: ˆ
It∧σw n(X) = Itw(1[0,σn]X), ∀ t ∈ [0, T ], (3.40)
for all n ∈ N, where the process on the right-hand side of (3.40) is the weak stochastic integral of 1[0,σn]X .
Proof. To prove (1), for each n ∈ N define τn by
τn(ω) = inf t ∈ [0, T ] : Z t 0 Z U
qr,u(X(r, ω, u))2µ(du)λ(dr) ≥ n
, ∀ ω ∈ Ω, (3.41) with the convention inf ∅ = 0 . Then, {τn}n∈N is an increasing sequence of {Ft}-
stopping times satisfying limn→∞τn= T , P-a.e. The proof that 1[0,τn]X ∈ Λ
2
w(T ) , for
all n ∈ N follows from standard arguments (e.g see the proof of Proposition 2.3.8 of Pr´evˆot and R¨ockner [87]).
To prove (2). Let {σn}n∈N be a sequence of stopping times satisfying the assumptions
of the statement. Such a sequence exists by part (1).
Now, define ˆIw(X) = { ˆItw(X)}t∈[0,T ] by means of the following prescription: for t ∈
[0, T ] , let
ˆ
3.2. The Weak Stochastic Integral 59
where n ∈ N is such that σn ≥ t. Notice that if m ≥ n is such that σm ≥ t, then it
follows from Proposition 3.2.14 that P-a.e.
It∧σw n(1[0,σm]X) = Itw(1[0,σn](1[0,σm]X)) = Itw(1[0,σn]X). (3.43) Therefore the definition (3.42) is consistent. Moreover, it follows from (3.42) and (3.43) that ˆIw(X) satisfies (3.40).
The fact that ˆIw(X) is a c`adl`ag real-valued locally zero-mean square integrable mar- tingale follows from (3.40) and Theorem 3.2.8.
Finally, let {θn}n∈N is another sequence of stopping times satisfying the properties of
the statement. A similar argument to that used to obtain (3.43) shows that the defi- nition (3.42) given with respect to the sequence {θn}n∈N leads to an indistinguishable
processes. This proves the uniqueness of ˆIw(X) , and its independence of the sequence
of stopping times satisfying (3.40).
Definition 3.2.21. For every X ∈ Λ2,locw (T ) , we will call the process ˆIw(X) given in
Theorem (3.2.20) the weak stochastic integral of X . We will sometimes denote the process ˆIw(X) by nRt 0 R UX(r, u)M (dr, du) : t ∈ [0, T ] o .
The property (3.40) allow us to “transfer” the properties of the weak stochastic integral for integrands in Λ2w(T ) (see Section 3.2.2) to those in Λ2,locw (T ) . We summarize this
in the following result:
Proposition 3.2.22. Let X ∈ Λ2,locw (T ) . Then, all the assertions in Propositions
3.2.14, 3.2.15 and 3.2.16 are valid for the weak stochastic integral ˆIw(X) of X . As was shown for the weak stochastic integral for integrands in Λ2
w(T ) , we can also
prove that the extended weak stochastic integral map ˆIw : Λ2,locw (T ) → M2,locT (R),
X 7→ ˆIw(X) , is linear and continuous, where we recall that M2,locT (R) is the space of all locally zero-mean square integrable c`adl`ag martingales (see Section 1.2.2).
The linearity of the map ˆIw follows from (3.40) and the corresponding linearity of the map Iw : Λ2
w(T ) → M2T(R). The continuity follows from the following estimate that
can by proved by similar arguments to those used in the proof of Proposition 4.16 of Da Prato and Zabczyk [20], p.104-5.
Proposition 3.2.23. Assume X ∈ Λ2,locw (T ) . Then, for arbitrary a > 0 , b > 0 ,
P sup t∈[0,T ] Iˆ w t (X) > a ! ≤ b a2 + P Z T 0 Z U
qr,u(X(r, u))2µ(du)λ(dr) > b
.
Proposition 3.2.24. The extended weak stochastic integral mapping ˆIw : Λ2,locw (T ) →
M2,locT (R) is linear and continuous.
Proof. As the map ˆIw is linear, we need only to show its continuity. Let {X n}n∈N
be a sequence converging to X in Λ2,locw (T ) . As both Λ2,locw (T ) and M2,locT (R) are
metrizable, it is sufficient to prove that { ˆIw(Xn)}n∈N converges to ˆIw(X) in M2,locT (R).
Let , δ > 0 . As {Xn}n∈N converges to X in Λ2,locw (T ) , then there exists some N,δ ∈ N
such that for all n ≥ N,δ,
P Z T
0
Z
U
qr,u(X(r, u) − Xn(r, u))2µ(du)λ(dr) >
δ2
2
≤ δ
Chapter 3. Stochastic Integration in Duals of Nuclear Spaces 60
By linearity of the integral map, Proposition 3.2.23 and (3.44), for all n ≥ N,δ, we
have P sup t∈[0,T ] Iˆ w t (X) − ˆItw(Xn) > ! ≤ δ 2 + P Z T 0 Z U
qr,u(X(r, u) − Xn(r, u))2µ(du)λ(dr) >
δ2 2
≤ δ. And hence (see (1.17)) { ˆIw(Xn)}n∈N converges to ˆIw(X) in M2,locT (R).