Finitely additive supermartingales

GIANLUCA CASSESE

Abstract. The concept of finitely additive supermartingales, originally due to Bochner, is revived and developed. We exploit it to study measure decompositions over filtered probability spaces and the properties of the associated Dol´eans-Dade measure. We obtain versions of the Doob Meyer decomposition and, as an application, we establish a version of the Bichteler and Dellacherie theorem with no exogenous probability measure.

1. Introduction

In the classical theory of probability one often encounters situations in which countable additivity fails.

Broadly speaking, these fall into two main classes of problems: those involving duality on the spaceL^∞and those in which the underlyingσ algebra needs to be extended, e.g. to overcome the lack of measurability of some random quantity. Both situtations are well documented in applications. [18] is a recent example of the former kind of situations arising in mathematical finance. In the area of weak convergence it is well known that even the classical empirical process is not Borel measurable in the spaceD[0,1] when the latter is equipped with the non separable topology induced by the supremum norm. Dudley [14] illustrates a number of situations relevant for empirical processes in which measurability fails. To overcome these drawbacks, a new approach, based on outer expectation, was developed by Hoffmann-Jorgensen and, in a systematic way, in the book by van der Vaart and Wellner [26]. More recently Berti and Rigo [4] have shown that such notion of weak convergence has an exact translation in the language of finite additivity.

In this paper we fix an algebraAof subsets of some set Ω and an increasing family (At:t∈R+) of sub algebras ofA, a filtration (FandFtwill hereafter denote theσalgebras generated byAandAtrespectively).

To illustrate our topic, consider the quantity m(ft) wheref = (ft :t ∈R+) is an adapted process and m a finitely additive probability onA, that ism is a positive, finitely additive set function on A(in symbols m ∈ ba(A)₊) and m(Ω) = 1. When dealing with finitely additive expectation, of special importance are the structural properties of m such as decompositions, particularly the one of Yosida and Hewitt [27]. In our setting, however, what matters are the properties ofmconditional onAtand the focus then shifts from the finitely additive measurem to the finitely additive process (m_t : t ∈ R+) where m_t = m|A_t; or even (m^c_t:t∈R+), wherem^c_t andm^⊥_t designate the countably and purely finitely additive components ofmt(in the sequel the spaces of countably and purely finitely additive set functions on an algebraGwill be indicated with the symbolsca(G) and pf a(G) respectively whileP(G) will be used to designate full probabilities, i.e.

countably additive, on G). The inclusion At ⊂ Au for u ≥ t implies m^c_u|At ≤m_t and m^⊥_t ⊥ m^c_u|At i.e.

m^c_u|At =mt∧(m^c_u|At)≤m^c_t+ m^⊥_t ∧(m^c_u|At)

=m^c_t, a conclusion which extends to any decomposition

Date: March 21, 2008.

2000Mathematics Subject Classification. Primary 28A12,60G07, 60G20.

Key words and phrases. Bichteler Dellacherie theorem, Conditional expectation, Dol´eans-Dade measure, Doob Meyer de- composition, finitely additive measures, supermartingales, Yosida Hewitt decomposition.

I am in debt with an anonymous referee for several helping suggestions. All remaining erros are my own.

1

m_t =m^e_t+m^p_t, such that m^p_t ⊥m^e_u|At – see e.g. Lemma 2. (m_t : t∈ R+) and (m^c_t :t ∈ R+) turn thus out being finitely additive supermartingales, a concept introduced by Bochner, in a number of little known papers – [6], [7] and [8] – and later revived by Armstrong in [2] and [3].

More formally, a finitely additive stochastic process ξ = (ξ_t : t ∈ R+) is an element of the vector lattice Q

t∈R+ba(At) endowed with the order induced by each coordinate space. ξ is a finitely additive supermartingale if

(1.1) ξt(F)≥ξu(F) F ∈ At, t≤u

The symbolS designates the set of finitely additive supermartingales such that kξk ≡ sup_t∈R

+kξtk <∞.

With no loss of generality we put A0 =T

t∈R+Atand A=S

t∈R+At and define, for F ∈ Aand G∈ A0, ξ_∞(F) = inf_{t:F∈At}ξt(F) and ξ0(G) = sup_tξt(G) a choice that will allow us to replace R+ by R+ = R+∪ {∞}when necessary. We also use the symbols ¯Ω = Ω×R+ and ¯F ≡ F ⊗ {∅,R+}.

We use repeatedly the following corollary of the Hahn-Banach theorem (see e.g. [5, 3.2.3(b) and 3.2.10]).

Lemma 1. IfΣ₀⊂Σare algebras of subsets of some setS andµ₀∈ba(Σ₀)₊ then there existsµ∈ba(Σ)₊ such that µ|Σ0=µ0.

Although all processes in this paper are indexed by R+, we often do not use but the order properties of the real numbers so that some of the results that follow carry over almost unchanged to the case of a linearly ordered index set.

2. Finitely Additive Conditional Expectation

The absence of a satisfactory concept of conditional expectation in the finitely additive setting, a major argument in favour of countable additivity, is a direct consequence of the failure of the Radon Nikodym theorem. The operator defined hereafter, e.g., provides an extension of such fundamental concept which is suitable for many analytical purposes but lacks some of the properties which matter for the sake of its statistical interpretation (a different proof of the following result appears in [10, proposition 2.1, p. 27]).

Theorem 1. Let H be an algebra of subsets of Ω, G a sub σ algebra of H, and µ ∈ ba(H)+. Let µ|G decompose as λ+π, withλ∈ca(G)+,π∈ba(G)+ andλ⊥πand define

(2.1) Iπ={G∈ G:π(G) = 0}

If f ∈L¹(µ)there exists a uniqueµ(f| I_π)∈L¹(λ)such that

(2.2) µ(f1I) =µ(µ(f| Iπ)1I) =λ(µ(f| Iπ)1I) I∈ Iπ

and

(2.3) µ(f1_G| I_π) =µ(f| I_π)1_G G∈ G

µ(·| Iπ) :L¹(µ)→L¹(λ)is a positive, linear operator withkµ(·| Iπ)k= 1.

