The Itô integral for martingales in vector latticesvtroitsky/positivity2017/talks/Grobler.pdf ·...

The Itô integral for martingales in vectorlattices

JJ Grobler, CCA Labuschagne

JJ Grobler, CCA Labuschagne The Itô integral for martingales in vector lattices

Motivation

In the classical setting of measure spaces, the Itô integral isclosely related to the Doob-Meyer decomposition ofsubmartingales:

If (Xt) is a submartingale, then (under certainassumptions),

Xt = Mt + At,

where (Mt) is a martingale, (At) is an increasing systemand the decomposition is unique.The measure involved with accompanying integral thatyields the “Itô integral" is determined by the increasingsystem (At).


Motivation

What is this measure and accompanying integral?


Notation and definitions

We use the same notation, definitions and assumptions asin the preceding lecture by Koos Grobler.A stochastic process in E is a function t 7→ Xt ∈ E, for t ∈ J,with J ⊂ R+ an interval.The stochastic process (Xt)t∈J is adapted to the filtration ifXt ∈ Ft for all t ∈ J.


The vector measure defined by an increasing process

DefinitionA stochastic process (At)t∈J is called an adapted increasing processif

1 (At)t∈J is adapted to the filtration (Ft,Ft)t∈J;

2 Aa = 0 if a is the left endpoint of J and As ≤ At for s ≤ t,s, t ∈ J.

3 (At)t∈J is right-continuous, i.e., At ↓ As as t ↓ s.

If J = [a,∞) the adapted increasing process (At) is called integrableif A∞ := supt∈J At ∈ E.


Given an increasing right-continuous bounded process 0 ≤ At

in E, we define a Stieltjes-Lebesgue measure µA on the algebraF(J) generated in J by the left-open and right-closed intervalsas follows:

1 µA(a, b] := Ab − Aa, a, b ∈ J.2 For any disjoint union

⋃nk=1 Ik of left-open and right-closed

intervals,

µA(

n⋃k=1

Ik) :=

n∑k=1

µA(Ik).

We can define µA(∅) = 0, but it actually follows from ∅ = (a, a].


TheoremLet (At)t∈J be an integrable adapted increasing process. ThenµA is an order countably additive E-valued measure on thealgebra F(J) of all finite unions of left-open and right-closedintervals in J. This means that if (En) is a sequence of disjointsubsets in F(J) such that

⋃∞n=1 En ∈ F(J) then

k∑n=1

µA(En) ↑ µA(

∞⋃n=1

En).


The next step is to extend this measure to the Borelσ-algebra B(J) generated by F(J).

The steps are exactly that of the Carathéodory extensionprocedure for extending a real-valued measure.

TheoremLet E be a Dedekind complete Riesz space separated by itsorder continuous dual E∼00. If (At)t∈J is an integrable adaptedincreasing process, then µA can be extended uniquely to acountably additive E-valued measure on the sigma-algebra B(J)of all Borel subsets of J.


Vector integration of vector-valued functions

Throughout the remainder of the presentation ourassumptions will be that E is a Dedekind complete vectorlattice with a weak order unit E, with separating ordercontinuous dual E∼00 and with a conditional expectation Fdefined on E, satisfying F(E) = E.

In addition, we will assume that E = (E∼00)∼00, i.e., we willassume that E is a perfect Riesz space, and that E isF-universally complete in Eu.


Let Φ be the set of all elements φ ∈ E∼00 such that|φ|(E) = 1.

We note that F and |φ| can be extended to Es (the proof forthe functional |φ| follows in the same manner as the prooffor F) and therefore, allowing the value +∞, the seminormswe are about to define make sense for all elements of Es.


The following constructions and facts are to be found in [20].For φ ∈ Φ, we define the Riesz seminorm

pφ(X) := |φ|(F(|X|))

and denote the set of all these seminorms by P.

Similarly, for such φ we define the Riesz seminorm

qφ(X) := (|φ|(F(|X|2)))1/2,

where for X ∈ E the product is formed in Eu.

We denote the set of all these seminorms by Q.

All these seminorms are order continuous Rieszseminorms.


The space L1 is defined to be the space (E, σ(P)) and wehave that L1 is the set of all X ∈ Es such that pφ(X) <∞ forall pφ ∈ P, equipped with the weak topology σ(P).

