On the structure of finitely additive martingales

Gianluca Cassese

On the structure of finitely additive martingales

Quaderno N. 02-04

Decanato della Facoltà di Scienze economiche Via G. Buffi , 13 CH-6900 Lugano

On the Structure of Finitely Additive Martingales

Gianluca Cassese Istituto di Economia Politica Universit`a Commerciale “Luigi Bocconi”

via U. Gobbi, 5, I-20136 Milan, Italy.

e-mail: gianluca.cassese@uni-bocconi.it July 16, 2003

Abstract

Finitely additive martingales are the counterpart offinitely additive measures overfiltered probability space. We study the structure of the Yosida Hewitt decomposition in such setting and obtaing a full characterisation. Based on this result we introduce a “conditional expectation” operator for finitely additive measures which has some properties in common with ordinary conditional expectation. We address then the problem of computing the expectation of random elements generated by a given class of stochastic processes. On the basis of a notion of coherence for processes, akin to the no arbitrage principle in mathematical finance, we give conditions under which such expectation may be computed explicitely.

Keywords: coherence, finitely additive measures, finitely additive martingales, semi- martingale, Doob-Meyer decomposition, equivalent martingale measure, fundamental the- orem of asset pricing

MSC codes: 60G48, 28A12, 28A25.

Short title: F.a. martingales.

1. Introduction.

In the foundations of subjective probability, finitely additive (henceforth f.a.) measures arise from an attitude of coherence in the face of uncertainty (see, among others, [10], [11], [12], [14]

and [17]). This is defined with reference to the class Kof bets that an individual, viewed as a bookie, stands ready to accept: coherence prescribes that no such bet may produce a strictly positive net outcome. In the space of bounded functions on a set Ω, a coherent probability is associated with the functional separating K and B(Ω)₊. The mathematical properties of probability follow then from the structure of the class of bets considered.

In this paper we consider individuals betting on the relative increase of a process K over some interval — typically over time. At a point of their choice bettors enter the game placing

a stake and at any successive point they may choose to walk off, receiving the amount of the stake times the increment ofK over that interval. Stakes are simple and bettors may only place

a finite though arbitrary number of them (a more precise definition of coherence will be given

in section 7, see (7.1)). An example of this framework is offered by financial markets, in which case K is the price of an asset.

In any setting in which information evolves according to some parameter, the structure of the separating measure (or rather its restrictions), viewed itself as a process, becomes important and a study of its properties leads us straight to the theory of f.a. stochastic processes. Despite its interest and importance, this theory did not receive but little attention since the first at- tempts made by Bochner (see e.g. [4]) — especially so if compared to the well established theory of stochastic processes. It is clear that, interpreting stochastic processes as ordered collections of random elements, and viewing in turn (if possible) random elements as densities of countably additive (henceforth c.a.) measures, most results on stochastic processes can be read as state- ments on families of c.a. measures. It is then challenging to investigate whether these claims extend to the f.a. setting. One of the purposes of this paper is to show that indeed this is not a purecuriosum.

In the next section we shall introduce a few definitions together with a key result on this sub- ject, due to Armstrong: this essentially provides a f.a. analogous of the well known Doob-Meyer decomposition. In the following section 3 we shall further characterize such decomposition. In section 4, based on the preceding results, we introduce an operator acting on f.a. measures but akin in some suitable sense to ordinary conditional expectation for c.a. measures. This tool will play a key role in the analysis of the following section 5 in which we shall apply our findings to the computation of the f.a. expectation of random elements, generated by a stochastic pro- cess. We show that such expectation can be approximated fairly well by a functional for which there exists a fully analytical formulation. In section 6 we treat a special case in which such formulation can be further refined. Eventually in section 7 we shall return to the initial problem of a coherent assessment of probability i.e. of separation and we shall prove that the expected value of the elements of our set of bets can be computed explicitly at least in one special case of interest, and can be approximated conveniently well in more general cases.

Although the focus of the paper is on subjective probability, there is a strong, and per- haps little known, analogy with a much debated issue in mathematical finance (the so called Fundamental Theorem of Asset Pricing) to which we shall briefly refer in the last section.

2. Definitions and preliminaries

The setting for the following sections shall be as follows. Ω and ∆ are arbitrary sets, with ∆ linearly ordered. A and A^δ will be algebras of subsets of Ω; by F^δ, F we shall denote their

induced σ algebras respectively. A filtration is a collection (A^δ:δ ∈∆) of sub algebras of A which increases withδ. The collection ˜λ= (λ_δ :δ∈∆), withλ_δ∈ba(Ω,Aδ) for eachδ ∈∆, is a f.a. stochastic process; a c.a. stochastic process ˜λis a f.a. process suchλ_δ is c.a. for eachδ∈∆.

λ˜ is bounded whenever

°°

°˜λ

°°

°, _

δ∈∆

kλ_δk<∞. ˜α≥β˜ stands forα_δ ≥β_δ for each δ ∈∆; ˜α >β˜ for ˜α≥β˜ but ˜β £α. By˜ µ| C we mean the restriction of the measure µto the algebra C⊂A; by µ_b we mean the measureµ_b(A) =µ(b1_A) defined for eachb∈L¹(Ω,A, µ) and A∈A.

At times it shall be convenient to “complete” ∆ by replacing it with ∆^∗ = ∆∪{δ0, δ_∞} and defining the order ≥^∗ on ∆^∗ by letting δ_∞ ≥^∗ ε ≥^∗ δ ≥^∗ δ₀ for any δ, ε ∈ ∆ such that ε≥δ, where ≥is the order on ∆. Toδ_∞ and δ0 we shall associate the algebrasAδ_∞ , _

δ∈∆

Aδ

and Aδ0 , ^

δ∈∆

Aδ respectively. Such completion — bearing no loss of generality — will often be implicit.

