aa r X i v : . [ m a t h . P R ] A p r FINITELY ADDITIVE SUPERMARTINGALES
GIANLUCA CASSESE
Abstract.
The concept of finitely additive supermartingales, originally due to Bochner, is revived anddeveloped. We exploit it to study measure decompositions over filtered probability spaces and the propertiesof the associated Dol´eans-Dade measure. We obtain versions of the Doob Meyer decomposition and, as anapplication, we establish a version of the Bichteler and Dellacherie theorem with no exogenous probabilitymeasure. 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 space L ∞ andthose in which the underlying σ algebra needs to be extended, e.g. to overcome the lack of measurabilityof some random quantity. Both situtations are well documented in applications. [18] is a recent exampleof the former kind of situations arising in mathematical finance. In the area of weak convergence it is wellknown that even the classical empirical process is not Borel measurable in the space D [0 ,
1] when the latter isequipped with the non separable topology induced by the supremum norm. Dudley [14] illustrates a numberof situations relevant for empirical processes in which measurability fails. To overcome these drawbacks, anew 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 notionof weak convergence has an exact translation in the language of finite additivity.In this paper we fix an algebra A of subsets of some set Ω and an increasing family ( A t : t ∈ R + ) of subalgebras of A , a filtration ( F and F t will hereafter denote the σ algebras generated by A and A t respectively).To illustrate our topic, consider the quantity m ( f t ) where f = ( f t : t ∈ R + ) is an adapted process and m a finitely additive probability on A , that is m 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 arethe structural properties of m such as decompositions, particularly the one of Yosida and Hewitt [27]. Inour setting, however, what matters are the properties of m conditional on A t and the focus then shifts fromthe finitely additive measure m to the finitely additive process ( m t : t ∈ R + ) where m t = m |A t ; or even( m ct : t ∈ R + ), where m ct and m ⊥ t designate the countably and purely finitely additive components of m t (inthe sequel the spaces of countably and purely finitely additive set functions on an algebra G will be indicatedwith the symbols ca ( G ) and pf a ( G ) respectively while P ( G ) will be used to designate full probabilities, i.e.countably additive, on G ). The inclusion A t ⊂ A u for u ≥ t implies m cu |A t ≤ m t and m ⊥ t ⊥ m cu | A t i.e. m cu |A t = m t ∧ ( m cu |A t ) ≤ m ct + (cid:0) m ⊥ t ∧ ( m cu |A t ) (cid:1) = m ct , a conclusion which extends to any decomposition Date : November 7, 2018.2000
Mathematics 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. m t = m et + m pt , such that m pt ⊥ m eu |A t – see e.g. Lemma 2. ( m t : t ∈ R + ) and ( m ct : t ∈ R + ) turn thusout being finitely additive supermartingales, a concept introduced by Bochner, in a number of little knownpapers – [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 vectorlattice Q t ∈ R + ba ( A t ) endowed with the order induced by each coordinate space. ξ is a finitely additivesupermartingale if(1.1) ξ t ( F ) ≥ ξ u ( F ) F ∈ A t , t ≤ u The symbol S designates the set of finitely additive supermartingales such that k ξ k ≡ sup t ∈ R + k ξ t k < ∞ .With no loss of generality we put A = T t ∈ R + A t and A = S t ∈ R + A t and define, for F ∈ A and G ∈ A , ξ ∞ ( F ) = inf { t : F ∈A t } ξ t ( F ) and ξ ( G ) = sup t ξ t ( 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. If Σ ⊂ Σ are algebras of subsets of some set S and µ ∈ ba (Σ ) + then there exists µ ∈ ba (Σ) + such that µ | Σ = µ . Although all processes in this paper are indexed by R + , we often do not use but the order properties ofthe real numbers so that some of the results that follow carry over almost unchanged to the case of a linearlyordered index set. 2. Finitely Additive Conditional Expectation