The proof of this fact in [20, Section 3] depends on theassumption that E = domF, which holds in this case by ourassumption that E is F-universally complete in Eu.

Similar to the case of L1, it follows that if L2 is the set of allX ∈ Es such that qφ(X) <∞ for all qφ ∈ Q equipped withthe topology σ(Q), then L2 = {X ∈ Es : |X|2 ∈ E = L1}.


As noted in [19], a standard computation with the bilinear form〈X,Y〉φ := φF(XY) defined on L2 × L2 yields the Cauchyinequality

pφ(XY) ≤ qφ(X)qφ(Y), for all X,Y ∈ L2. (CS)

The spaces L1 and L2 with their respective topologies aretopologically complete and the topologies are Lebesguetopologies (see [1] and [20]).Both pφ and qφ, restricted to the carrier band of φ, arenorms.We need to show that they are complete norms as requiredin the definition of the Dobrakov integral given below.We need the following result.


LemmaLet ψ be a strictly positive order continuous linear functional onE. If 0 ≤ X ∈ Es and if the extension of ψ to Es satisfiesψ(X) <∞, then X ∈ Eu.

TheoremUnder the assumptions stated in the beginning of the section,the norms pφ and qφ restricted to the carrier bands of φ arecomplete norms.

The spacesL1φ := {PφX ∈ Es : pφ(X) <∞}

andL2φ := {PφX ∈ Es : qφ(X) <∞}

are therefore Banach lattices (even a Hilbert lattice in the caseof qφ).


Dobrakov integral

We need an integral for vector valued functions relative toa vector measure.Two such integrals are known in the literature, namely theBartle integral ([5, 9]) and the Dobrakov integral ([10]).For countably additive measures the latter integral is themore general one and we shall use it in the sequel.The Dobrakov integral is defined for functions havingvalues in a Banach space G, and a measure that mapssets into the space L(G,H) of all bounded linear operatorsfrom G into a Banach space H, where the measure iscountably additive in the strong operator topology onL(G,H).


Special case

We shall firstly consider the case in which we have a fixedstrictly positive order continuous linear functional0 ≤ φ ∈ E∼00 on E. In this case the spaces L1

φ and L2φ are

Banach spaces (and the latter is of course a Hilbert space).If (At)t∈J is an integrable increasing right-continuousprocess, we have the vector measure µA on the Borelsubsets B = B(J) of J = [a, b] and we will assume itsvalues to be in L2

φ.

The stochastic processes we want to integrate will also beassumed to take values in L2

φ and so their product (in thef -algebra Eu) will have values in L1

φ.


Define, for any S ∈ B, the multiplication operatorTS : L2

φ → L1φ, corresponding to µA, by TSX = µA(S)X.

The map S 7→ TS is then an L(L2φ,L1

φ)-valued measure,where L(L2

φ,L1φ) denotes the space of all continuous linear

operators from L2φ into L1

φ and ‖TS‖ = qφ(µA(S)).

Since µA is countably additive in order, we have for eachdisjoint sequence of sets (Si) that

|n∑

i=1

µA(Si)− µA(

∞⋃i=1

Si)| → 0

in order, and since qφ is order continuous, the convergenceis also in the norm qφ.


This means that

‖n∑

i=1

TSi − T⋃∞i=1 Si‖ → 0.

Thus, the operator valued measure is uniformly (andtherefore strongly) σ-additive.We identify the measure µA with the operator valuedmeasure S 7→ TS.


We shall adapt Dobrakov’s definition of integrability slightlyto take into account the fact that we have a lattice structure.This will have the advantage that we can prove a Lebesguedominated convergence theorem, something the Dobrakovintegral in general lacks.For the benefit of the reader we will recall the relevantdefinitions of measurability and integrability.Finally we will proceed to define the integral for the casewhere we do not assume the existence of a strictly positivelinear functional.


The following facts and definitions are necessary to define theintegral:

The φ-semivariation of µA is defined as

µA(S) = sup pφ

(r∑

i=1

XiµA(S ∩ Si)

)the supremum taken over all measurable partitions (Si) of[a, b] and all Xi ∈ L2

φ with qφ(Xi) ≤ 1. We denote by B0 theclass of all sets in B with finite φ-semivariation and byσ(B0) the σ-algebra generated in B by B0.

A measurable step function is a function of the form

Xt =

k∑i=1

XiISi(t), Si ∈ B, Si ∩ Sj = ∅ for i 6= j,Xi ∈ L2φ.