The most important result on f.a. processes was obtained by T. Armstrong in a noteworthy paper ([1], but see [2] as well) in which it was shown that some form of the core theorem of stochastic processes, the Doob Meyer decomposition, indeed holds in the f.a. model. More precisely:

Lemma 1 (Armstrong [1], Proposition 6.1 and Corollary 6.1.1, p. 288). Let the pos- itive, bounded f.a. stochastic processλ˜ satisfy

λ_δ≤ λ_ε| Aδ (2.1)

(resp. λ_δ≥ λ_ε| Aδ) (2.2)

for eachδ ≤ε. Then

1. there exists a collection λˆ =

³ˆλ_δ:δ∈∆

´

of bounded, positive, f.a. measures on (Ω,A) that increase (resp. decrease) withδ and such thatλ_δ= ˆλ_δ

¯¯

¯Aδ for eachδ∈∆;

2. λ˜ admits the decomposition

λ_δ=µ_δ+α_δ (2.3)

(resp. λ_δ=µ_δ−α_δ) (2.4)

where µ_δ and α_δ are positive, bounded f.a. measures on (Ω,Aδ) satisfying (i). for eachδ ≤ε

µ_δ= µ_ε| Aδ (2.5)

(ii). α_δ= ˆα_δ| Aδwhereαˆ ,(ˆα_δ :δ∈∆)is a collection of positive f.a. measures on(Ω,A) increasing withδ and such that infδ∈∆kαˆδk= 0.

This lemma is the backbone of what follows and we shall therefore adopt its notation and terminology. If a f.a. process satisfies (2.1) (resp. (2.2), resp. (2.5)) it will be referred to as a f.a. submartingale (resp. supermartingale resp. martingale) and in this case ˆλ_δ will always denote the extension of λ_δ to (Ω,A) with the properties described in the first claim: we shall refer to ˆλas the extensionof ˜λ. A f.a. process with the properties of ˜αin lemma 1 claim 2 is an increasing f.a. processes. By analogy with the theory of stochastic processes, the f.a. processes

˜

µand ˜αappearing in the decompositions (2.3) and (2.4) will be indicated as thecharacteristics of ˜λ.

When λ ∈ ba(Ω,A), λ =λ^c+λ^⊥ will be its Yosida and Hewitt decomposition into a c.a.

(λ^c) and a purelyfinitely additive (henceforth p.f.a.,λ^⊥) part. ByIA we shall indicate the ideal of all A measurable λ^⊥ null sets. It is well known (see [3], theorem 10.3.3, p. 244) that when A is a σ algebra and P ∈ ca(Ω,A) then for each ε > 0 there exists a set F ∈ IA such that P(F^c)≤ε.

Throughout the paper we adopt the convention ⁰₀ = 0.

3. The Structure of F.A. Martingales

F.a. martingales are the counterpart of f.a. measures on filtered f.a. measurable spaces and this is what makes them so important. If λ∈ba(Ω,A) we denote ˜λ= (λ| Aδ:δ ∈∆) which is clearly a f.a. martingale. The converse is also true: the equation

λ(F) =λ_δ(F) where F ∈ Aδ defines in fact a positive, f.a. measure on ¡

Ω,S

δ∈∆Aδ

¢ which may in turn be extended to the whole of (Ω,A) (a multiplicity of such extensions is known to exist, see [3], chapter 3). In the setting of ordinary stochastic processes, a full correspondence can only be established between c.a. measures and uniformly integrable martingales: this will be a bijection (see, for example, [13], proposition III.3.5, p. 154). L¹ bounded martingales, which are a fortiori f.a. martingales, do generate a measure, but only a f.a. one, unless uniform integrability is satisfied. The extreme difficulty of establishing this property is a good reason why f.a. should be considered with interest in these models. For the rest of the paper, letλbe a f.a. probability on (Ω,A), λ_δ = λ| Aδ and ˜λ = (λ_δ:δ ∈∆). If G is a sub algebra of A we write λ^c_G , (λ| G)^c (resp. λ^⊥_G = (λ| G)^⊥) and λ^c_δ,λ^c_A

δ (resp. λ^⊥_δ =λ^⊥_A

δ).

Projecting a f.a. measure on its c.a. part and restricting it to some sub algebra are linear operations that do not commute, i.e. in general λ^c_δ 6= λ^c| A^δ. To see this point more clearly, take the opposite case: then, λ^c is the c.a. extension of λ^c_δ to (Ω,A). This situation is quite exceptional, especially if A is considerably “larger” thanA^δ. In general one would expect that c.a. is not preserved by taking arbitrary extensions, or, in other terms, that a p.f.a. component

comes in and the probability mass assigned in accordance to the c.a. criterion shrinks the larger the algebra of events considered. More precisely, when ε ≥ δ it is clear that λ^c_ε| Aδ is a c.a., positive measure on (Ω,Aδ); furthermore, given that 0≤λ^⊥_δ,

λ^c_ε| Aδ≤ λ^c_ε| Aδ+λ^⊥_ε

¯¯

¯Aδ=λ_δ i.e.