The absence of a satisfactory concept of conditional expectation in the finitely additive setting, a majorargument in favour of countable additivity, is a direct consequence of the failure of the Radon Nikodymtheorem. The operator defined hereafter, e.g., provides an extension of such fundamental concept whichis suitable for many analytical purposes but lacks some of the properties which matter for the sake of itsstatistical 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 , with λ ∈ ca ( G ) + , π ∈ ba ( G ) + and λ ⊥ π and define (2.1) I π = { G ∈ G : π ( G ) = 0 } If f ∈ L ( µ ) there exists a unique µ ( f | I π ) ∈ L ( λ ) such that (2.2) µ ( f I ) = µ ( µ ( f | I π ) I ) = λ ( µ ( f | I π ) I ) I ∈ I π and (2.3) µ ( f G | I π ) = µ ( f | I π ) G G ∈ G µ ( ·| I π ) : L ( µ ) → L ( λ ) is a positive, linear operator with k µ ( ·| I π ) k = 1 .Proof. Being closed with respect to finite unions, I π is a directed set relatively to inclusion. Since λ ⊥ π and G is a σ algebra, for each ǫ > I ∈ I π such that λ ( I c ) ≤ ǫ : i.e. λ ( G ) = lim I ∈I π λ ( IG ), G ∈ G .Let f ∈ L ( µ ). Any solution p ( f ) ∈ L ( λ ) to (2.2) must then satisfy λ ( p ( f ) G ) = lim I ∈I π λ ( p ( f ) G ∩ I ) = lim I ∈I π µ ( f G ∩ I ) G ∈ G INITELY ADDITIVE SUPERMARTINGALES 3 and is therefore unique P a.s.: by considering f + and f − separately we can (and will) thus restrict to thecase in which f ∈ L ( µ ) + .Let µ f ∈ ba ( G ) + be defined implicitly by letting(2.4) µ f ( G ) = lim I ∈I π µ ( f G ∩ I ) G ∈ G The limit in (2.4) exists uniformly with respect to G ∈ G . In fact, I ∈ I π implies µ f ( I ) = µ ( f I ) andlim I ∈I π µ f ( I c ) = 0 so that 0 ≤ µ f ( G ) − µ ( f G ∩ I ) = µ f ( G ) − µ f ( G ∩ I ) ≤ µ f ( I c )Let h G n i n ∈ N ⊂ G be such that lim n λ ( G n ) = 0 and I ∈ I π . Then lim n µ ( G n ∩ I ) = lim n λ ( G n ∩ I ) = 0 sothat lim n µ ( f G n ∩ I ) = 0, by absolute continuity of the finitely additive integral [15, III.2.20(b)]. Then [15,I.7.6] lim n µ f ( G n ) = lim n lim I ∈I π µ ( f G n ∩ I ) = lim I ∈I π lim n µ ( f G n ∩ I ) = 0i.e. µ f ≪ λ . (2.2) follows by letting µ ( f | I π ) ∈ L ( λ ) + be the corresponding Radon Nikodym derivative;(2.3) from IG ∈ I π whenever I ∈ I π and G ∈ G . µ ( ·| I π ) is linear and positive as µ is. If f ∈ L ( µ ), λ ( | µ ( f | I π ) | ) ≤ lim I ∈I π λ ( µ ( | f || I π ) I ) = lim I ∈I π µ ( | f | I ) ≤ k f k with equality if f is the indicator of some I ∈ I π i.e. k µ ( ·| I π ) k = 1. (cid:3) Referring to µ ( ·| I π ) as “conditional expectation” is just a convenient abuse of terminology as the law oftotal probability µ ( f ) = µ ( µ ( f | I π )), 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 ) . Of course, if µ ∈ ca ( H ) the above concept ofconditional 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], supermartingaleswere associated with measures over predictable rectangles. We address this issue in the present setting. Theclaims and the proofs of this this section remain true if we replace R + by any linearly ordered index set.Denote by R the collection of all sets of the form(3.1) F × { } ∪ N [ n =1 F n × ] t n , ∞ [where F ∈ A , N ∈ N and, for each N ≥ n > m ≥ F n ∈ A t n and F n ∩ F m = ∅ . R is closed with respectto intersection and contains ¯Ω and ∅ . We denote by P the algebra generated by R : each F ∈ P takes thenthe form of a disjoint union(3.2) F × { } ∪ N [ n =1 F n × ] t n , u n ]with F ∈ A , t n , u n ∈ R + , F n ∈ A t n . We also denote by ¯ P the collection defined as in (3.2) but with A t replaced by A for each t ∈ R + . Let(3.3) M = (cid:26) ¯ x ∈ ba (cid:16) ¯Ω (cid:17) + : lim t ¯ x (Ω × ] t, ∞ [) = 0 (cid:27) The failure of this property for reasonable definitions of finitely additive conditional expectation is well known since thework of Dubins [13]
GIANLUCA CASSESE
Theorem 2. ξ ∈ S if an only if there exists ¯ x ∈ M and λ ∈ ba ( A ∞ ) such that (3.4) ξ t ( F ) = λ ( F ) + ¯ x ( F × ] t, ∞ [) t ∈ R + , F ∈ A t Proof.
Assume that ξ ∈ S and, replacing ξ t with ξ t − ξ ∞ , assume also that ξ ∞ = 0. For each F × { } ∪ S Nn =1 F n × ] t n , ∞ [ ∈ R define the quantity(3.5) x F × { } ∪ N [ n =1 F n × ] t n , ∞ [ ! = N X n =1 ξ t n ( F n )Let F = F × { } ∪ S Nn =1 F n × ] t n , ∞ [ and G = G × { } ∪ S Kk =1 G k × ] u k , ∞ [ be sets in R . To provethat R is a lattice, write F ′ = F ∩ G c , G ′ = G and, for n, k > F ′ n = F n ∩ T { k> u k ≤ t n } G ck and G ′ k = G k ∩ T { n> t n