Proof. Being closed with respect to finite unions,Iπis a directed set relatively to inclusion. Sinceλ⊥πand G is aσalgebra, for each >0 there existsI∈ Iπ such that λ(I^c)≤: i.e. λ(G) = lim_I∈Iπλ(IG),G∈ G.

Letf ∈L¹(µ). Any solutionp(f)∈L¹(λ) to (2.2) must then satisfy λ(p(f)1G) = lim

I∈Iπ

λ(p(f)1_G∩I) = lim

I∈Iπ

µ(f1_G∩I) G∈ G

and is therefore uniqueP a.s.: by consideringf⁺ andf⁻ separately we can (and will) thus restrict to the case in whichf ∈L¹(µ)+.

Letµf ∈ba(G)+ be defined implicitly by letting

(2.4) µf(G) = lim

I∈Iπ

µ(f1_G∩I) G∈ G

The limit in (2.4) exists uniformly with respect to G ∈ G. In fact, I ∈ Iπ implies µ_f(I) = µ(f1_I) and lim_I∈Iπµf(I^c) = 0 so that

0≤µ_f(G)−µ(f1_G∩I) =µ_f(G)−µ_f(G∩I)≤µ_f(I^c) LethG_ni_n∈

N ⊂ Gbe such that lim_nλ(G_n) = 0 and I ∈ I_π. Then lim_nµ(G_n∩I) = lim_nλ(G_n∩I) = 0 so that limnµ(f1Gn∩I) = 0, by absolute continuity of the finitely additive integral [15, III.2.20(b)]. Then [15, I.7.6]

limn µ_f(G_n) = lim

n lim

I∈Iπ

µ(f1_Gn∩I) = lim

I∈Iπ

limn µ(f1_Gn∩I) = 0

i.e. µf λ. (2.2) follows by letting µ(f| Iπ)∈ L¹(λ)+ be the corresponding Radon Nikodym derivative;

(2.3) fromIG∈ I_π wheneverI∈ I_π andG∈ G. µ(·| I_π) is linear and positive asµis. Iff ∈L¹(µ), λ(|µ(f| Iπ)|)≤ lim

I∈Iπ

λ(µ(|f|| Iπ)1I) = lim

I∈Iπ

µ(|f|1I)≤ kfk

with equality iff is the indicator of someI∈ Iπ i.e. kµ(·| Iπ)k= 1.

Referring toµ(·| I_π) as “conditional expectation” is just a convenient abuse of terminology as the law of total probability µ(f) =µ(µ(f| Iπ)), which is at the basis of the statistical interpretation of this concept [17, p. 1229], will in general not hold unless µ|G ∈ca(G)¹. Of course, if µ ∈ ca(H) the above concept of conditional expectation would coincide (by uniqueness) with the traditional one.

3. The Dol´eans-Dade Measure

In the early works of Dol´eans-Dade [12], F¨ollmer [16] and Metivier and Pellaumail [21], supermartingales were associated with measures over predictable rectangles. We address this issue in the present setting. The claims and the proofs of this this section remain true if we replaceR+ by any linearly ordered index set.

Denote byRthe collection of all sets of the form

(3.1) F0× {0} ∪

N

[

n=1

Fn×]tn,∞[

whereF0∈ A0,N ∈Nand, for eachN ≥n > m≥1,Fn∈ At_n andFn∩Fm=∅. Ris closed with respect to intersection and contains ¯Ω and∅. We denote by P the algebra generated byR: eachF ∈ P takes then the form of a disjoint union

(3.2) F0× {0} ∪

N

[

n=1

Fn×]tn, un]

with F0 ∈ A0, tn, un ∈R+,Fn ∈ At_n. We also denote by ¯P the collection defined as in (3.2) but with At

replaced byAfor eacht∈R⁺. Let

(3.3) M=

¯ x∈ba

2^Ω¯

+

: lim

t x¯(Ω×]t,∞[) = 0

1The failure of this property for reasonable definitions of finitely additive conditional expectation is well known since the work of Dubins [13]

Theorem 2. ξ∈S if an only if there existsx¯∈Mandλ∈ba(A∞)such that (3.4) ξ_t(F) =λ(F) + ¯x(F×]t,∞[) t∈R+, F ∈ At

Proof. Assume that ξ∈S and, replacing ξt with ξt−ξ_∞, assume also that ξ_∞ = 0. For each F0× {0} ∪ SN

n=1F_n×]t_n,∞[∈ Rdefine the quantity

(3.5) x F₀× {0} ∪

N

[

n=1

F_n×]t_n,∞[

!

=

N

X

n=1

ξ_tn(F_n) Let F = F0× {0} ∪SN

n=1Fn×]tn,∞[ and G = G0× {0} ∪SK

k=1Gk×]uk,∞[ be sets in R. To prove that R is a lattice, write F₀⁰ = F₀∩G^c₀, G⁰₀ = G₀ and, for n, k > 0, F_n⁰ = F_n∩T

{k>0:u_k≤t_n}G^c_k and G⁰_k=Gk∩T

{n>0:tn<u_k}F_n^c. We obtain F∪G= (F₀⁰∪G⁰₀)× {0} ∪

N

[

n=1

F_n⁰×]t_n,∞[

!

∪

K

[

k=1

G⁰_k×]u_k,∞[

!