λ^c_ε| Aδ≤sup©

µ∈ca(Ω,Aδ)₊ :µ≤λ_δª ,λ^c_δ

λ˜^c = (λ^c_δ :δ∈∆) is then a f.a. supermartingale and, according to lemma 1, it admits the decomposition λ^c_δ = µ^c_δ −α^c_δ. Analogously ˜λ^⊥ = ¡

λ^⊥_δ :δ∈∆¢

is a f.a. submartingale and decomposes asλ^⊥_δ =µ^⊥_δ +α^⊥_δ. The inequalities λ^⊥_δ ≥µ^⊥_δ ∨α^⊥_δ ≥0 imply that both µ^⊥_δ andα^⊥_δ are p.f.a. measures, absolutely continuous with respect to λ^⊥_δ. The analogous conclusion for ˜λ^c is more indirect and to establish we shall use the following lemma

Lemma 2. Let ˜ξ be a f.a. positive supermartingale and m∈ ca(Ω,A)₊. Then, ˜ξ^m , ˜ξ∧m˜ is a positive, c.a. supermartingale and admits an extension ˆξ^m (as of lemma 1) such that 0≤ˆξ^m_δ ≤m.

Proof. Letm∈ca(Ω,A)₊. Clearly, ifε≥δ

ξ^m_ε | Aδ = (ξ_ε∧mε)| Aδ

≤ ξ_ε| A^δ∧mε| A^δ

≤ ξ_δ∧mδ

= ξ^m_δ

LetDbe the collection offinite subsets of∆(directed by inclusion),d={δ₁ ≤δ₂ ≤. . .≤δ_N}∈ D, and defineφ_N ,ξ^m_δN and φ_n , ³

ξ^m_δn−ξ^m_δn+1´¯¯¯Aδn forn= 1, . . . , N −1. Clearly 0 ≤φ_n ≤ m_δn−PN

j=n+1φ_j¯¯Aδn. By Hahn Banach applied recursively,φ_nadmits a positive extension ˆφ_nto (Ω,A) dominated bym−PN

j=n+1φˆ_j and therefore c.a.. Forn= 1, . . . , N, define ˆξ^m_δn ,PN j=nφˆ_j so that 0≤ˆξ^m_δn+1 ≤ˆξ^m_δn ≤m. Let now ˆξ^m,d=³

ˆξ^m,d_δ :δ∈∆´

be defined by ˆξ^m,d_δ ,ˆξ^m_δ11_{δ≤δ1}(δ) +

NX−1 n=1

ˆξ^m_δn+11_]δn,δn+1](δ) (3.1)

ξ^m,d_δ , ˆξ^m,d_δ

¯¯

¯Aδ, and ˜ξ^m,d , ³

ξ^m,d_δ :δ∈∆´

. From (3.1) we deduce the following properties of ˆξ^m,d and ˜ξ^m,d:

(i) ˆξ^m,d is a decreasing collection of c.a. measures dominated bym(i.e. ˜ξ^m,d is a c.a. super- martingale and ˜ξ^m,d ≤m);˜

(ii) ˜ξ^m,d ≤˜ξ^m and ξ^m,d_δ =ξ_δ when δ∈d, since ξ^m,d_δ = ξ^m_δ1¯¯A^δ1_{δ≤δ1}+X

n

ξ^m_δn+1

¯¯

¯A^δ1_]δn,δn+1]≤ξ^m_δ Endow the spaceQ

δ∈∆ba(Ω,A) with the product topology obtained after assigning to each coordinate space the vague topology. For each d∈D, the setU^d of elements satisfying (i) and (ii) above is closed and its intersection with the sphere of radius

°°

°˜ξ^m

°°

° is non empty. It follows that T

d∈DU^dis itself non empty: if ˆξ^m ∈T

d∈DU^d then necessarily ˜ξ^m =

³ˆξ^m_δ

¯¯

¯A^δ:δ ∈∆

´ . Theorem 1. Let˜ξ be a f.a. positive supermartingale with decompositionµ˜−α.˜

1. If ξ_δ is c.a., then there exists a decomposition whereµ_δ and α_δ are c.a.;

2. Ifξ_δ¿P_δfor someP ∈ca(Ω,F)₊, then there exists a decomposition whereµ_δ¿P_δand α_δ¿P_δ.

Proof. Consider the set ˆZ of decreasing families ˆζ ∈ Q

δ∈∆ca(Ω,A)₊ such that ˆζ_δ

¯¯

¯Aδ ≤ ξ_δ for each δ∈∆and let ˆZ₀ be a linearly ordered subset. IfA is the collection of allfinite subsets of ˆZ₀ — directed by inclusion, as usual — let ˆζ^a = sup_ˆζ∈aˆζ: ˆζ^a∈Zˆ₀. Then ˆζ^∗ ,limaˆζ^a≥ˆζ for each ˆζ ∈Zˆ₀, furthermore

ˆζ^∗_δ = lim

a

ˆζ^a_δ

≥ lim

a

ˆζ^a_ε

= ˆζ^∗_ε forε≥δ and

ˆζ^∗_δ

¯¯

¯Aδ = ³ lima

ˆζ^a_δ´¯¯¯Aδ

= lim

a

³ˆζ^a_δ

¯¯

¯Aδ

´

≤ ξ_δ

so that ˆζ^∗ ∈ Zˆ. By Zorn’s lemma, we can therefore find a maximal element ˆζ for ˆZ: let ζ˜=³

ˆζ_δ

¯¯

¯Aδ :δ∈∆´

. With the notation of the previous lemma, from ˆξ^m ∈Zˆwe get ˜ξ ≥˜ζ ≥˜ξ^m. But then, since ˜ξ=W

m∈ca(Ω,A)₊˜ξ^m, we conclude that ˆζ is a c.a. extension of ˜ξ and we denote it consequently by ˆξ.