0, i.e. ˆ ξ t n ≥ ˆ ξ u k whenever t n ≤ u k . For 0 < k ≤ K , S Nn =1 ( G k ∩ F n ) × ] u k , t n ] ⊂ F c ∩ G (as F n ∩ F m = ∅ for n > m > G ⊂ F implies G k ∩ F n = ∅ for all 1 ≤ k ≤ K and 1 ≤ n ≤ N such that u k < t n that is G k = S { n : t n ≤ u k } ( G k ∩ F n ). Therefore,(3.6) K X k =1 ξ u k ( G k ) = X { ≤ n ≤ N, ≤ k ≤ K : t n ≤ u k } ξ u k ( G k ∩ F n ) ≤ X { ≤ n ≤ N, ≤ k ≤ K : t n ≤ u k } ˆ ξ t n ( G k ∩ F n ) ≤ N X n =1 ξ t n ( F n )The set function x defined in (3.5) is then monotonic and a fortiori well defined. If F = ∅ then S Nn =1 F n = ∅ so that x ( F ) = 0; moreover x ( F ∪ G ) = N X n =1 ξ t n ( F ′ n ) + K X k =1 ξ u k ( G ′ k )= N X n =1 ξ t n ( F n ) − X { ≤ n ≤ N, ≤ k ≤ K : t n ≥ u k } ξ t n ( F n ∩ G k )+ K X k =1 ξ u k ( G k ) − X { ≤ n ≤ N, ≤ k ≤ K : t n
Any ξ ∈ S admits a decomposition (3.8) ξ = µ − α where µ is a finitely additive martingale and α a positive, finitely additive increasing process (as defined in [2, p. 287] ). Moreover, the following are equivalent: ( i ) ξ ∈ S uc ; ( ii ) µ and α in (3.8) may be chosen such that µ ∞ , α ∞ ∈ ca ( A ) (and thus so that µ and α are countablyadditive processes); ( iii ) there exists λ ∈ ca ( A ) + such that | ξ t | ≤ λ |A t for each t ∈ R + .Proof. Let ¯ x ∈ M ( ξ ) and define(3.9) µ t ( F ) = ξ ∞ ( F ) + ¯ x ( F × R + ) and α t ( F ) = ¯ x ( F × [0 , t ]) F ∈ A t Then (3.8) follows from (3.4). In fact µ is a finitely additive martingale while α extends to an increasingfamily (¯ α t : t ∈ R + ) of measures on A such that inf t k ¯ α t k = k α k = 0. If ( i ) holds, then upon choosing¯ x ∈ M ( ξ ) such that ¯ x | ¯ F , (3.9) implies ( ii ). If ( ii ) holds let λ = | µ ∞ | + 2 α ∞ . Then in restriction to A t weobtain | ξ t | ≤ | ξ ∞ | + ( ξ t − ξ ∞ ) ≤ | µ ∞ | + α ∞ + ( α ∞ − α t ) ≤ λ and ( iii ) follows. Assume ( iii ), then ξ ∞ ∈ ca ( A ).Let U = { t ≤ . . . ≤ t N } and define ¯ ζ Ut N ∈ ca ( A ) + to be an extension of ξ t N − ξ ∞ |A t N to A dominatedby λ − ξ ∞ and set ¯ ξ Ut N = ξ ∞ + ζ Ut N ; likewise, for n < N let ¯ ζ Ut n ∈ ca ( A ) + be an extension of ξ t n − ξ t n +1 |A t n to A dominated by λ − ¯ ξ Ut n +1 and set ¯ ξ Ut n = ¯ ξ Ut n +1 + ¯ ζ Ut n . Define ¯ ξ U = P Nn =1 ¯ ξ Ut n [ t n ,t n +1 [ a map from R + to ba ( A ).One easily establishes that ξ ∞ ≤ ¯ ξ Ut ≤ λ for each t ∈ R + , i.e. ¯ ξ U ∈ [ ξ ∞ , λ ] R + , that ¯ ξ U is decreasing and that¯ ξ Ut |A t = ξ t when t ∈ U . If ba ( A ) R + is equipped with the product topology obtained after endowing eachcoordinate space with the weak ∗ topology, we conclude that [ ξ ∞ , λ ] R + is compact and that the net (cid:10) ¯ ξ U (cid:11) U ∈U ,with U denoting the collection of finite subsets of R + directed by inclusion, admits a cluster point ¯ ξ . Thennecessarily, ¯ ξ is decreasing and ¯ ξ t |A t = ξ t for each t ∈ R + . The same argument used in the proof of Theorem2 shows that the quantity P Nn =1 ( ¯ ξ t n − ¯ ξ u n )( F n ), where F × { } ∪ S Nn =1 F n × ] t n , u n ] ∈ ¯ P , implicitly definesa measure on ¯ P which admits an extension ¯ x ∈ M ( ξ ) such that ¯ x ( F × R + ) = ( ¯ ξ − ¯ ξ ∞ )( F ) ≤ λ ( F ) so that¯ x | ¯ F ∈ ca ( ¯ F ). (cid:3) Corollary 1 establishes a general version of the Doob Meyer decomposition. In addition it characterisesexactly those processes ξ admitting a countably additive version of such decomposition. This characterisationimplies a condition hinging on the uniform countable additivity of the process ξ or, equivalently, a weak form GIANLUCA CASSESE of countable additivity of the Dol´eans-Dade measure, namely ¯ x | ¯ F ∈ ca ( ¯ F ). We shall return on this issue inthe 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.For H ⊂ ¯Ω let H ω denote the section { t ∈ R + : ( ω, t ) ∈ H } . Corollary 2.
Consider an increasing family ( A τ : τ ∈ T ) of algebras of subsets of Ω where T ⊂ ¯Ω isordered by reverse inclusion and let ξ ∈ S and ¯ x ∈ M ( ξ ) . There exists a finitely additive supermartingale ξ ∗ on ( A τ : τ ∈ T ) such that ¯ x ∈ M ( ξ ∗ ) . As a consequence ( i ) If τ ( t ) ≡ Ω × ] t, ∞ [ ∈ T and F ∈ A t ∩ A τ ( t ) then ξ ∗ τ ( t ) ( F ) = ξ t ( F )( ii ) If τ, υ ∈ T , F ⊂ Ω and F τ,υ ≡ { ω ∈ F : υ ω ⊂ τ ω } ∈ A τ ∩ A υ then ξ ∗ τ ( F τ,υ ) ≥ ξ ∗ υ ( F τ,υ ) Proof.