∈ R

Rearrange the collection {tn, uk : 1≤n≤N,1≤ k≤K} as hγii^I_i=1 with γi≥γi+1 and set conventionally ξγ₀ = 0. For 1≤i≤I, let ˆψi be a positive extension of (ξγ_i −ξγi−1)|Aγ_i to A and ˆξγ_i =Pi

j=1ψˆj. Then, ξˆ_γi|A_γi=ξ_γiand ˆξ_γi+1 ≥ξˆ_γi ≥0, i.e. ˆξ_tn ≥ξˆ_ukwhenevert_n ≤u_k. For 0< k≤K,SN

n=1(G_k∩F_n)×]u_k, t_n]⊂ F^c∩G(asFn∩Fm=∅forn > m >0). ThenG⊂F impliesGk∩Fn =∅for all 1≤k≤Kand 1≤n≤N such thatuk< tn that is Gk=S

{n:tn≤uk}(Gk∩Fn). Therefore, (3.6)

K

X

k=1

ξu_k(Gk) = X

{1≤n≤N,1≤k≤K:tn≤uk}

ξu_k(Gk∩Fn)≤ X

{1≤n≤N,1≤k≤K:tn≤uk}

ξˆtn(Gk∩Fn)≤

N

X

n=1

ξtn(Fn) The set functionxdefined in (3.5) is then monotonic anda fortioriwell defined. IfF =∅thenSN

n=1Fn =∅ so thatx(F) = 0; moreover

x(F∪G) =

N

X

n=1

ξt_n(F_n⁰) +

K

X

k=1

ξu_k(G⁰_k)

=

N

X

n=1

ξt_n(Fn)− X

{1≤n≤N,1≤k≤K:tn≥uk}

ξt_n(Fn∩Gk)

+

K

X

k=1

ξu_k(Gk)− X

{1≤n≤N,1≤k≤K:tn<uk}

ξu_k(Fn∩Gk)

=

N

X

n=1

ξ_tn(F_n) +

K

X

k=1

ξ_uk(G_k)− X

k,n≥1

ξ_tn∨uk(F_n∩G_k)

= x(F) +x(G)−x(F∩G)

In other words, xis a strongly additive set function on a lattice of sets which contains ∅as well as ¯Ω: as such [5, 3.1.6, 3.2.1 and 3.2.5] it admits an extension ˆxto P. IfF ∈ Pis as in (3.2), then

ˆ x(F) =

N

X

n=1

{x(Fn×]tn,∞[)−x(Fn×]un∨tn,∞[)}

ˆ

xis thus unique and, in view of (3.6), positive. But then, ˆxadmits a positive extension ¯xto 2^Ω¯, by Lemma 1. If ¯x∈Mandλ∈ba(A∞) are given andξis defined as in (3.4) then it is clear thatξ∈Swithξ∞=λ

If ξ ∈ S, then denote by M(ξ) the collections of those ¯x ∈ M meeting (3.4). Each ¯x ∈ M(ξ) will be referred to as a Dol´eans-Dade measure associated toξ. Disregarding the apparent arbitrariness implicit in the existence of a multiplicity of such measures, there are several remarkably simple implications of Theorem 2 on such relevant issues as the decomposition and extension of supermartingales that are spelled out in the next corollaries where the following notation is used: let M^uc,M^up ⊂M consist of measuresm such that m|F ∈¯ ca( ¯F) andm|F ∈¯ pf a( ¯F), respectively. We set

(3.7) S^uc={ξ∈S:ξ_∞∈ca(A), M(ξ)∩M^uc6=∅} and S^up={ξ∈S:ξ_∞∈pf a(A), ⊂M^up} A supermartingale belonging toS^uc(resp. S^up) will be called uniformly countably additive (resp. uniformly purely finitely additive).

Corollary 1. Anyξ∈Sadmits a decomposition

(3.8) ξ=µ−α

whereµ is a finitely additive martingale andα a positive, finitely additive increasing process (as defined in [2, p. 287]). Moreover, the following are equivalent:

(i) ξ∈S^uc;

(ii) µandαin (3.8) may be chosen such thatµ_∞, α_∞∈ca(A)(and thus so that µandαare countably additive processes);

(iii) there existsλ∈ca(A)+ such that |ξt| ≤λ|Atfor each t∈R⁺. Proof. Let ¯x∈M(ξ) and define

(3.9) µt(F) =ξ∞(F) + ¯x(F×R+) and αt(F) = ¯x(F×[0, t]) F ∈ At

Then (3.8) follows from (3.4). In fact µ is a finitely additive martingale while αextends to an increasing family ( ¯α_t : t ∈ R+) of measures on Asuch that inf_tkα¯_tk = kα₀k = 0. If (i) holds, then upon choosing

¯

x∈M(ξ) such that ¯x|F¯, (3.9) implies (ii). If (ii) holds let λ=|µ∞|+ 2α∞. Then in restriction toAt we obtain|ξt| ≤ |ξ_∞|+ (ξt−ξ_∞)≤ |µ_∞|+α_∞+ (α_∞−αt)≤λand (iii) follows. Assume (iii), thenξ_∞∈ca(A).

LetU ={0 =t₁≤. . .≤t_N} and define ¯ζ_t^U

N ∈ca(A)₊ to be an extension of ξ_tN−ξ_∞|A_tN toAdominated byλ−ξ∞and set ¯ξ_t^U_N =ξ∞+ζ_t^U_N; likewise, forn < N let ¯ζ_t^U_n ∈ca(A)+be an extension ofξtn−ξtn+1|Atn to Adominated byλ−ξ¯_t^U_n+1and set ¯ξ_t^U_n= ¯ξ_t^U_n+1+ ¯ζ_t^U_n. Define ¯ξ^U =PN

n=1ξ¯_t^U_n1_[tn,tn+1[a map fromR+ toba(A).