As in [1],µ,W

δ∈∆ˆξ_δ∈ca(Ω,A)₊,µ_δ ≥ξ_δand ˜µis obviously a c.a. martingale. ˆα,µ−ˆξ is a collection of positive, c.a. measures on (Ω,F) increasing with δ and

^

δ∈∆

kαˆδk = ^

δ∈∆

ˆ αδ(Ω)

= µ(Ω)− _

δ∈∆

ξ_δ(Ω)

= 0

By definition, ˜α,µ˜−˜ξ is a c.a. increasing process and the decomposition ˜µ−α˜ satisfies the claim.

The flow of information — as parametrized by δ — causes a shift of the probability mass

between the c.a. and p.f.a. components of λ_δ and it is important to understand wether this phenomenon is smooth or not, i.e. if the interplay between the characteristics ofλis continuous with respect to the filtration. To this end we introduce the following definition

Definition 1. The characteristics of λ are left (resp. right) continuous with respect to the filtration if for any monotone increasing (resp. decreasing) sequencehδ_nin∈N in∆,

∆λ^⊥W

nA_δn ,lim

n

³ λ^⊥W

nA_δn −λˆ^⊥_δn´¯¯¯_

n

Aδn = 0

(resp. ∆λ^⊥V

nAδn ,lim_n³ λ^⊥_δ

n−λ^⊥V

nAδn

´¯¯¯V

nAδn = 0).

Since it is not difficult to establish (as will clearly emerge from the proof of the following lemma) that∆λ^⊥W

nA_δn =−∆λ^cW

nA_δn and∆λ^⊥V

nA_δn =−∆λ^cV

nA_δn, the above definition is indeed a condition on λcharacteristics.

Lemma 3. The characteristics ofλ are left continuous with respect to thefiltration. IfAand Aδ are σ algebras, then the characteristics of λ are also right continuous with respect to the filtration.

Proof. Clearly, 0≤∆λ^⊥W

nA_δn ≤λ^⊥W

nA_δn. If F ∈Aδn0 andn > n₀

∆λ^⊥W

nAδn (F) ≤ λ^⊥W

nAδn(F)−λ^⊥_δn(F)

= −λ^cW

nAδn(F) +λ^c_δn(F)

≤ λ^c_δn(F)

= µ^c_δn(F)−α^c_δn(F)

≤ µ^c(F) i.e. ∆λ^⊥W

nAδn ≤λ^⊥W

nAδn ∧µ^cW

nA_δn proving thefirst claim.

Let nowhδ_nin∈N be a monotone decreasing sequence and F ∈V

nAδn

∆λ^⊥V

nAδn(F) ≤ λ^⊥_δn(F)−λ^⊥V

nAδn(F)

= λ^cV

nAδn(F)−λ^c_δn(F)

≤ λ^cV

nAδn(F) (3.2)

≤ µ^cV

nAδn(F)

= µ^c(F)

By (3.2), ∆λ^⊥V

nAδn is then c.a. and, by the Hahn Banach theorem, it admits a c.a. extension, φ, to the whole of (Ω,A). LetN be the collection of subsets ofΩwhich are null with respect to the inner measure generated by∆λ^⊥V

nAδn, i.e.

N =n

F ⊂Ω:∆λ^⊥V

nAδn(E) = 0, E⊂F, E ∈Aδ

o

Clearly,[

n

IAδn ⊂N andG∩{φ(G| Aε) = 0}∈N for each G∈A andε≥δ. It is well known that the collection

B= (

F∆N :F ∈^

n

Aδn, N ∈N )

is an algebra with the property that if F₁∆N₁ =F₂∆N₂ ∈B then F₁∆F₂ ∈N. An extension β of ∆λ^⊥V

nAδn to (Ω,B) can then be defined as usual by lettingβ(F∆N) =∆λ^⊥V

nAδn (F). For any F ∈V

nAδn and Fn∈IA_δn

φ(F) = ∆λ^⊥V

nAδn(F)

= β Ã

F ∩F_n^c∩ (

φ Ã

F_n^c

¯¯

¯¯

¯

^

n

Aδn

!

>0 )!

= ∆λ^⊥V

nAδn

à F ∩

( φ

à F_n^c

¯¯

¯¯

¯

^

n

Aδn

!

>0 )!

= φ

à F∩

( φ

à F_n^c

¯¯

¯¯

¯

^

n

Aδn

!

>0 )!

i.e. φ(F_n^c|V

nAδn) is positiveφa.s.: letG= (

φ Ã [

n>m

F_n^c

¯¯

¯¯

¯ V

nAδn

!

>0 )

. Let

fm ,φ Ã [

n>m

F_n^c

¯¯

¯¯

¯

^

n

A^δn

!₋1

1_G∩^S

n>mF_n^c

The sequencehfmim∈Nconvergesφa.s. to the indicator ofG∩¡T

m

S

n>mF_n^c¢

∈V

nAδn: indeed f_m = 1_G onT

m

S

n>mF_n^c while ifω∈T

n>nωF_n for somen_ω, thenf_m = 0 whenm > n_ω. Then f^∗ ,sup_nf_n<∞,φ a.s.. But then forF ∈V

nAδn

φ(F) = lim

m φ

Ãφ¡S

n>mF_n^c¯¯ V_nAδn

¢ φ¡S

n>mF_n^c¯¯ V_nAδn

¢1_G∩F

!

= lim

m φ(fm1F)

= lim

r lim

m φ(1_F(f_m∧r)) + lim

r lim

m φ¡

1_F(f_m−r)⁺¢

= φ

à F ∩\

m

[

n>m

F_n^c

! + lim

r lim

m φ¡

1F(fm−r)⁺¢ In other words, φvanishes outside the set¡T

m

S

n>mF_n^c¢

∪{r < f^∗ <∞}for any r. However, for any c.a. probability measure Q and anyη >0, the sequence hF_nin∈N and the integer r can

be chosen such that Q¡T

m

S

n>mF_n^c¢

+Q(r < f^∗<∞) < η, or, equivalently, φ is p.f.a.. It follows thatφ= 0.

