Quasi-sure analysis, aggregation and dual representations of sublinear expectations in general spaces

El e c t ro nic J

o f

Pr

ob a bi l i t y

Electron. J. Probab.17(2012), no. 62, 1–15.

ISSN:1083-6489 DOI:10.1214/EJP.v17-2224

Quasi-sure analysis, aggregation and dual representations of sublinear expectations

in general spaces

Samuel N. Cohen

Abstract

We consider coherent sublinear expectations on a measurable space, without assum- ing the existence of a dominating probability measure. By considering a decomposi- tion of the space in terms of the supports of the measures representing our sublinear expectation, we give a simple construction, in a quasi-sure sense, of the (linear) con- ditional expectations, and hence give a representation for the conditional sublinear expectation. We also show an aggregation property holds, and give an equivalence between consistency and a pasting property of measures.

Keywords:sublinear expectation; capacity; aggregation; dual representation.

AMS MSC 2010:60A10; 60A86; 91B06.

Submitted to EJP on October 26, 2011, final version accepted on August 1, 2012.

SupersedesarXiv:1110.2592v2.

1 Introduction

Decision making in the presence of uncertain outcomes is a fundamental human ac- tivity. In many cases, we need to make decisions, not only when we do not know what the outcome of our decision will be, but when we do not even know the probabilities of different outcomes. In this setting (commonly known as Knightian uncertainty, follow- ing [11]) the classical mathematical approach based on the mathematical expectation is insufficient. An alternative approach in this context is to take the ‘worst case’ under a range of different probability measures, which leads to a form of risk-averse decision making. This approach has strong axiomatic support (see Theorem 2.3) and is amenable to mathematical analysis.

When all the probability measures we consider agree on what events will occur with probability zero, this approach is, from a mathematical perspective, a relatively straightforward generalisation of the classical theory. On the other hand, when the measures do not agree in this manner (and more generally, when there is no domi- nating probability measure), then many difficulties arise, cutting to the heart of the mathematical theory of probability. In particular, results which are known to hold ‘with

∗Mathematical Institute, University of Oxford, UK. E-mail:[email protected]

probability one’ in the classical setting (for example, the existence and uniqueness of the conditional expectation, martingale convergence results, the martingale represen- tation theorem, etc...) may cease to be true in this more general setting.

In some ways, this issue may seem unreasonably abstract, however it arises even in the common case of the analysis of a Brownian motion, where the volatility is known only to lie within a given bound. This problem has been studied in various frameworks by various authors, for example, Lyons [12], Peng and coauthors [15, 6, 3], Soner, Touzi and Zhang [16, 17], Bion-Nadal and Kervarec [2] and Nutz [13], amongst many others.

In this type of analysis, the detailed structure of the mathematical spaces under consideration comes to the fore, and some technical details are needed. One option is to assume that the underlying measurable space can be viewed as a separable topo- logical space (Ω,B(Ω)), and then to only consider those random variables which are quasi-continuous as functionsΩ → R. This is the approach taken in Denis et al. [6].

This is in some ways unsatisfactory, as it implies that there are events (which can be easily assigned probabilities in the classical setting) which we refuse to consider when in the setting of uncertainty, purely due to insufficient continuity. Furthermore, by re- sults of Bion-Nadal and Kervarec [2], for random variables in this class there exists a dominating probability measure, that is, there exists a measureθ^∗ such that a (quasi- continuous) set is null for every test measure if and only if it is θ^∗-null. In this sense, the problem is avoided, as classical methods can be used.

A different assumption is made in Soner, Touzi and Zhang [17], where the set of test measures is assumed to be made up of measures in a particular separable class. In par- ticular, they consider the measures induced on Wiener space by right-constant volatility processes satisfying some further restrictions (see Example 3.8). Under this assump- tion, they prove an aggregation property, with which much of the desired analysis can be performed. This approach is possibly unsatisfying as it is restricted to the problem of volatility uncertainty, and it is not apparent how this would generalize to other situa- tions. For example, in discrete time (as one might obtain simply by taking theδ-skeleton of their setting), there is no process analogous to the volatility of the Wiener process with which to parameterize the test measures, yet some regularity assumptions are needed.