Fix ¯ x ∈ M ( ξ ) and define ξ ∗ τ ∈ ba ( A τ ) implicitly by letting(3.10) ξ ∗ τ ( F ) = ¯ ξ ∞ ( F ) + ¯ x (( F × R + ) ∩ τ ) F ∈ A τ 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 ∈ A τ and τ ≤ υ imply ξ ∗ τ ( F ) ≥ ξ ∗ υ ( F ) so that ξ ∗ is a finitely additive supermartingaleon ( A τ : τ ∈ T ). Moreover, if F ⊂ Ω and F τ,υ ∈ A τ ∩ A υ then ξ ∗ τ ( F τ,υ ) ≥ ξ ∗ υ ( F τ,υ ) is equivalent to¯ x (( F τ,υ × R + ) ∩ τ ) ≥ ¯ x (( F τ,υ × R + ) ∩ υ ) which follows from ¯ x being positive. (cid:3) Whenever τ ( t ) ∈ T and A t ⊂ A τ ( t ) for all t ∈ R + , Corollary 2 suggests that any ξ ∈ S may be consistentlyextended to any filtration indexed by T . Corollary 2 is an illustration of the importance of finite versuscountable additivity. 4. Two Decompositions
We shal prove in this section that all finitely additive supermartingales have a component that may berepresented as a classical supermartingale with respect to some P ∈ P ( F ). It should be highlighted that theprobability measure P involved here emerges endogenously, rather than being given from the outset, as inthe classical theory. We start with a preliminary result. Lemma 2.
Let
G ⊂ H be two algebras of subsets of Ω and denote by ca ( G , H ) and pf a ( G , H ) the subspaces of ba ( G ) consisting of set functions which admit a countably additive extension to H and whose norm preservingextensions to H are all purely finitely additive, respectively. For each λ ∈ ba ( G ) there exists a unique way ofwriting (4.1) λ = λ e + λ p where λ e ∈ ca ( G , H ) , λ p ∈ pf a ( G , H ) and λ e , λ p ≥ if and only if λ ≥ .Proof. With the aid of the Radon Nikodym theorem it is easily seen that λ ∈ ca ( G , H ) if and only if λ ≪ ¯ λ |G for some ¯ λ ∈ ca ( σ H ) and, thus, that ca ( G , H ) is an ideal. Let h λ α i α ∈ A be an increasing net in ca ( G , H ) + with W α ∈ A λ α = λ ∈ ba ( G ). Fix α ∈ A arbitrarily and, for given α n − , let α n ≥ α n − be such that λ α n (Ω) ≥ λ (Ω) − − n . If F ∈ G , λ ( F ) ≥ lim n λ α n ( F ) = λ (Ω) − lim n λ α n ( F c ) ≥ λ (Ω) − π ( F c ) = λ ( F ) INITELY ADDITIVE SUPERMARTINGALES 7
But then, λ ≪ P n − n λ α n ∈ 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 ) extend λ ∈ ca ( G , H ) ⊥ . Then there exists G n ∈ G such that | λ | ( G cn ) + | ¯ λ c | ( G n ) < − n . If G ∈ G| λ ( G ) | = lim n | λ ( G ∩ G n ) | = lim n | ¯ λ ⊥ ( G ∩ G n ) | ≤ | ¯ λ ⊥ | ( G )i.e. k ¯ λ c k + k ¯ λ ⊥ k = k ¯ λ k = k λ k ≤ k ¯ λ ⊥ k . In other words, λ ∈ pf a ( G , H ). (cid:3) Lemma 2 is a slight generalization of the classical decomposition of Yosida and Hewitt (by uniquenessthe two decompositions coincide for G = H ). It has though an important implication here as it implicitlysuggests that finitely additive supermartingales may admit a component that can be represented as a classicalsupermartingale with respect to some P ∈ P ( F ). Proposition 1.
Let ξ ∈ S + . For each t ∈ R + let ξ t = ξ et + ξ pt with ξ et ∈ ca ( A t , F ) and ξ pt ∈ pf a ( A t , F ) andset ξ e = ( ξ et : t ∈ R + ) and ξ p = ( ξ pt : 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 ) while ξ p is positive and orthogonal to all finitely additive processes admitting suchrepresentation. We say that ξ e is representable and that the pair ( P, X ) is a representation of ξ e .Proof. The inclusion ξ e ∈ S + was shown in the Introduction. As ξ p is clearly orthogonal to any classicalstochastic process, we only need to prove that ξ e admits a representation. Define the function T ( t ) = k ξ et k and the set J = { t ∈ R + : T ( t ) > sup u>t T ( 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 thereis a decreasing sequence h t k i k ∈ N in C such that lim k T ( t k ) = T ( t ). Let h t n i n ∈ N be an explicit enumerationof D = C ∪ J , choose ¯ ξ et n ∈ ca ( F ) such that ¯ ξ et n (cid:12)(cid:12) A t n = ξ et n , fix Q ∈ P ( F ) and let ¯ P = Q + P n − n ¯ ξ et n and P = k ¯ P k − ¯ P . Clearly, P ∈ P ( F ), P ≫ ¯ ξ et n for each n ∈ N . By construction, for each t ∈ R + and k > t k ∈ D such that t ≤ t k and ( ξ et − ξ et k )(Ω) ≤ − k . Remark that ( ¯ ξ et − ¯ ξ et k ) (cid:12)(cid:12) σ A t ∈ ca ( σ A t )is the (unique) countably additive extension of ξ et − ξ et k |A t to σ A t and is therefore positive. We concludethat ¯ ξ et ( F ) = lim k ¯ ξ et k ( F ) for each F ∈ σ A t . By Vitali Hahn Saks theorem and its corollaries [15, III.7.2-3],¯ ξ et | σ A t ≪ P | σ A t , i.e. ξ e is representable. (cid:3) Uniformly countably additive supermartingales play a special role in the following section.