One easily establishes thatξ_∞≤ξ¯_t^U ≤λfor eacht∈R+, i.e. ¯ξ^U ∈[ξ_∞, λ]^R+, that ¯ξ^U is decreasing and that ξ¯^U_t|At =ξ_t when t ∈U. If ba(A)^R+ is equipped with the product topology obtained after endowing each coordinate space with the weak^∗topology, we conclude that [ξ_∞, λ]^R+ is compact and that the netξ¯^U

U∈U, withU denoting the collection of finite subsets ofR⁺ directed by inclusion, admits a cluster point ¯ξ. Then necessarily, ¯ξis decreasing and ¯ξ_t|At=ξ_tfor eacht∈R+. The same argument used in the proof of Theorem 2 shows that the quantityPN

n=1( ¯ξt_n−ξ¯u_n)(Fn), whereF0× {0} ∪SN

n=1Fn×]tn, un]∈P¯, implicitly defines a measure on ¯P which admits an extension ¯x∈M(ξ) such that ¯x(F×R+) = ( ¯ξ0−ξ¯_∞)(F)≤λ(F) so that

x|¯F ∈¯ ca( ¯F).

Corollary 1 establishes a general version of the Doob Meyer decomposition. In addition it characterises exactly those processesξadmitting a countably additive version of such decomposition. This characterisation implies a condition hinging on the uniform countable additivity of the processξor, equivalently, a weak form

of countable additivity of the Dol´eans-Dade measure, namely ¯x|F ∈¯ ca( ¯F). We shall return on this issue in the following sections.

The existence of Dol´eans-Dade measures easily translates into that of extensions of finitely additive su- permartingales, a result which may prove useful in problems involving changes of the underlying filtration.

ForH ⊂Ω let¯ Hω denote the section{t∈R+: (ω, t)∈H}.

Corollary 2. Consider an increasing family (Aτ : τ ∈ T) of algebras of subsets of Ω where T ⊂ 2^Ω¯ is ordered by reverse inclusion and letξ∈Sandx¯∈M(ξ). There exists a finitely additive supermartingaleξ^∗ on(A_τ :τ∈T) such thatx¯∈M(ξ^∗). As a consequence

(i) If τ(t)≡Ω×]t,∞[∈TandF ∈ At∩ A_τ(t)then ξ_τ(t)^∗ (F) =ξt(F)

(ii) If τ, υ∈T,F ⊂ΩandF_τ,υ≡ {ω∈F :υ_ω⊂τ_ω} ∈ Aτ∩ Aυ then ξ_τ^∗(F_τ,υ)≥ξ_υ^∗(F_τ,υ) Proof. Fix ¯x∈M(ξ) and defineξ^∗_τ∈ba(Aτ) implicitly by letting

(3.10) ξ_τ^∗(F) = ¯ξ_∞(F) + ¯x((F×R+)∩τ) F ∈ Aτ

where ¯ξ_∞ is an extension of ξ_∞ to 2^Ω: (i) is immediate from (3.5). Given that τ ≤ υ is equivalent to υ ⊂ τ then F ∈ Aτ and τ ≤ υ imply ξ_τ^∗(F) ≥ ξ^∗_υ(F) so that ξ^∗ is a finitely additive supermartingale on (Aτ : τ ∈ T). Moreover, if F ⊂ Ω and Fτ,υ ∈ Aτ ∩ Aυ then ξ_τ^∗(Fτ,υ) ≥ ξ^∗_υ(Fτ,υ) is equivalent to

¯

x((F_τ,υ×R+)∩τ)≥x((F¯ _τ,υ×R+)∩υ) which follows from ¯xbeing positive.

Wheneverτ(t)∈TandAt⊂ Aτ(t)for allt∈R+, Corollary 2 suggests that anyξ∈Smay be consistently extended to any filtration indexed by T. Corollary 2 is an illustration of the importance of finite versus countable additivity.

4. Two Decompositions

We shal prove in this section that all finitely additive supermartingales have a component that may be represented as a classical supermartingale with respect to someP∈P(F). It should be highlighted that the probability measure P involved here emerges endogenously, rather than being given from the outset, as in the classical theory. We start with a preliminary result.

Lemma 2. LetG ⊂ Hbe two algebras of subsets ofΩand denote byca(G,H)andpf a(G,H)the subspaces of ba(G)consisting of set functions which admit a countably additive extension toHand whose norm preserving extensions toHare all purely finitely additive, respectively. For eachλ∈ba(G)there exists a unique way of writing

(4.1) λ=λ^e+λ^p

whereλ^e∈ca(G,H),λ^p∈pf a(G,H)andλ^e, λ^p≥0 if and only if λ≥0.

Proof. With the aid of the Radon Nikodym theorem it is easily seen thatλ∈ca(G,H) if and only ifλλ|G¯ for some ¯λ∈ca(σH) and, thus, that ca(G,H) is an ideal. Let hλαi_α∈A be an increasing net in ca(G,H)+

with W

α∈Aλα = λ ∈ ba(G). Fix α1 ∈ A arbitrarily and, for given αn−1, let αn ≥ αn−1 be such that λα_n(Ω)≥λ(Ω)−2⁻ⁿ. IfF ∈ G,

λ(F)≥lim

n λα_n(F) =λ(Ω)−lim

n λα_n(F^c)≥λ(Ω)−π(F^c) =λ(F)

But then,λP

n2⁻ⁿλ_αn∈ca(G,H) i.e. λ∈ca(G,H). We obtain from Riesz theorem the decomposition ba(G) = ca(G,H) +ca(G,H)^⊥. The inclusion pf a(G,H) ⊂ca(G,H)^⊥ is clear. To prove the converse, let λ¯∈ba(H) extendλ∈ca(G,H)^⊥. Then there exists Gn∈ G such that|λ|(G^c_n) +|¯λ^c|(Gn)<2⁻ⁿ. IfG∈ G

|λ(G)|= lim

n |λ(G∩Gn)|= lim

n |λ¯^⊥(G∩Gn)| ≤ |λ¯^⊥|(G)

i.e. kλ¯^ck+kλ¯^⊥k=kλk¯ =kλk ≤ kλ¯^⊥k. In other words,λ∈pf a(G,H).