Remark 1. Since lim_n³ λˆ^⊥W

nAδn −λˆ^⊥_δn´

> 0 if an only if 0 < lim_n³ λˆ^⊥W

nAδn −ˆλ^⊥_δn´ (Ω) =

∆λ^⊥W

nAδn(Ω), it follows that continuity with respect to the filtration carries over to the ex- tension ˆλ^⊥ as well.

Remark 2. Consider the case when thefiltration is discontinuous and, in particular, letδn↑δ but Aδ ! _

n

Aδn. As illustrated in the proof of lemma 3, lim_n³

λ^⊥_δ −λˆ^⊥_δn´¯¯¯Aδ ≤ λ^⊥_δ, while lim_n³

λ^⊥_δ −λˆ^⊥_δn´¯¯¯_

n

Aδn ≤µ^cV

nAδn. Although in principle the “distance” between_

n

Aδn and Aδ may be small — even Aδ = σ

Ã_

n

Aδn

!

— nevertheless we have here a general situation in which a c.a. measure lim_n³

λ^⊥_δ −λˆ^⊥_δn´¯¯¯_

n

Aδn admits a p.f.a. extension (see [5] for explicit examples of c.a. measures on an algebra admitting p.f.a. extensions to the generatedσalgebra).

In the case of supermartingales or submartingales driven by a given, c.a. probability measure P the property ofcontinuity with respect to thefiltration simply translates into that of continuity in mean. Clearly, martingales are filtration continuous and it is well known that when∆=R+,

the filtration is right continuous and X is either a supermartingale or a submartingale, this

property is equivalent to the existence of a right continuous modification of the processX (see [16]).

We summarize the results of this section in the following

Proposition 1. Let (Ω,A) be a probability space, (Aδ:δ ∈∆) a filtration of algebras (resp.

σ algebras) withV

δAδ finite andλa positive, bounded, f.a. set function on(Ω,A). Letλ_δ,λ^c_δ and λ^⊥_δ be defined as above; let λ^c_δ be the c.a. extension ofλ^c_δ to(Ω,Fδ). Then there exist:

1. a c.a. probability measure P such that

P| Aδ Àλ^c_δ (3.3)

2. a positive, filtration left continuous (resp. continuous) P supermartingale

X=M −A (3.4)

on (Ω,F, P; (Fδ :δ∈∆)), such that

λ^c_δ= P_Xδ| Fδ

where

(i). M is a positive, uniformly integrable martingale and (ii). A is an increasing, integrable process such that A0= 0.

3. a positive, filtration left continuous (resp. continuous), bounded f.a. submartingale λ^⊥_δ =µ^⊥_δ +α^⊥_δ

where

(i). µ^⊥ is a p.f.a. martingale and (ii). α^⊥ is a p.f.a., increasing process;

Furthermore, forF ∈A^δ and each δ, ε∈∆,ε > δ,

³

λ^⊥_ε −λ^⊥_δ

´

(F) = (λ^c_δ−λ^c_ε) (F) =P(1F(Aε−A_δ)) (3.5) Proof. Replacing∆by∆^∗, (λ^c_δ:δ ∈∆^∗) and¡

λ^⊥_δ :δ ∈∆^∗¢

are easily seen to be a c.a. positive supermartingale and a positive f.a. submartingale respectively. By lemma 1 and theorem 1, we obtain the decompositionsλ^⊥_δ =µ^⊥_δ +α^⊥_δ andλ^c_δ=µ^c_δ−α^c_δ with ˜µ^⊥(resp. ˜µ^c) and ˜α^⊥ (resp. ˜α^c) a positive, p.f.a. (resp. c.a.) martingale and increasing process respectively. In case µ^c_δ

∞ = 0, λ˜^c= 0 so that ˜λ^⊥is a p.f.a. martingale; since Aδ0 isfinite 0 =λ^⊥_δ0(Ω) =λ^⊥_δ (Ω), i.e. λ= 0. Let P = µ^c_δ

∞(Ω)⁻¹µ^c_δ

∞; it is obvious that P_δ Àµ^c_δ and the inequality µ^c_δ ≥ λ^c_δ∨α^c_δ implies (3.3) and P_δÀα^c_δ. Let ˆα^c_δ be the extension ofα^c_δ to the whole of (Ω,A). Since 0≤αˆ^c_δ ≤αˆ^c_δ

∞ =α^c_δ

∞

we conclude that ˆα^c_δ is c.a. and absolutely continuous with respect to P. Let M_δ = _dP^dµcδ

δ and Aδ = ^dˆα

c δ

dP . By construction for each F ∈ F^δ, P(1FMδ) = P(1FMδ_∞) so that {Mδ:δ∈∆^∗} is uniformly integrable. A is clearly a.s. increasing. Continuity with respect to the filtration follows from lemma 3. (3.5) is a direct consequence of the martingale nature of ˜λby which

0 = (λ_ε−λ_δ) (F)

= (λ^c_ε−λ^c_δ) (F) +³

λ^⊥_ε −λ^⊥_δ´ (F)

= −P(1_F(A_ε−A_δ)) +³

λ^⊥_ε −λ^⊥_δ´ (F) an the proof is complete.