Proposition 2.
Each ξ ∈ S admits a unique decomposition ξ = ξ uc + ξ up where ξ uc ∈ S uc and ξ 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 ¯ x ′ ∈ M by letting ¯ x ′ ( H ) = ¯ x c ¯ F (cid:16) ¯ x (cid:16) H (cid:12)(cid:12)(cid:12) I ¯ x ⊥ ¯ F (cid:17)(cid:17) H ⊂ ¯ΩThen, by (2.3), ¯ x ′ ¯ F = ¯ x c ¯ F – so that ¯ x ′ ∈ M uc . Letting I n ∈ I ¯ x ⊥F be such that ¯ x c F ( I cn ) < − n ¯ x ′ ( H ) = lim n ¯ x c ¯ F (cid:16) I n ¯ x (cid:16) H (cid:12)(cid:12)(cid:12) I ¯ x ⊥ ¯ F (cid:17)(cid:17) = lim n ¯ x ( I n H ) ≤ ¯ x ( H ) H ⊂ ¯ΩClearly, ¯ x ′′ = ¯ x − ¯ x ′ ∈ M up . Thus the set M ∗ ( ξ ) = { ¯ y ∈ M uc : ¯ y ≤ ¯ x for some ¯ x ∈ M ( ξ ) } is nonempty and, we claim, it admits a maximal element with respect to the partial order ≥ ¯ F defined by letting The property defined here was called the Kolmogoroff property by Bochner [8, p. 164]
GIANLUCA CASSESE ¯ y ≥ ¯ F ¯ y ′ whenever ¯ y ¯ F ≥ ¯ y ′ ¯ F . In order to apply Zorn lemma, consider an increasing net h ¯ y α i 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 ∈ A ¯ x 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 ∈ A ¯ x α ( H ) ≥ lim inf α ∈ A ¯ y α ( H ) which holds for any H ⊂ ¯Ω implies that ¯ x ◦ ≥ ¯ F ¯ y α for all α ∈ A i.e. that ¯ x ◦ is an upper bound for h ¯ y α i α ∈ A . Let ¯ x uc be a maximal element of M ∗ ( ξ ), let ¯ x ∗ ∈ M ( ξ )be such that ¯ x uc ≤ ¯ x ∗ and define ¯ x up = ¯ x ∗ − ¯ x uc ∈ M . Let ξ uc , ξ up ∈ S be uniquely defined by the condition¯ x uc ∈ M ( ξ uc ), ¯ x up ∈ M ( ξ up ), ξ uc ∞ = ξ c ∞ and ξ up ∞ = ξ ⊥∞ . By construction, ξ uc ∈ S uc . Decompose ¯ y ∈ M ( ξ up )as ¯ y ′ + ¯ y ′′ where ¯ y ′ ∈ M uc and ¯ y ′′ ∈ M up , as in the first step of this proof. From ¯ x uc + ¯ y ′ ≤ ¯ x uc + ¯ y ∈ M ( ξ ) andthe fact that ¯ x uc is ≥ ¯ F maximal, we deduce ¯ y ′ = 0 or, equivalently, ξ up ∈ S up . If ξ = κ uc + κ up were anothersuch 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 , thisimplies ¯ k up ≤ ¯ x up while the converse is obtained mirrorwise. In other words κ up and ξ up induce the sameDol´eans-Dade measure; in addition, κ up ∞ = ξ up ∞ = ξ ⊥∞ . The claim follows from Theorem 2( iii ). (cid:3) Increasing Processes
Fix P ∈ P ( F ) and let A ( P ) denote the set of processes ( A t : t ∈ R + ) such that A ∞ ∈ L ( P ) and P (0 = A ≤ A t ≤ A u ) = 1 for each 0 ≤ t ≤ u < ∞ . Of course, if A ∈ A ( P ) and A ′ is a modification of A (i.e. P ( A ′ t = A t ) = 1 for all t ∈ R + ) then A ′ ∈ A ( P ). Put A = S P ∈ P ( F ) A ( P ). Lemma 3.
Let A ∈ A ( P ) . Then there is F ∈ F with P ( F c ) = 0 and a modification A ′ of A such that foreach t ≤ u , A ′ ≤ A ′ t ≤ A ′ u on F . If in addition P ( A t ) = lim n P ( A t +2 − n ) then A ′ and F may be chosento be right continuous at each t ∈ R + and for each ω ∈ F .Proof. As in the proof of Proposition 1, there exists a countable subset D of R + with the property that foreach t ∈ R + and ǫ > d ∈ D such that d ≥ t and P ( A t ) > P ( A d ) − ǫ . Define F = T { d,d ′ ∈ D : d>d ′ } { A d ≥ A d ′ } : clearly, P ( F c ) = 0. Let D ( t ) = { d ∈ D : d ≥ t } and A ′ t = inf d ∈ D ( t ) A d . By definition of D , A ′ t ≥ A t but P ( A ′ t ) = P ( A t ) so that P ( A t = A ′ t ) = 1. If A is right continuous in mean the same conclusionholds even if we replace D ( t ) with D + ( t ) = { d ∈ D : d > t } . However A ′ is right continuous on F since D + ( t ) = T u>t D + ( u ). (cid:3) For H = ( F ×{ } ) ∪ S Nn =1 ( F n × ] t n , u n ]) ∈ ¯ P and A ∈ A ( P ) the integral R H dA has an obvious definition,namely P Nn =1 F n ( A u n − A t n ). In the following Theorem we obtain an extension of this integral togetherwith a characterization of increasing processes in terms of their Dol´eans-Dade measure. Theorem 3.