Lemma 2 is a slight generalization of the classical decomposition of Yosida and Hewitt (by uniqueness the two decompositions coincide for G =H). It has though an important implication here as it implicitly suggests that finitely additive supermartingales may admit a component that can be represented as a classical supermartingale with respect to someP ∈P(F).

Proposition 1. Letξ∈S+. For eacht∈R+ letξt=ξ_t^e+ξ^p_t withξ_t^e∈ca(At,F)andξ_t^p∈pf a(At,F)and setξ^e= (ξ_t^e:t∈R+)andξ^p= (ξ^p_t :t∈R+). Then

(4.2) ξ=ξ^e+ξ^p

is the unique decomposition of ξ such that ξ^e ∈ S+ may be represented as a classical P supermartingale X for some P ∈P(F) ² whileξ^p is positive and orthogonal to all finitely additive processes admitting such representation. We say that ξ^e is representable and that the pair(P, X)is a representation ofξ^e.

Proof. The inclusionξ^e∈ S+ was shown in the Introduction. As ξ^p is clearly orthogonal to any classical stochastic process, we only need to prove thatξ^e admits a representation. Define the function T(t) =kξ_t^ek and the set J={t∈R+:T(t)>sup_u>tT(u)} (with sup∅=−∞). As T is monotone,J is countable; let C be a countable subset of R+ such that T[C] is dense in T[R+]. For each t ∈ R+ either t ∈ J or there is a decreasing sequence htki_k∈

Nin Csuch that limkT(tk) =T(t). Let htni_n∈

N be an explicit enumeration of D = C∪J, choose ¯ξ^e_t

n ∈ ca(F) such that ¯ξ_t^e

n

A_tn = ξ^e_t

n, fix Q ∈ P(F) and let ¯P = Q+P

n2⁻ⁿξ¯_t^e

n

and P = kP¯k⁻¹P. Clearly,¯ P ∈ P(F), P ξ¯_t^e_n for each n ∈ N. By construction, for each t ∈ R+ and k >0 there is tk ∈D such that t ≤tk and (ξ_t^e−ξ_t^e_k)(Ω) ≤2^−k. Remark that ( ¯ξ_t^e−ξ¯_t^e_k)

σAt ∈ ca(σAt) is the (unique) countably additive extension of ξ^e_t −ξ^e_t

k|A_t to σA_t and is therefore positive. We conclude that ¯ξ_t^e(F) = limkξ¯_t^e_k(F) for eachF ∈σAt. By Vitali Hahn Saks theorem and its corollaries [15, III.7.2-3],

ξ¯^e_t|σAtP|σAt, i.e. ξ^e is representable.

Uniformly countably additive supermartingales play a special role in the following section.

Proposition 2. Eachξ∈Sadmits a unique decompositionξ=ξ^uc+ξ^up whereξ^uc ∈S^uc andξ^up∈S^up. Proof. Let ¯x∈M(ξ) and let ¯x^c_¯

F and ¯x^⊥_¯

F be the countably and purely finitely additive parts of ¯x|F, respec-¯ tively. Define ¯x⁰ ∈Mby letting

¯

x⁰(H) = ¯x^c_F¯

¯ x

H I_x¯⊥

F¯

H ⊂Ω¯ Then, by (2.3), ¯x⁰_F¯ = ¯x^c_F¯ – so that ¯x⁰ ∈M^uc. LettingIn ∈ I_x¯⊥

F be such that ¯x^c_F(I_n^c)<2⁻ⁿ

¯

x⁰(H) = lim

n x¯^cF¯

Inx¯

H I_x¯⊥

F¯

= lim

n x(I¯ nH)≤¯x(H) H ⊂Ω¯

Clearly, ¯x⁰⁰ = ¯x−x¯⁰ ∈ M^up. Thus the set M^∗(ξ) = {¯y ∈ M^uc : ¯y ≤ x¯ for some ¯x ∈ M(ξ)} is non empty and, we claim, it admits a maximal element with respect to the partial order≥F¯ defined by letting

2The property defined here was called the Kolmogoroff property by Bochner [8, p. 164]

¯

y ≥F¯ y¯⁰ whenever ¯yF¯ ≥y¯⁰_¯

F. In order to apply Zorn lemma, consider an increasing neth¯y^αi_α∈A in M^∗(ξ) and let ¯x^α ∈ M(ξ) be such that ¯y^α ≤ x¯^α for all α ∈ A. Define ¯x^◦(H) = LIM_α∈Ax¯^α(H), H ⊂ Ω –¯ where LIM denotes the Banach limit functional introduced in [1]. By linearity, ¯x^◦∈M(ξ). The inequality

¯

x^◦(H)≥lim inf_α∈Ax¯^α(H)≥lim inf_α∈Ay¯^α(H) which holds for any H ⊂Ω implies that ¯¯ x^◦ ≥F¯ y¯^α for all α∈Ai.e. that ¯x^◦ is an upper bound for h¯y^αi_α∈A. Let ¯x^uc be a maximal element ofM^∗(ξ), let ¯x^∗∈M(ξ) be such that ¯x^uc≤x¯^∗and define ¯x^up= ¯x^∗−x¯^uc∈M. Letξ^uc, ξ^up∈Sbe uniquely defined by the condition

¯

x^uc∈M(ξ^uc), ¯x^up ∈M(ξ^up),ξ_∞^uc=ξ_∞^c andξ^up_∞ =ξ_∞^⊥. By construction,ξ^uc∈S^uc. Decompose ¯y∈M(ξ^up) as ¯y⁰+ ¯y⁰⁰where ¯y⁰∈M^ucand ¯y⁰⁰∈M^up, as in the first step of this proof. From ¯x^uc+ ¯y⁰≤x¯^uc+ ¯y∈M(ξ) and the fact that ¯x^ucis≥F¯ maximal, we deduce ¯y⁰ = 0 or, equivalently,ξ^up∈S^up. Ifξ=κ^uc+κ^upwere another such decomposition, and k^up and k^uc the associated Dol´eans-Dade measures, then from k^up ≤ x^uc+x^up and Hahn Banach one may find ¯k^up ∈M(κ^up) such that ¯k^up ≤x¯^uc+ ¯x^up. However, since ¯k^up ⊥x¯^uc, this implies ¯k^up ≤x¯^up while the converse is obtained mirrorwise. In other words κ^up and ξ^up induce the same Dol´eans-Dade measure; in addition,κ^up_∞ =ξ_∞^up=ξ_∞^⊥. The claim follows from Theorem 2(iii).

