3. Probability Essentials 21
3.4. Generalized stochastic integration
In Section 3.2 we presented a quick overview on Itˆo-integration in R1. A natural extension of that concept should be the integration w.r.t. a random field (Xt)t∈T, whereT ⊂R1+q.As in the previous chapter we distinguish one variable the “time.”
When defining integration there are two natural questions: Which random fields can be chosen as good integrators? Which functions are good integrands?
Here, we will develop the classical theory of Walsh as in [Wal86]. First, we will speak about the integrators and give the definitions of martingale measures.
We will show that the Gaussian noises defined in Definition 3.3.15 are martingale measures. Secondly, we will speak about the integrands and the definition of the integral. However, we will do this definition only in the case of the Gaussian noises presented in the previous section.
Martingale Measures
We will start with an abstract setting: let E be a Polish space,E the Borelσ-field on E andA ⊂ E a certain subset.
Definition 3.4.1. Let (Ω,F,P) be a probability space. A mappingU : Ω× A →R is called an L2(P)-valued, σ-finite, signed measure on Aif
(a) there exist En%E with En=E|En ⊂ A,s.t. U(En)<∞ a.s. for alln∈N, (b) for all A∈ Athe random variableU(A) is inL2(P),
(c) for eachn∈Nthe mapping U is countably additive in an L2-sense:
U(A) +U(B) =U(A∪B) (a.s.), lim
j→∞U(Aj)−→L2 0 forAj & ∅and A, B, Aj ∈ En, j ∈N and A, B disjoint.
3.4 Generalized stochastic integration 39 It is always possible to extend U : Ω × A → R to all of Ω× E by setting U(A) = limn→∞U(A∩En) if this is a well-definedL2-limit for a given A∈ E.Set the value to ∞if this limit is not well-defined.
Let us show that the Gaussian noises from Definition 3.3.15 satisfy these condi-tions in the following sense:
Proposition 3.4.2. Let W˙ be a Gaussian noise, which is white in time. Then for anyt≥0 it is true thatWt: Ω× Bb(Rq)→Ris aσ-finite L2-valued measure on E and En(t) can be chosen independently of t≥0.
Proof. Fix t≥0.DefiningEn=Bq(0, n) ={x∈Rq:|x| ≤n}, observe E[(Wt(En))2] =tLk(1En,1En)<∞,
byk∈L1loc(R2q),which implies (a) of Definition 3.4.1. Finite additivity and count-able L2 additivity follow from Lemma 3.3.13. This shows (c). Then also (b) fol-lows, since by finite additivity for any A ∈ A there is an n∈ N with A ⊂En, so Wt(A)≤Wt(En) and by the aboveWt(En)∈L2, so also Wt(A)∈L2.
Remember that theR1-stochastic integral is defined for martingales. The concept of martingales also plays a crucial role here. We give the following definition.
Definition 3.4.3. Let (Ω,F,Ft,P) be a filtered probability space. A random variable M :R+× A ×Ω→Ris called an L2-valued martingale measure, if
(a) M0(A) = 0 almost surely for allA∈ A.
(b) (Mt(A))t≥0 is a martingale for all bounded sets A ∈ A w.r.t. the filtration (Ft)t≥0.
(c) Mt(·) is anL2-valued signed finite measure on Afor all t≥0.
For the Gaussian noise ˙W as in Definition 3.3.15 we define fort≥0:
F˜t= ˜FtW =σ(Ws(A) : 0≤s≤t, A∈ B(Rq))
as the natural filtration ( ˜Ft)t≥0.This filtration is not necessarily the one we want to work with. CallN theP-null sets and define
Ft=FtW = \
u>t
F˜u∪ N. (3.9)
We call the filtration (Ft)t≥0thenatural filtrationand note that it satisfies the usual conditions. Then we can define on the filtered probability space (Ω,F∞,Ft,P) the random variable
W˙ :R+× Bb(Rq)×Ω→R,
40 Probability Essentials by setting
W˙ (t, A, ω) := ˙W([0, t]×A)(ω).
For this random variable we can state the following lemma.
Lemma 3.4.4. The Gaussian noisesW˙ defined in Definition 3.3.15 are martingale measures with respect to their natural filtration and the martingales are continuous.
Proof. We want to check the three conditions of Definition 3.4.3. Proposition 3.4.2 gives (c); (a) is obvious and (b) is true, since ˙W is Gaussian and
E[(Wt(A)−Ws(A))Wu(B)] = 0 ∀u≤s, B ∈ Bb(Rq).
The continuity of the martingales follows from Lemma 3.3.14.
This ends the paragraph on martingale measures, which will be the integrators and we can go on to the definition of the integrands and the integral itself.
Stochastic Integral
In this subsection we use the results of the previous paragraph and assume that {Mt(A),Ft : t ≥ 0, A ∈ Bb(Rq)} is always a martingale measure obtained from a Gaussian noise from Definition 3.3.15. As in the Itˆo-theory first define the integral for a small class of functions.
Definition 3.4.5.