Let ¯ x ∈ M . The following are equivalent: ( i ). There exists P ∈ P ( F ) and, given P , a unique (up to modification) A ∈ A ( P ) such that (5.1) ¯ x ( H ) = P Z H dA H ∈ ¯ P ( ii) . ¯ x ∈ M uc and ¯ x ( { } ) = 0 ; ( iii ). There exists P ∈ P ( F ) such that for each h ∈ L (¯ x ) the equation (5.2) ¯ x ( bh ) = P ( bI ¯ x ( h )) b ∈ B ( F ) admits a unique solution I ¯ x ( h ) ∈ L ( P ) such that I ¯ x ( { } ) = 0 . INITELY ADDITIVE SUPERMARTINGALES 9 I ¯ x : L (¯ x ) → L ( P ) as defined in (5.2) is a positive, continuous, linear functional such that k I ¯ x k = k ¯ x k and that lim n I ¯ x ( h n ) = 0 whenever sup n | h n | ∈ L (¯ x ) and lim n P ∗ ( h ∗ n > η ) = 0 for each η > – where h ∗ n ≡ sup t | h n,t | and P ∗ 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 ( { } ) = P R { } dA = 0 and ¯ x ( F ) = ¯ x ( F × ]0 , ∞ [) = P ( F A ∞ ) for any F ∈ F . Assume ( ii ) and fix P = ( k Q k + k ¯ x k ) − ( Q + ¯ x | ¯ F ) for some Q ∈ P ( F ). By Theorem 1, for each h ∈ L (¯ x ) we may define I ¯ x ( h ) = ¯ x ( h | ¯ F ) d ¯ x | ¯ F dP ∈ L ( 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 ¯ x k , byTheorem 1; ¯ x ( { } ) = 0 implies I ¯ x ( { } ) = 0, P a.s.. Any other solution J ( h ) ∈ L ( P ) to (5.2) satisfies P ( bJ ( h )) = P ( bI ¯ x ( h )) for all b ∈ L ∞ ( P ) i.e. P ( J ( h ) = I ¯ x ( h )) = 1. Assume ( iii ) and define A t = I ¯ x ( ]0 ,t ] ), A = ( A t : t ∈ R + ) and let H = ( F × { } ) ∪ S Nn =1 ( F n × ] t n , u n ]) ∈ ¯ P . Then A ∈ A ( P ) and, up to a P null set I ¯ x ( H ) = N X n =1 ¯ x (cid:0) F n × ] t n ,u n ] (cid:12)(cid:12) ¯ F (cid:1) d ¯ x | ¯ F dP = N X n =1 F n ¯ x (cid:0) ] t n ,u n ] (cid:12)(cid:12) ¯ F (cid:1) d ¯ x | ¯ F dP = N X n =1 F n ( A u n − A t n )= Z H dA But then (5.1) follows from (5.2). If B ∈ A ( P ) also meets (5.1) then for h = F × ]0 ,t ] and F ∈ F we concludethat P ( F A t ) = P ( F B t ) from which we deduce uniqueness. It is clear from (5.1) that I ¯ x is linear, positiveand that k I ¯ x k = k ¯ x k .If h h n i n ∈ N is a sequence in L (¯ x ) with the above properties then so is h| h n |i n ∈ N . Given that I ¯ x is positive,it is enough to prove the claim for h n ≥
0. Observe that ¯ x ( h n ≥ η ) ≤ ¯ x ( h ∗ n ≥ η ); moreover, ¯ x | ¯ F ≪ P impliesthat, in restriction to 2 Ω ⊗ { ∅ , R + } , ¯ x ≪ P ∗ . But then, h n converges to 0 in ¯ x measure and, by [15, theoremIII.3.6], in L (¯ x ). Given (5.2) this is equivalent to I ¯ x ( h n ) converging to 0 in L ( P ). (cid:3) The equivalence of ( i ) and ( ii ) establishes a correspondence between A and M uc which compares to theclassical (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 Meyerdecomposition. Although there are connections between these two properties, it is noticeable that the latteris independent of the given filtration. On should also remark that we do not assume the existence of anunderlying probability P ∈ P ( F ) but rather deduce it.Each ¯ x ∈ M may be considered in restriction to special classes of functions such as the set C of functions f : ¯Ω → R with continuous sample paths and bounded support (i.e. such that f ( t ) = 0 for all t larger thansome T ). Let C be the σ algebra on ¯Ω generated by C . Lemma 4.
Let ¯ x ∈ M c . There exist α c ∈ ca ( C ) + , P ∈ P ( F ) and A 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 sequence h h n i n ∈ N in the vector lattice L (¯ x ) ∩ C decreasingto 0 and fix T such that ¯ x ( | h − h T | ) < ǫ , where h Tn = h n ]0 ,T ] . Let h T, ∗ n = sup t h Tn ( t ). A simple applicationof Dini’s theorem for each ω ∈ Ω guarantees that the sequence (cid:10) h T, ∗ n (cid:11) n ∈ N converges to 0 pointwise; moreover,by continuity of the sample paths, h T, ∗ n is in fact F measurable. Thus Theorem 3 implies that lim n ¯ x ( h n ) ≤ ǫ + lim n ¯ x ( h Tn ) = ǫ + lim n P ( I ¯ x ( h Tn )) = ǫ . In other words, the restriction of ¯ x to C is a Daniell integraland 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 ct ∈ ba ( F ) as α c ( F × ]0 , t ]) for each F ∈ F . Since α ct ≤ ¯ x ¯ F ≪ P , denote by A ct the Radon Nikodym derivative of α ct withrespect to P . We deduce that P (( A cu − A ct ) F ) = α c ( F × ] t, u ]) = ¯ x ( F × ] t, u ]) ≥
0, so that A c ∈ A ( P ), andthat lim n P ( F ( A ct +2 − n − A ct )) = lim α c ( F × ] t, t + 2 − n ]) = 0 (by countable additivity) for each F ∈ F so that A ct = lim n A ct +2 − n up to a P null set. By Lemma 3, we obtain that A c admits a modification which is rightcontinuous. (cid:3) Theorem 4.