5. Increasing Processes

Fix P ∈ P(F) and let A(P) denote the set of processes (At : t ∈ R+) such that A∞ ∈ L¹(P) and P(0 =A0 ≤At≤Au) = 1 for each 0≤t ≤u <∞. Of course, ifA∈A(P) and A⁰ is a modification ofA (i.e. P(A⁰_t=A_t) = 1 for allt∈R+) thenA⁰∈A(P). PutA=S

P∈P(F)A(P).

Lemma 3. Let A∈A(P). Then there isF ∈ F with P(F^c) = 0 and a modification A⁰ ofA such that for each t≤u,0 =A⁰₀≤A⁰_t≤A⁰_u onF. If in additionP(At) = limnP(A_t+2⁻ⁿ) thenA⁰ andF may be chosen to be right continuous at each t∈R+ and for each ω∈F.

Proof. As in the proof of Proposition 1, there exists a countable subsetDofR⁺with the property that for eacht∈R+and >0 there isd∈Dsuch thatd≥tandP(A_t)> P(A_d)−. DefineF =T

{d,d⁰∈D:d>d⁰}{Ad≥ Ad⁰}: clearly,P(F^c) = 0. LetD(t) ={d∈D:d≥t} and A⁰_t= inf_d∈D(t)Ad. By definition of D, A⁰_t ≥At

but P(A⁰_t) = P(At) so that P(At = A⁰_t) = 1. If A is right continuous in mean the same conclusion holds even if we replace D(t) with D⁺(t) = {d ∈ D : d > t}. However A⁰ is right continuous on F since D⁺(t) =T

u>tD⁺(u).

ForH= (F0×{0})∪SN

n=1(Fn×]tn, un])∈P¯ andA∈A(P) the integralR

1HdAhas an obvious definition, namely PN

n=11_Fn(A_un−A_tn). In the following Theorem we obtain an extension of this integral together with a characterization of increasing processes in terms of their Dol´eans-Dade measure.

Theorem 3. Let x¯∈M. The following are equivalent:

(i). There existsP ∈P(F)and, given P, a unique (up to modification) A∈A(P)such that

(5.1) x(H¯ ) =P

Z

1_HdA H∈P¯ (ii). x¯∈M^uc andx({0}) = 0;¯

(iii). There existsP ∈P(F)such that for eachh∈L¹(¯x)the equation

(5.2) x(bh) =¯ P(bI_x¯(h)) b∈B(F)

admits a unique solutionI¯x(h)∈L¹(P)such that Ix¯(1_{0}) = 0.

I_¯x : L¹(¯x) → L¹(P) as defined in (5.2) is a positive, continuous, linear functional such that kIx¯k = k¯xk and that limnIx¯(hn) = 0 whenever sup_n|hn| ∈ L¹(¯x) and limnP^∗(h^∗_n > η) = 0 for each η > 0 – where h^∗_n≡sup_t|hn,t|andP^∗ is the outer measure generated by P.

Proof. Let us start remarking that one may easily identify ¯F with F, as we shall now do. Under (i), x({0}) =¯ PR

1_{0}dA = 0 and ¯x(F) = ¯x(F×]0,∞[) = P(1_FA_∞) for any F ∈ F. Assume (ii) and fix P = (kQk+kxk)¯ ⁻¹(Q+ ¯x|F¯) for someQ∈P(F). By Theorem 1, for eachh∈L¹(¯x) we may define

I¯x(h) = ¯x(h|F¯)d¯x|F¯

dP ∈L¹(P)

By (2.2), I_x¯(h) is a solution to (5.2); moreover the operator I_x¯ is positive, linear and has norm k¯xk, by Theorem 1; ¯x({0}) = 0 implies Ix¯(1_{0}) = 0, P a.s.. Any other solution J(h) ∈ L¹(P) to (5.2) satisfies P(bJ(h)) =P(bIx¯(h)) for allb∈L^∞(P) i.e. P(J(h) =Ix¯(h)) = 1. Assume (iii) and define At=I¯x(1_]0,t]), A= (A_t:t∈R+) and letH = (F₀× {0})∪SN

n=1(F_n×]t_n, u_n])∈P. Then¯ A∈A(P) and, up to aP null set I_x¯(1_H) =

N

X

n=1

¯

x 1_Fn×]tn,un]

F¯d¯x|F¯ dP

=

N

X

n=1

1F_nx¯ 1_]tn,un]

F¯d¯x|F¯ dP

=

N

X

n=1

1Fn(Aun−Atn)

= Z

1HdA

But then (5.1) follows from (5.2). IfB∈A(P) also meets (5.1) then forh=1_F×]0,t]andF∈ F we conclude that P(F At) = P(F Bt) from which we deduce uniqueness. It is clear from (5.1) thatIx¯ is linear, positive and thatkI_x¯k=kxk.¯

Ifhhni_n∈Nis a sequence inL¹(¯x) with the above properties then so ish|hn|i_n∈N. Given thatIx¯is positive, it is enough to prove the claim forhn ≥0. Observe that ¯x(hn≥η)≤x(h¯ ^∗_n≥η); moreover, ¯x|F ¯ P implies that, in restriction to 2^Ω⊗ {∅,R+}, ¯xP^∗. But then,h_nconverges to 0 in ¯xmeasure and, by [15, theorem III.3.6], inL¹(¯x). Given (5.2) this is equivalent toI_x¯(h_n) converging to 0 inL¹(P).