Remark 3. The c.a. probability measure P mentioned in the first claim of the preceding proposition will be referred to as the c.a. probability generated by λ. By theorem 1, if Q is any arbitrary c.a. probability measure that satisfies (3.3), then the characteristics of λ˜^c may be constructed to be absolutely continuous with respect to Q and so will be the measure P generated by λ. Clearly the supermartingale X defined in (3.4) will have in this case a Q version, namely X^Q = U X where U is the uniformly integrable martingale generated by the Radon Nikod´ym derivative of P with respect toQ. UnderP we have M_δ =µ^c_δ

∞(Ω),P a.s..

Proposition 1 can be considered as the extension to the f.a. case of the theorem stating that the density of one c.a. probability with respect to another is described by a uniformly integrable martingale (see e.g. [13], proposition III.3.5, p. 154). One of the implications of this proposition is that lack of c.a. receives a rather convenient characterization in filtered probability spaces, where it is associated with the process A. This makes f.a. measures quite tractable in this context, as the following sections will confirm.

In section 5 we shall restrict our model to the usual setting for which proposition 1 specializes as follows.

Corollary 1. Let(Ω,F; (Ft:t∈R+))be afiltered probability space withFt= \

u>t

Fu for each

t and F0 finite. Let λbe a f.a. probability on (Ω,F) generating the c.a. probabilityP. Then

under P:

1. there exists a uniformly integrable martingale M and a predictable, increasing process of integrable variation A such that the supermartingaleX=M −A satisfies

λ^c_τ = P_Xτ| Fτ

for any stopping time τ with P(τ <∞) = 1.

2. If σ and τ are stopping times and F ∈F^σ then

³

λ^⊥_τ −λ^⊥_σ´

(F∩{τ ≥σ}) =P¡

1_{F∩{τ≥σ}}(A_τ−A_σ)¢

(3.6) and, for f ∈B(Ω,F)

λˆ^⊥_τ∧σ¡

f1_{τ≥σ}¢

= ˆλ^⊥_σ ¡

f1_{τ≥σ}¢

(3.7) for any pair of extensionsλˆ^⊥_τ∧σ and λˆ^⊥_σ to(Ω,F)of λ^⊥_τ∧σ and λ^⊥_σ satisfyingλˆ^⊥_τ∧σ ≤ˆλ^⊥_σ. Proof. If thefiltration is right continuous the supermartingaleXin (3.4) will be right continuous in mean since it is filtration continuous: i.e. it admits a c`adl`ag modification. Since 0 ≤X ≤ µ_δ

∞(Ω)⁻¹, this is clearly a process of classDso that the properties ofM andAfollow from the theorem of Doob Meyer.

LetF_kⁿ,{(k−1) 2⁻ⁿ< τ ≤k2⁻ⁿ}∈Fk2⁻ⁿ andτⁿ=P

kk2⁻ⁿ1_Fⁿ

k. This definition implies (i) τⁿ ↓τ,P a.s., (ii) F_kⁿ =F_2kⁿ⁺¹₋₁∪F_2kⁿ⁺¹ and (iii) F ∈F^τn if and only ifF ∩F_kⁿ ∈Fk2⁻ⁿ for any k. LetF ⊂F_kⁿ,F ∈Fτn. Then

λ^c_τn(F) , inf (X

i

λ(G_i) :G_i∈Fτⁿ_k; [

i

G_i =F;G_i∩G_j =∅ )

= inf (X

i

λ(Gi) :Gi∈Fk2⁻ⁿ; [

i

Gi =F;Gi∩Gj =∅ )

(by (iii)) , λ^c_k2−n(F)

For arbitraryF ∈F^τ

λ^c_τn(F) = X

k

λ^c_τn(F ∩F_kⁿ) +λ^c_τn(F ∩{τ =∞})

= X

k

λ^c_k2−n(F ∩F_kⁿ)

= X

k

P¡

X_k2−n1F∩F_kⁿ

¢

= P(X_τⁿ1_F)

(i) implies Fτⁿ⁺¹ ⊂Fτⁿ. From (ii) we deduce that onF_kⁿ⁺¹,X_τⁿ⁺¹ =X_k2−n−1 while X_τⁿ =

( X_k2−n−1 k even X_(k+1)2−n−1 τ odd It follows that when F ∈Fτⁿ⁺¹

P(Xτⁿ1F) = X

k

P

³

Xτⁿ1_F∩Fⁿ⁺¹

k

´

≤ X

k

P³

X_τⁿ⁺¹1_F∩Fⁿ⁺¹

k

´

= P(X_τⁿ⁺¹1_F)

Let Gn=Fτⁿ and X_n=X_τⁿ, then P◦X_n =λ^c_Gn so that (X_n:n∈N) is filtration continuous.

From lemma 3 and the fact thatF^τ =^

n

Gⁿ we conclude that forF ∈F^τ λ^c_τ(F) = lim

n λ^c_Gn(F)

= lim

n P_Xn(F)

= P³ 1_F lim

n X_τⁿ´

= P(1_FX_τ) the last line following fromX being a.s. right continuous.

LetF ∈F^σ. Since F∩{τ ≥σ}∈F^σ∩F^τ, then

³

λ^⊥_τ −λ^⊥_σ´

(F∩{τ ≥σ}) = −(λ^c_τ−λ^c_σ) (F ∩{τ ≥σ})

= P¡

1_{F∩{τ≥σ}}(X_σ−X_τ)¢

= P¡

1_{F∩{τ≥σ}}(A_τ−A_σ)¢ Let f ∈B(Ω,F) and ˆλ^⊥_σ and ˆλ^⊥_τ∧σ be the above mentioned extensions. Then

0 ≤

¯¯

¯

³ˆλ^⊥_σ −λˆ^⊥_τ∧σ´ ¡

f1_{τ≥σ}¢¯¯¯

≤ kfk³

λ^⊥_σ −λ^⊥_τ∧σ´

(τ ≥σ)

= 0

by (3.6).