The function t 7→ X(t) is called B-measurable if there existsa sequence (Xn(t)) of measurable step functions thatconverges in L2

φ to X(t) for every t ∈ J.


A measurable step function Xt =∑r

i=1 XiISi(t) is called aµA-integrable step function if µA(Si) <∞ for all i. Thus ameasurable step function is µA-integrable if it is a B0-simplefunction (i.e., each Si ∈ B0).A function X(t) is called µA-measurable if there exists asequence (Xn(t)) of µA-integrable step functions thatconverges to X(t) in every point t.

We call a set N ∈ B a µA-null set if µA(N) = 0 andµA-almost convergence of a sequence means convergencein all points except in those belonging to a µA-null set.The integral of a measurable step functionXt =

∑ri=1 XiISi(t) is defined as∫

SX dµA :=

r∑i=1

XiµA(S ∩ Si), S ∈ B.


A sequence of vector measures (µk) with values in aBanach space is called uniformly countably additivewhenever, for every ε > 0 and every sequence of setsSn ↓ ∅ in B there exists some n0 ∈ N such thatsupk ‖µk(Sn)‖ < ε for all n ≥ n0.

We note that if Xt =∑r

i=1 XiISi(t) is a measurable step function,then, since |Xt| =

∑ri=1 |Xi|ISi(t), we have that |Xt| is also a

measurable step function. Moreover,∣∣∣∣∫S

X dµA

∣∣∣∣ ≤ r∑i=1

|Xi|µA(S ∩ Si) =

∫S|X| dµA, S ∈ B.


DefinitionThe µA-measurable function X is said to be (Dobrakov)integrable if

1 there exists a sequence (Xn) of µA-integrable stepfunctions that converges µA-almost everywhere to X;

2 the sequence of set functions (|ν|n)∞n=1 defined by

|ν|n(S) =

∫S|Xn| dµA, S ∈ B,

is uniformly σ-additive on B. We call (Xn) a definingsequence for X.


The fact that |∫

S Xn dµA| ≤∫

S |Xn| dµA for all S ∈ B, impliesthat for the sequence (Xn) in the definitions above, we havethat the sequence of set functions

νn(S) =

∫S

Xn dµA, S ∈ B,

is also uniformly σ-additive.In the Dobrakov integration theory a function X is calledintegrable if the first condition above holds and if instead ofthe second condition one has that the sequence (νn)∞n=1 isuniformly σ-additive.Thus our condition of integrability implies that of Dobrakov.


Lemma

Using the notation above we have:(1) If X is integrable then |X| is integrable.(2) For each S ∈ B the limits

ν(S) := limn→∞

∫S

Xn dµA and |ν|(S) := limn→∞

∫S|Xn| dµA

exist and |ν(S)| ≤ |ν|(S).

(3) The limits are independent of the choice of (Xn) and areuniform in S.


DefinitionWith the definitions and notation as above, we define theDobrakov integral of an integrable function X : J → L2

φ withdefining sequence (Xn) as∫

SX dµA = lim

n→∞

∫S

Xn dµA, S ∈ B.


Proposition

(1) We have ∫S|X| dµA = lim

n→∞

∫S|Xn| dµA, S ∈ B

for every defining sequence (Xn) of X.

(2) If 0 ≤ X we have∫

S X dµA ≥ 0 for all S ∈ B. Thus, if X and Yare integrable, and if X ≤ Y, then

∫S X dµA ≤

∫S Y dµA for all

S ∈ B.(3) If X ≥ 0 then there exists a defining sequence of X

consisting of positive integrable functions.(4) If X is integrable, then∣∣∣∣∫

SX dµA

∣∣∣∣ ≤ ∫S|X| dµA for all S ∈ B.


We have that integrability of X implies that of |X|.The converse is also true and it follows from the followingresults that show that the integrable functions is an ideal inthe space of measurable functions.

Proposition

If X is a measurable function and if |X| ≤ Y with Y an integrablefunction, then X is integrable. In particular, if X is measurableand if |X| is integrable, then X is also integrable.


In the general case of Banach spaces a direct analogue ofLebesgue’s dominated convergence theorem is not readilyavailable.However, in the case of lattices with our definition ofintegrability, we have a Lebesgue theorem that is easy toderive from Dobrakov’s theory.