The equivalence of (i) and (ii) establishes a correspondence between Aand M^uc which compares to the classical (and well known) characterization of increasing processes as measures given by Meyer [11, VI.65, p. 128] (see also [22, p. 6]). Meyer’s result, which ultimately delivers the Doob Meyer decomposition, focuses however on countable additivity over F ⊗ B(R⁺); we rather require this property relatively to F. In Theorem 4 below we show that indeed this is enough to obtain a suitable version of Doob Meyer¯ decomposition. Although there are connections between these two properties, it is noticeable that the latter is independent of the given filtration. On should also remark that we do not assume the existence of an underlying probabilityP ∈P(F) but rather deduce it.

Each ¯x∈Mmay be considered in restriction to special classes of functions such as the setC of functions f : ¯Ω→Rwith continuous sample paths and bounded support (i.e. such thatf(t) = 0 for allt larger than someT). Let Cbe theσalgebra on ¯Ω generated byC.

Lemma 4. Let x¯∈M^c. There existα^c∈ca(C)+,P ∈P(F)andA^c ∈A(P)right continuous such that

(5.3) x(f¯ ) =α^c(f) =P

Z

f dA^c f ∈L(¯x)∩C Proof. In order to apply Daniell theorem, consider a sequencehhni_n∈

Nin the vector latticeL(¯x)∩Cdecreasing to 0 and fixT such that ¯x(|h₁−h^T₁|)< , whereh^T_n =h_n1_]0,T]. Leth^{T ,∗}_n = sup_th^T_n(t). A simple application of Dini’s theorem for eachω∈Ω guarantees that the sequence

h^{T ,∗}_n

n∈Nconverges to 0 pointwise; moreover, by continuity of the sample paths,h^{T ,∗}_n is in factF measurable. Thus Theorem 3 implies that limnx(h¯ n)≤ + lim_nx(h¯ ^T_n) = + lim_nP(I_¯x(h^T_n)) = . In other words, the restriction of ¯x to C is a Daniell integral and as such it admits the representation as the integral with respect to some α^c ∈ ca(C). Observe that F×]t,∞[∈ C for all F ∈ F and t ∈ R+. Fix P ∈ P(F) as in Theorem 3(i) and define α^c_t ∈ ba(F) as α^c(F×]0, t]) for each F ∈ F. Sinceα^c_t ≤x¯F¯ P, denote byA^c_t the Radon Nikodym derivative of α^c_t with respect toP. We deduce thatP((A^c_u−A^c_t)1_F) = α^c(F×]t, u]) = ¯x(F×]t, u])≥0, so that A^c ∈A(P), and that limnP(1F(A^c_t+2−n−A^c_t)) = limα^c(F×]t, t+ 2⁻ⁿ]) = 0 (by countable additivity) for eachF ∈ F so that A^c_t = limnA^c_t+2−n up to aP null set. By Lemma 3, we obtain thatA^c admits a modification which is right

continuous.

Theorem 4. Let ξ∈S. Then ξ∈ S^uc if and only if there exist P ∈P(F), M ∈L¹(P) andA^p ∈A(P) which is adapted, right continuous in mean and such that

(5.4) ξt(F) =P(1F(M−A^p_t)) t∈R+ andF ∈ Ft

and that

(5.5) P

b

Z hdA^p

=P Z

M(b)−hdA^p b∈L^∞(P), h∈B(σP) whereM(b) = (P(b|Ft) :t∈R+).

Proof. We use the notation of Lemma 4 and the inclusionσP ⊂ C. Letd={t₁≤t₂≤. . .≤t_N}be a finite sequence inR+ and define

(5.6) P^d(f) =

N−1

X

n=1

P(ft_n|Ft_n)1_]tn,tn+1] and x¯^d(f) = ¯x(P^d(f)) f :R⁺→L¹(P) Denote by α^d the restriction of ¯x^d to F ⊗2^R+. On the one hand it is easily seen that ¯x^d_¯

F P so that, as in the proof of Theorem 3, we can associate to ¯x^da process A^d∈A(P), by lettingA^d_tdP = ¯x^d(1_]0,t]|F)d¯¯ x^d_F¯. On the other hand, (5.3) impliesPR

f dA^d = ¯x(P^d(f)) =α^c(P^d(f)). Consider the case in which f =bh where b ∈ L^∞(P) and his bounded, adapted and left continuous. Let d_n ={k2⁻ⁿ : k = 0, . . . ,2²ⁿ} and observe that, by [11, VI.2, p. 67], there exists aP null setF ∈ F outside of which limnP^dn(b)t=M(b)t−

and limnP^dn(h)t =ht for each t∈ R+ (as his left continuous and adapted). Given that α^c is countably additive in restriction toC, we conclude

(5.7) lim

n P

b Z

hdA^dn

= lim

n α^c(P^dn(bh)) = lim

n α^c(P^dn(b)P^dn(h)) =α^c(M(b)₋h) Define thenα^p∈ba(F ⊗2^R+) implicitly asα^p(H) =α^c(M(1H)₋). Then from (5.7) we deduce that

α^dn

n∈N

converges to α^p and, by [15, III.7.3, p. 159], that α^p_¯

F P. Let A^p ∈ A(P) be the increasing process associated toα^p. Thus for every bounded, adapted and left continuous process hand everyb∈L^∞(P) we have

P

b Z

hdA^p

= ¯x(M(b)₋h) =α^c(M(b)₋h) =P Z

M(b)₋hdA^p

which delivers (5.4) if we only letM =dξ_∞/dP +A^p_∞,b=1_F withF ∈ Ft andh=1_]t,∞[. In addition, if F ∈ F,s≤t andhF,t=1F−P(1F|Ft), thenM(hF,t)_s−= 0 so that

P(hF,tA^p_t) =α^p(hF,t1_[0,t]) =P Z t

0

M(hF,t)₋dA^p= 0

Therefore, replacing A^p_t with P(A^p_t|Ft), we may assume that A^p ∈ A(P) is adapted. Eventually, letting h_n = 1_]t,t+2−n] we conclude that 0 = lim_nα^c(h_n) = lim_nP(A^p_t+2−n −A^p_t) and, thus, that A^p is right

continuous in mean. That (5.4) impliesξ∈S^uc is obvious.