Property (3.5) suggests that A should be viewed as the predictable “compensator” of the f.a. process ˜λ^⊥. This simple fact shall have deep consequences — at least for the case covered in the above corollary 1. Let in factP⁰ be the algebra offinite unions of sets of the formF×]s, t], where F ∈F_s,s < t and s, t∈R+. For any f.a. submartingale ˜ξ consider the Dol´eans measure V˜ξ on

³Ω,˜ P⁰

´

associated to ˜ξ via the equation V˜ξ

à _N [

n=1

Fn×]sn, tn]

!

= XN n=1

³

ξ^⊥_tn−ξ^⊥_sn

´ (Fn)

By corollary 1,Vλ˜^⊥admits a c.a. extension toP,σ(P0), the predictableσ algebra (as proved in [7] and [15]). In other words, any probability measurem, though f.a. when acting on random elements, is c.a. when viewed as acting on processes (through the mapV) and it is obvious that this transposition from random elements to processes is likely to be crucial when computing the expectation of random elements generated by stochastic processes. In the next sections we shall establish this loose relationship in clearer terms.

4. A F.A. Conditional Expectation

As is well known, conditional expectation, in the sense established by Kolmogorov, is not avail- able when c.a. fails. Nevertheless, we shall construct an operator that, while not possessing all properties of c.a. conditional expectation, still is almost as useful from a practical point of view.

LetIδ=IFδ and ε∈∆. ˆλ^⊥_ε ∈ba(Ω,F) has the following two noteworthy properties:

(a). for eachδ ≤εand any B∈Iδ, ˆλ^⊥_ε (B) =

³λˆ^⊥_ε −λˆ^⊥_δ

´

(B) =P((Aε−A_δ) 1B);

(b). for eachδ < ε,P_Aε−Aδ[Iδ] is dense inP_Aε−Aδ[Fδ].

We shallfix the above situation in the following definition:

Definition 2. Letξ ∈ba(Ω,F)₊,m∈ca(Ω,F)₊,G a subσ algebra ofF andI an ideal inG. mis a compensating measure forξ overGwith respect to I if (i)ξ(B) =m(B) for eachB∈I and (ii)m[I]is dense inm[G].

For the rest of this section let ξ ∈ ba(Ω,F)₊ and let m ∈ ca(Ω,F)₊ be a compensating measure forξ overG with respect toI.

Let Π be the family of all finite, disjoint collections of elements of I. For π ∈ Π and f ∈B(Ω,F) , define the map ξ(·|π) :B(Ω,F)→B(Ω,G) by setting

ξ(f|π) =X

F∈π

ξ(f1_F) m(F) 1_F We then have the following properties

1. |ξ(f|π)|≤kfk;

2. For any F ∈π₀ ≤π (the order being defined by refinement)

ξ(f1_F) =ξ(ξ(f|π) 1_F) =m(ξ(f|π) 1_F) (4.1) LetSf be the subset of L¹(Ω,G, m) consisting of those elements U such that|U|≤kfk,m a.s.. Being closed in the weak topology, Sf is in fact a compact set (see [8], theorem IV.8.9, p.

292). Furthermore, for eachπ ∈Π the subsets

U^π(f) ={U ∈S^f :m(U1F) =ξ(f1F), F ∈π} (4.2) is weakly closed and non empty. By the finite intersection property we conclude that the intersection U(f) = \

π∈Π

U^π(f) is non empty. Let ξ(f| I) ∈U(f): since for each F ∈I there exists π ∈ Π such that F ∈ π, ξ(f| I) satisfies (4.1) for any set F ∈ I; furthermore, since

|ξ(f| I)|≤kfk, ma.s. we can choose a version satisfying such inequality pointwise. We have therefore proved the existence claim in the following proposition.

Proposition 2. Let f ∈B(Ω,F). Then there exists ξ(f| I) ∈B(Ω,G) such that|ξ(f| I)|≤ kfk, ma.s. and

ξ(f1_F) =ξ(ξ(f| I) 1_F) =m(ξ(f| I) 1_F) (4.3) for eachF ∈I. Furthermore, the conditional expectation operator

ξ(·| I) :B(Ω,F)→B(Ω,G) satisfies the following properties

1. ξ(·| I) is positive;

2. ξ(f1_F| I) =ξ(f| I) 1_F for anyF ∈G;

Proof. (Existence). An alternative proof can also be given, based on the Radon Nikod´ym approach to conditional expectation. This will help clarifying some other aspects. Let K be the closed linear subspace of B(Ω,G) generated by the indicators of the sets in I: plainly, if f ∈B(Ω,G) andF ∈I, thenf1F ∈K. Clearly ξ and mcoincide onK. For any f ∈B(Ω,F)₊, define the positive linear functionalξ_f overKby writing ξ_f(K) =ξ(f K).

ξ_f(K) ≤ ξ(|f K|)

≤ kfkξ(|K|)

= kfkm(|K|)

Let ψ(K) =kfkm(|K|) forK ∈B(Ω,G). ξ_f is then dominated by the sub additive functional ψ overK and, by the Hahn Banach theorem, it admits an extension, ˆξ_f, toB(Ω,G) that still satisfies 0 ≤ˆξ_f ≤ψ. ˆξ_f is therefore a bounded linear functional on B(Ω,G) and the equation β_f(A) = ˆξ_f(1A) defines β_f ∈ ba (Ω,G)₊. Since, by construction, 0 ≤ β_f(A) ≤ kfkm(A), β_f is c.a. and absolutely continuous with respect to m| G. Let b_f be its Radon Nikod´ym derivative: clearly |b_f| ≤ kfk, m a.s. so that, replacing, if necessary, b_f by b_f1{|^bf|^≤kfk}, we have b_f ∈B(Ω,G). For F ∈I,b_f1_F ∈I and so

ξ(f1_F) = ξ_f(F)

= ˆξ_f(F)

= β_f(F)

= m(b_f1F)

= ξ(b_f1F) For f ∈B(Ω,F), let

ξ(f| Iδ) =b_f∨0−b_f∧0

(4.3) is therefore proved. Properties (1) and (2) are indeed obvious.