Let ξ ∈ S . Then ξ ∈ S uc if and only if there exist P ∈ P ( F ) , M ∈ L ( P ) and A p ∈ A ( P ) which is adapted, right continuous in mean and such that (5.4) ξ t ( F ) = P ( F ( M − A pt )) t ∈ R + and F ∈ F t and that (5.5) P (cid:18) b Z hdA p (cid:19) = P Z M ( b ) − hdA p b ∈ L ∞ ( P ) , h ∈ B ( σ P ) where M ( b ) = ( P ( b |F t ) : t ∈ R + ) .Proof. We use the notation of Lemma 4 and the inclusion σ P ⊂ C . Let d = { t ≤ t ≤ . . . ≤ t N } be a finitesequence in R + and define(5.6) P d ( f ) = N − X n =1 P ( f t n |F t n ) ] t n ,t n +1 ] and ¯ x d ( f ) = ¯ x ( P d ( f )) f : R + → L ( P )Denote by α d the restriction of ¯ x d to F ⊗ R + . On the one hand it is easily seen that ¯ x d ¯ F ≪ P so that, asin the proof of Theorem 3, we can associate to ¯ x d a process A d ∈ A ( P ), by letting A dt dP = ¯ x d ( ]0 ,t ] | ¯ F ) d ¯ x d ¯ F .On the other hand, (5.3) implies P R f dA d = ¯ x ( P d ( f )) = α c ( P d ( f )). Consider the case in which f = bh where b ∈ L ∞ ( P ) and h is bounded, adapted and left continuous. Let d n = { k − n : k = 0 , . . . , n } andobserve that, by [11, VI.2, p. 67], there exists a P null set F ∈ F outside of which lim n P d n ( b ) t = M ( b ) t − and lim n P d n ( h ) t = h t for each t ∈ R + (as h is left continuous and adapted). Given that α c is countablyadditive in restriction to C , we conclude(5.7) lim n P (cid:18) b Z hdA d n (cid:19) = lim n α c ( P d n ( bh )) = lim n α c ( P d n ( b ) P d n ( h )) = α c ( M ( b ) − h )Define then α p ∈ ba ( F ⊗ R + ) implicitly as α p ( H ) = α c ( M ( H ) − ). Then from (5.7) we deduce that (cid:10) α d n (cid:11) 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 processassociated to α p . Thus for every bounded, adapted and left continuous process h and every b ∈ L ∞ ( P ) wehave P (cid:18) b Z hdA p (cid:19) = ¯ x ( M ( b ) − h ) = α c ( M ( b ) − h ) = P Z M ( b ) − hdA p INITELY ADDITIVE SUPERMARTINGALES 11 which delivers (5.4) if we only let M = dξ ∞ /dP + A p ∞ , b = F with F ∈ F t and h = ] t, ∞ [ . In addition, if F ∈ F , s ≤ t and h F,t = F − P ( F |F t ), then M ( h F,t ) s − = 0 so that P ( h F,t A pt ) = α p ( h F,t [0 ,t ] ) = P Z t M ( h F,t ) − dA p = 0Therefore, replacing A pt with P ( A pt |F t ), we may assume that A p ∈ A ( P ) is adapted. Eventually, letting h n = ] t,t +2 − n ] we conclude that 0 = lim n α c ( h n ) = lim n P ( A pt +2 − n − A pt ) and, thus, that A p is rightcontinuous in mean. That (5.4) implies ξ ∈ S uc is obvious. (cid:3) With a complete filtration Theorem 4 implies that A p may 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 underlyingfiltration. Corollary 3.
Let ξ ∈ S and let D ⊂ R + be such that ξ t = sup d ∈ D ( t ) ξ d |A t where D ( t ) = { d ∈ D : d ≥ t } .Then, ξ admits a Doob Meyer decomposition if and only if ξ D = ( ξ d : d ∈ D ) does.Proof. Given that, by Theorem 4, the Doob Meyer decomposition is equivalent to ξ ∈ S uc , the directimplication is obvious. As for the converse, choose ¯ x D ∈ M ( ξ D ) to be countably additive in restriction to F ⊗ { ∅ , D } . If t ∈ R + and F ∈ A t then(5.8) | ξ t | ( F ) ≤ sup d ∈ D ( t ) | ξ d | ( F ) ≤ | ξ ∞ | ( F ) + ¯ x D ( F × D ) ≡ λ ( F )The claim then follows from Corollary 1. (cid:3) Corollary 3 makes clear that decomposition (5.4) is a property that involes any subset D which is densefor the range of ξ and we know from the proof of Proposition 1 that this may be taken to be countable. Theclass D property may thus be replaced by a corresponding property, the class D σ , in which the stoppingtimes 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 = (cid:26)Z hdf : h is P simple , | h | ≤ (cid:27) and C = K − B ( F ) + Bichteler and Dellacherie start from the assumption that K is bounded in L ( P ) for some given P ∈ P ( F )and that f is right continuous with left limits outside some P null set. These two properties are then shownto imply that for given η > δ > d F / ∈ C F for all F ∈ F such that P ( F ) > η . Wetake inspiration from this separating condition to define a concept of boundedness suitable for our setting.To this end we denote by U a collection of subets of Ω with the following properties: Assumption 1.