With a complete filtration Theorem 4 implies thatA^pmay be chosen to be adapted and right continuous.

We want to emphasize that the existence of the decomposition (5.4) does not depend on the underlying filtration.

Corollary 3. Let ξ∈S and let D⊂R⁺ be such that ξt = sup_d∈D(t)ξd|At whereD(t) ={d∈ D:d≥t}.

Then, ξadmits a Doob Meyer decomposition if and only ifξ^D= (ξ_d:d∈D)does.

Proof. Given that, by Theorem 4, the Doob Meyer decomposition is equivalent to ξ ∈ S^uc, the direct implication is obvious. As for the converse, choose ¯x^D ∈M(ξ^D) to be countably additive in restriction to F ⊗ {∅,D}. Ift∈R+ andF∈ Atthen

(5.8) |ξt|(F)≤ sup

d∈D(t)

|ξd|(F)≤ |ξ∞|(F) + ¯x^D(F×D)≡λ(F)

The claim then follows from Corollary 1.

Corollary 3 makes clear that decomposition (5.4) is a property that involes any subset Dwhich is dense for the range ofξand we know from the proof of Proposition 1 that this may be taken to be countable. The class D property may thus be replaced by a corresponding property, the class Dσ, in which the stopping times are restricted to have countable range, see [9].

6. The Bichteler Dellacherie Theorem without Probability Let f : ¯Ω → R be adapted to the filtration, define f^∗ = sup_t∈R

+|f_t| and let F be such that f^∗ is F measurable. The starting point of this section are the sets

(6.1) K=

Z

hdf :hisP simple,|h| ≤1

and C=K −B(F)+

Bichteler and Dellacherie start from the assumption thatK is bounded in L⁰(P) for some given P ∈P(F) and thatf is right continuous with left limits outside someP null set. These two properties are then shown to imply that for givenη >0 there existsδ >0 such thatd1F ∈ C1/ F for allF∈ F such thatP(F)> η. We take inspiration from this separating condition to define a concept of boundedness suitable for our setting.

To this end we denote byU a collection of subets of Ω with the following properties:

Assumption 1. There existsλ₀>0 such that

(6.2) {λ1U :λ≥λ0} ∩ C1U =∅ U ∈ U

Moreover,U, V ∈ U implyU ⊂ {f^∗< n} for somen andU∪V ∈ U.

A violation of (6.2) indicates that the setK is unbounded relatively to some U ∈ U. Both sets in (6.2) are convex subsets of B(2^U) and 1UC contains −1U as an internal point. By ordinary properties of the support functional [15, lemma V.I.8(f), p. 411], the Hahn Banach theorem and [15, lemma V.II.7, p. 417]

we conclude that for eachU ∈ U there is ˆm_U ∈ ba(2^U) such that sup_x∈1

UCmˆ_U(x) ≤1 =λ₀mˆ_U(U). The inclusion−B(2^U)+⊂1UCimplies that ˆmU ≥0. By definingmU ∈ba+implicitly by

m_U(F) = mˆ_U(F∩U) ˆ

mU(U) F ⊂Ω we have completed the proof of the following:

Lemma 5. Let f : ¯Ω→R+ satisfy Assumption 1 and define the set

(6.3) M=

m∈ba₊:kmk= 1, sup

x∈C

m(x)≤λ₀

For each U∈ U there existsmU ∈ M such thatmU(U) = 1.

Fix now m ∈ M (so that f^∗ ∈ L¹(m)) and let ξ^e and ξ^p be the components of the finitely additive supermartingale (m|F_t:t∈R+) as of (4.2). Set alsoI_t=I_ξ^p

t (see (2.1)), let (P, X) be a representation for ξ^e and observe that−ξ^p ∈S and that M(ξ^e) = M(−ξ^p). Fix an extension ¯ξ∞ ∈ba(F) ofξ∞ to F and

¯

x∈M(ξ^e) and define

ξ¯_t^p(F) = ¯ξ_∞^p (F)−x(F¯ ×]t,∞[) F ∈ F

The collection ( ¯ξ_t^p : t ∈ R+) is then increasing with t. For each b ∈ B(F) and u ≥ t, F ∈ It implies ξ^p_∞(b1F) = ¯x(b1_F×]t,∞[) and thus

(6.4) ξ¯_u^p(b1_F) =ξ_∞^p (b1_F)−x(b1¯ _F×]u,∞[) = ¯x(b1_F×]t,u]) Let nowd={t1≤. . .≤tN},

(6.5) P^d=

(

F₀× {0} ∪

N−1

[

n=1

F_n×]t_n, t_n+1] :F₀∈ F₀, F_n∈ F_tn,1≤n≤N−1 )

and chooseF_n∈ I_tn 1≤n < N and set

F^d=

N−1

[

n=1

Fn×]tn, tn+1] and f^d=f01_{0}+

N−1

X

n=1

ft_n+11_]tn,tn+1]

By (6.4)

N−1

X

n=1

ξ^p_t

n+1((f_tn+1−f_tn)1_Fn) =

N−1

X

n=1

¯

x((f_tn+1−f_tn)1_{Fn×]tn,tn+1]})

= x(f¯ ^d1_Fd)−x

N−1

X

n=1

ft_n1_{Fn×]tn,tn+1]}

!

= x(f¯ ^d1_Fd) +P

N−1

X

n=1

ftn1Fn(Xtn+1−Xtn)