5. F.A. Expectation of Stochastic Processes

In this section we shall consider an ordinary filtered measurable space (Ω,F; (Ft:t∈R+)) endowed with a f.a. probability λ, the filtration being right continuous and F⁰ finite. P shall denote the c.a. probability generated byλ(see remark 3). Let ˜Ω,Ω×R+and ˜F,F⊗B(R+).

Corollary 1 applies.

Let H be a pair ³

ht_ii^Ii=0,hF_ii^Ii=0

´

of finite sequences such that (i) t_i is a finite stopping time, (ii) 0 =t0 ≤t1 ≤. . .≤tI<∞,P a.s. and (iii)Fi ∈I^ti. Let Hbe the family of all such pairs of sequences H. By a stochastic process we simply mean a functionK : ˜Ω→Rsuch that K_t,K(·, t) is Ft measurable. A bounded process is an element ofB³

Ω˜´

. IfK is a stochastic process andτ a stopping timeK^τ is the stopped process defined viaK_t^τ =Kt∧τ. We assume that K_∞exists (otherwise, replaceK byK^tfort∈R+). ForH ∈H, letD^H_i (K),1_Fi¡

K_ti+1−K_ti¢ and

K_t^H=K₀1_F0 + XI i=0

D^H_i ¡ K^t¢

(5.1) where t_I+1 =∞. Choose extensions ˆλ^⊥_ti+1∧t ofλ^⊥_ti+1∧tto (Ω,F) such that ˆλ^⊥_ti∧t≤λˆ^⊥_ti+1∧t≤λˆ^⊥_t.

Let us begin by defining the following quantities I_λ^H_⊥(K)_t,λ^⊥_t (K₀1_F0) +

XI i=0

³λˆ^⊥_t −λˆ^⊥_ti+1∧t´ ¡

D^H_i (K)¢

(5.2)

and h

λ^⊥, KiH t ,

XI i=0

λˆ^⊥_t

i+1∧t

¡D_i^H(K)¢

(5.3) Then, clearly, ˆλ^⊥_t ¡

K_∞^H¢

=I^H

λ^⊥(K)_t+£

λ^⊥, K¤H

t . We aim at representing I^H

λ^⊥(K) and £

λ^⊥, K¤H

as explicitly as possible. For what concerns (5.2) we have I_λ^H_⊥(K)_t =

XI i=0

³λˆ^⊥_tI+1∧t−ˆλ^⊥_ti+1∧t´ ¡

D^H_i (K)¢

+λ^⊥_t (K₀1_F0)

= XI j=1

³λˆ^⊥_tj+1∧t−λˆ^⊥_tj∧t´Ã_j−1 X

i=0

D_i^H(K)

!

+λ^⊥_t (K₀1_F0)

= XI j=0

³λˆ^⊥_tj+1∧t−λˆ^⊥_tj∧t´ ³ K_t^H_j´

+λ^⊥_t1∧t(K₀1_F0)

= XI j=0

³

λ^⊥_tj+1∧t−λ^⊥_tj∧t´ ³

K_t^H_j1_{tj<t}´

+λ^⊥_t1∧t(K01_F0)

= P

XI j=0

³

A^t_tj+1−A^t_tj

´

K_t^H_j +P¡

A^t_t1K01F0

¢

i.e.

I_λ^H_⊥(K)_t=P Z t

0

H·K^HdA+P¡

A^t_t1K₀1_F0¢

(5.4) where H·X ,PI

j=0Xtj1_]]tj,tj+1]]. It is quite natural to investigate convergence properties as H increases according to some convenient notion. To this end let us introduce the following:

Definition 3. A Riemann sequenceH˜ inHis any sequencehH_nin∈N,H_n=³

htⁿ_ii^I_i=0ⁿ ,hF_iⁿi^I_i=0ⁿ ´

∈ H satisfying (i) for each t there exists a scalar δⁿ_t ≥W

i≤In

¡tⁿ_i ∧t−tⁿ_i−1∧t¢

such that δⁿ_t ↓ 0 and (ii)PIⁿ

i=0P(F_i^nc)→0.

Theorem 2. Let H be defined as above and I^H

λ^⊥(K)_t as in (5.2). Then, for any Riemann sequence H˜ = hHnin∈N and for any bounded process K, the limit I_λ⊥(K)_t , limnI^Hn

λ^⊥ (K)_t exists and is equal to

I_λ⊥(K)_t=P Z t

0

K₋dA (5.5)

Proof.

¯¯

¯K_t^H_j −Ktj

¯¯

¯≤2kKk1[

i≤I

F_i^c and, by (5.4),

¯¯

¯¯I^Hn

λ^⊥ (K)_t−P Z t

0

K₋dA

¯¯

¯¯≤P Z t

0

¡¯¯H_n·¡

K^Hn−K¢¯¯+|H_n·K−K₋|¢

dA+¯¯¯P³ A^t_tn

1K₀1_Fⁿ

0

´¯¯¯ The two terms on the right hand side are easily seen to converge to 0: thefirst by the definition of Riemann sequence and bounded convergence for stochastic integrals (see [13], theorem I.4.31, p. 46); the second one by right continuity ofA.