There exists λ > such that (6.2) { λ U : λ ≥ λ } ∩ C U = ∅ U ∈ U Moreover,
U, V ∈ U imply U ⊂ { f ∗ < n } for some n and U ∪ V ∈ U . A violation of (6.2) indicates that the set K is unbounded relatively to some U ∈ U . Both sets in (6.2)are convex subsets of B (2 U ) and U C contains − U as an internal point. By ordinary properties of thesupport 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 each U ∈ U there is ˆ m U ∈ ba (2 U ) such that sup x ∈ U C ˆ m U ( x ) ≤ λ ˆ m U ( U ). Theinclusion − B (2 U ) + ⊂ U C implies that ˆ m U ≥
0. By defining m U ∈ ba + implicitly by m U ( F ) = ˆ m U ( F ∩ U )ˆ m U ( U ) F ⊂ Ωwe have completed the proof of the following:
Lemma 5.
Let f : ¯Ω → R + satisfy Assumption 1 and define the set (6.3) M = (cid:26) m ∈ ba + : k m k = 1 , sup x ∈C m ( x ) ≤ λ (cid:27) For each U ∈ U there exists m U ∈ M such that m U ( U ) = 1 . Fix now m ∈ M (so that f ∗ ∈ L ( m )) and let ξ e and ξ p be the components of the finitely additivesupermartingale ( m |F t : t ∈ R + ) as of (4.2). Set also I t = I ξ pt (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 ) of ξ ∞ to F and¯ x ∈ M ( ξ e ) and define ¯ ξ pt ( F ) = ¯ ξ p ∞ ( F ) − ¯ x ( F × ] t, ∞ [) F ∈ F The collection ( ¯ ξ pt : t ∈ R + ) is then increasing with t . For each b ∈ B ( F ) and u ≥ t , F ∈ I t implies ξ p ∞ ( b F ) = ¯ x ( b F × ] t, ∞ [ ) and thus(6.4) ¯ ξ pu ( b F ) = ξ p ∞ ( b F ) − ¯ x ( b F × ] u, ∞ [ ) = ¯ x ( b F × ] t,u ] )Let now d = { t ≤ . . . ≤ t N } ,(6.5) P d = ( F × { } ∪ N − [ n =1 F n × ] t n , t n +1 ] : F ∈ F , F n ∈ F t n , ≤ n ≤ N − ) and choose F n ∈ I t n ≤ n < N and set F d = N − [ n =1 F n × ] t n , t n +1 ] and f d = f { } + N − X n =1 f t n +1 ] t n ,t n +1 ] By (6.4) N − X n =1 ξ pt n +1 (( f t n +1 − f t n ) F n ) = N − X n =1 ¯ x (( f t n +1 − f t n ) F n × ] t n ,t n +1 ] )= ¯ x ( f d F d ) − x N − X n =1 f t n F n × ] t n ,t n +1 ] ! = ¯ x ( f d F d ) + P N − X n =1 f t n F n ( X t n +1 − X t n ) INITELY ADDITIVE SUPERMARTINGALES 13 i.e. m (cid:18)Z F d df (cid:19) = N − X n =1 m (( f t n +1 − f t n ) F n )= N − X n =1 ( ξ pt n +1 + ξ et n +1 )(( f t n +1 − f t n ) F n )(6.6) = ¯ x ( f d F d ) + P N − X n =1 F n ( f t n +1 X t n +1 − f t n X t n )Assume that H = H { } + S N − n =1 H n ] t n ,t n +1 ] ∈ P d . Then by (3.5)¯ x ( H ) = P ( H ( X − X ∞ ) + N − X n =1 H n ( X t n − X t n +1 ) ) i.e. ¯ x |P d is countably additive. Replacing F d with a sequence (cid:10) F d,k (cid:11) k ∈ N such that lim k P (cid:16)T N − n =1 F kn − (cid:17) = 1we thus deduce then from (6.7)(6.7) lim k m (cid:18)Z F d,k df (cid:19) = ¯ x ( f d ) + P ( f ∞ X ∞ − f X )Replace f with R F × ] t,u ] df where t ≤ u and F ∈ F t and choose d such that F × ] t, u ] ∈ P d . We also deduce(6.8) lim k m (cid:18) F Z ut F d,k df (cid:19) = P ( F ( f u X u − f t X t )) + ¯ x ( f d F × ] t,u ] ) Theorem 5.
Let f ∈ R ¯Ω satisfy Assumption 1. Then there exists P ∈ P ( F ) and a P positive supermartingale X such that Xf is a P quasimartingale. If there is Q ∈ P ( F ) and η > such that Q ( f ∗ < ∞ ) = 1 and that F ∈ F and Q ( F ∩ { f ∗ < k } ) ≥ η imply F ∩ { f ∗ < k } ∈ U then for any δ > η the pair ( P, X ) above can bechosen such that P ( X ∞ = 0) < δ .Proof. By Lemma 5 for fixed n > n there is m ∈ M such that m ( f ∗ > n ) = 0 so that m ( f ∗ ) < ∞ . By(6.4), | ¯ x ( f d ) | ≤ N − X n =1 ( ¯ ξ pt n +1 − ¯ ξ pt n ) (cid:18) sup
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