Theorem

Let (Xn) be a sequence of integrable functions convergingµA-almost everywhere to a measurable function X. Supposethat there exists an integrable function Y such that |Xn| ≤ Y.Then X is integrable and∫

SX dµA = lim

n→∞

∫S

Xn dµA, S ∈ B.

This limit is uniform with respect to S ∈ B.


General case

Having defined the integral for the case that we have astrictly positive order continuous linear functional, we nowdefine the integral for the general case: Let therefore L1

and L2 be the locally solid spaces with topologiesgenerated by the sets of Riesz seminorms P and Qrespectively.We define B0 to be the class of all sets S ∈ B such that(µA)φ(S) <∞ for every order continuous φ ≥ 0 satisfyingφ(S) = 1.

An integrable simple function is then a φ-integrablefunction for all φ, i.e., it is a B0-simple function.


A function X : J → L2 is called measurable if there exists asequence (Xn) of integrable simple functions such thatqφ(Xn(t)− X(t))→ 0 for every qφ ∈ Q.

A measurable function X : J → L2 is called integrable ifthere exists a sequence of simple integrable functions (Xn)converging µA-almost everywhere to X in the σ(L2,Q)topology and for which the sequence of set functions(|ν|n(S)) defined by |ν|n(S) =

∫S |Xn| dµA is uniformly

σ-additive in B with reference to the topology σ(L1,P).

As before we call the sequence (Xn) of integrable simplefunctions used in the definition above, a defining sequencefor the integrable function X.


Proposition

Let X be an integrable function defined on J with values in L2

with defining sequence (Xn) of integrable simple functions.Then |X| is integrable and for each S ∈ B the limits

ν(S) := limn→∞

∫S

Xn dµA and |ν|(S) := limn→∞

∫S|Xn| dµA

exist in the topological space σ(L1,P). These limits areindependent of the defining sequence and are uniform in S.


DefinitionIf (Xn) is a defining sequence of integrable simple functions forthe integrable function X, we define for all S ∈ B,∫

SX dµA := lim

n→∞

∫S

Xn dµA,

with the limit taken in the space σ(L1,P).


Lebesgue’s theorem as formulated above holds for thegeneral case.


The notion of progressively measurable can be extendedas follows:The process (X(t)), with X measurable is calledprogressively measurable if for every t ∈ [a, b], we havethat X is measurable on [a, t].

We denote the set of all Dobrakov integrable functionsdefined on J = [a, b] by L1([a, b], µA).


We denote by L2([a, b], µA) the space of all µA-integrablefunctions X from [a, b] in Es satisfying

qφ(X)2 := |φ|F∫ b

a|X|2 dµA <∞ for all φ ∈ E∼00.

The set of all these semi-norms is denoted by Q.

It is easy to check that 〈X,Y〉 := |φ|F∫ b

a XY dµA is a bilinearform and that the Cauchy inequality holds:∣∣∣∣|φ|F ∫ b

aXY dµA

∣∣∣∣ ≤ qφ(X)qφ(Y).


Natural processes

DefinitionA right-continuous increasing process (At)t∈J satisfying

φF(MtAt) = φF∫ t

aMs dµA = φF

∫ t

aMs−dµA for all φ ∈ E∼00

and for all bounded martingales (Mt) is called a natural process.


The notion of a natural process was the instrument used toprove uniqueness of the Doob-Meyer decomposition(see [33]) of submartingales in the classical case.This is also the case in the abstract setting with thedefinition given above.

DefinitionLet (Xt,Ft)t∈J be a submartingale adapted to the filtration (Ft)t∈J. ADoob-Meyer decomposition of Xt is a decomposition

Xt = Mt + At

where (Mt) is a martingale and (At) is a right-continuous increasingprocess.


Proposition

Let X = (Xt,Ft)t∈J be a submartingale adapted to the filtration(Ft)t∈J. Then Xt admits only one Doob-Meyer decompositionwith natural (At).


The stochastic integral with reference to a martingale

Let (Mt,Ft) be a right-continuous martingale with respect to thefiltration (Ft) with left-hand limits.

We know that the submartingale M2t has a unique

Doob-Meyer decomposition M2t = Lt + At, where L is a

martingale and A is a right-continuous, increasingpredictable process and Aa = 0.

Use the process A to generate a vector measure in thedefinition of the Dobrakov integral to construct the Itôintegral.


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