(a) A function g:Rq×R+×Ω→Ris called elementary if g(x, t, ω) =X(ω)1(a,b](t)1A(x) where X∈ Fa is bounded, A∈ B(Rq), 0≤a < b <∞.
(b) Linear combinations of elementary functions are called simple. The class of simple functions is denoted by S.LetP =σ(S) be thepredictable σ-field.
(c) Let gbe elementary as above and M anL2-valued martingale measure. The stochastic integral of g w.r.t. M is a martingale measure defined by
(g·M)t(B) = ( Z t
0
g dM)(B)(ω) :=X(ω)[Mt∧b(A∩B)−Mt∧a(A∩B)](ω), where t≥0, B∈ Bb(Rq), ω ∈Ω.
3.4 Generalized stochastic integration 41 Let us verify the claim in (c) that (g·M) is an L2-valued martingale measure.
Conditions (a) and (c) in Definition 3.4.3 are trivial, so concentrate on (b): (g· M)(B) is the difference of two stopped martingales, so by Proposition 3.2.5 it is a martingale. When checking the square-integrability, we have
E[(g·M)t(B)2] =E[X2(hM·(A∩B)it∧b− hM·(A∩B)it∧a)] (3.10)
=E[Lk(g1[0,t]×B, g1[0,t]×B)]
where the brackets h·i stand for the quadratic variation of the real martingale (Ms(A∩B))s≥0.We can also show that the martingales are continuous. As in the classical Itˆo-theory, the notion of quadratic variation plays an important role for the extension of the integral. We define for g∈S the norm:
kgk0,t :=E
Lk(g1[0,t]×Rq, g1[0,t]×Rq)1/2
, t≥0 and the function
kgk0:=
∞
X
n=1
2−n(1∧ kgk0,n).
This functionk·k0induces a metric onSbyd0(g, h) :=kg−hk0; identifyingg, h∈S withkg−hk0 = 0,we observe that (S, d0) is a linear space. By (3.10), we have that the mapping
IM :
((S, d0) →(M2,c,k · k), g 7→((g·M)t(Rq))t≥0
(3.11) is an isometry. Remember that M2,c is complete (see Proposition 3.2.7).
Lemma 3.4.6. The mappingIM can be extended to the space of predictable, square integrable processes PM onR+×Rq×Ω :
PM ={f ∈ P :kfk0,t <∞ ∀t≥0}.
Proof. We only need to show thatSis dense inPM w.r.t.d0. For this, see Theorem 2.5 in [Wal86] or Theorem 2 in [Dal99].
Definition 3.4.7 (Stochastic Integral). The extension of the mapping IM to the space PM is called the stochastic integral. Forf ∈ PM, t ≥0 it is denoted by the process
(f·M)t= Z t
0
Z
Rq
f(s, y, ω)M(ds dy).
We list some properties of this generalized stochastic integral in the next lemma.
Lemma 3.4.8. For g, h∈ PM we have that
(a) (λg+h)·M =λ(g·M) +h·M, where λ∈R,
42 Probability Essentials (b) g·M is again a martingale measure,
(c) h(g·M)·(A),(h·M)·(B)it=Lk(g1[0,t]×A, h1[0,t]×B).
Proof. The proof for g, h ∈ S was given before and in (3.10). The extension to g, h∈ PM is straightforward by approximation.
It might be worth noting that the definition of d0 here and k · kα in (3.7) are similar. The extension of ˙W was also obtained by a similar procedure. However, in the martingale approach we obtained that the stochastic integrals are again martingale measures and that integrands can be predictable. This is a result which was not attainable in the previous section.
For the generalized stochastic integrals there is also an analogue of the Fubini Theorem.
Proposition 3.4.9 (Stochastic Fubini Theorem). Let M be a martingale measure and (A,A, µ) be a finite measure space. Additionally, let f :Rq×R+×Ω×A→R be PM ⊗ A-measurable and for a T >0 assume
EP
"
Z
Ω×Rq×Rq×[0,T]×A
|f(x, t, ω, u)||f(y, t, ω, u)||QM(dxdydt)|µ(du)
#
<∞. (3.12) Then it holds P-almost surely for any 0≤t≤T:
Z
A
Z
Rq×[0,t]
f(x, s,·, u)M(ds dx)
!
µ(du) = Z
Rq×[0,t]
Z
A
f(x, s,·, u)µ(du)
M(ds dx).
This result can be shown using elementaryf first and the full proof can be found on page 296 of [Wal86].
The reader may have noticed that for the Gaussian noises the letters W and W˙ are used both. The precise notation and distinction between the two is cho-sen according to the following heuristic. If B is Brownian motion, we denote its temporal increments by dBt. The noises defined in Definition 3.3.15 are martin-gale measurese Wt(dx) = W(t, dx). We could denote their spatial increment by
∂tWt(dx), but using physics notation, it is more convenient to write ˙W(t, dx). In a weak setting of an SPDE the notation ˙W(t, x) will be used, but note that it is only a formal notation. What we have well-defined in this section is the integration w.r.t. time and space simultaneously and the notation will be RT
0
R
RqW(dt dx).