In this paper we seek to provide such regularity assumptions, in a manner consistent with [17]. We shall assume thatΘ, the set of test measures, permits a Hahn-like decom- position of the underlying spaceΩ, uniformly in all the measures inΘ. A key step in the proof of the main aggregation result in [17] is to verify that a stronger version of our assumption holds (our Lemma 3.6 holds); we show that our weaker version is sufficient to guarantee their result holds (Theorem 3.16), and that with our assumption the proof is remarkably simple. On the other hand, our assumption has a natural interpretation in any space, rather than in the particular case of uncertain volatility. We shall also show that there are natural results regarding the pasting of measures and the representation of conditional sublinear expectations which follow directly from our assumption.

2 Sublinear expectations

The theory of sublinear expectations lies at the heart of our study. These opera- tors can either be defined on probability spaces, when they are related to the theory of BSDEs, or can be defined using the approach of quasi-sure analysis, for example the G-expectation of Peng [15] or the 2BSDEs of Soner, Touzi and Zhang [16, 17], amongst many others. In discrete time, the theory of sublinear expectations using quasi-sure analysis is discussed in [3]. In this work, we shall use the approach of quasi-sure analy- sis, and shall be quite general about the types of probability spaces under consideration.

Let (Ω,F)be a measurable space, and letmF denote theF/B(R)-measurable real valued functions. We wish to define a sublinear expectation on this space, that is, a map taking random variables toRsatisfying some useful properties. We begin by defining the space of random variables for which the expectation will be well defined.

Definition 2.1. LetHbe a linear space ofF-measurableR-valued functions onΩcon- taining the constants. We assume thatX ∈ H implies|X| ∈ Hand IAX ∈ Hfor any A∈ F.

Definition 2.2. A mapE:H →Rwill be called a coherent sublinear expectation if, for allX, Y ∈ H, it is

(i) (Monotone:) ifX≥Y (for allω) we haveE(X)≥ E(Y), (ii) (Constant invariant:) for constantsc,E(c) =c,

(iii) (Cash additive:) for constantsc,E(X+c) =E(X) +c, (iv) (Coherent:) for all constantsc >0,E(cX) =cE(X), and

(v) (Sublinear:)E(X+Y)≤ E(X) +E(Y),

(vi) (Monotone continuous:) forX_n a nonnegative sequence inHincreasing pointwise toX,E(X_n)↑ E(X).

Due to its convexity, a coherent sublinear expectation has a simple representation.

Theorem 2.3(See [4, Theorem 3.2], [15, Theorem I.2.1]). A coherent sublinear expec- tation has a representation

E(X) = sup

θ∈Θ

Eθ[X] (2.1)

whereΘ is a collection of (σ-additive) probability measures on Ω. For simplicity, we shall say thatΘrepresentsE.

Once we have this representation, it is natural to wonder how far we can extendEto functions not inH. Clearly we can defineE for every boundedF-measurable function.

As we will not, in general, know that our measures inΘwill be absolutely continuous (in fact, the focus of this paper is on the case where they are not), we cannot simply complete F under some reference measure, however this leads us to the following definition.

Definition 2.4. LetΘbe a collection of probability measures on(Ω,F). LetF^θdenote the completion ofF under the measureθ. We write

F^Θ= \

θ∈Θ

F^θ.

The collectionF^Θ is aσ-algebra, and everyθ∈Θhas a unique extension toF^Θ. Definition 2.5. A setN∈ F^Θis called a (Θ-)polar setifθ(N) = 0for allθ∈Θ.

Remark 2.6. A natural alternative to the use ofF^Θis to simply completeF by adding the polar sets. That is, ifN denotes the polar sets, functions which areF ∨N-measurable are the main objects of study. By considering the setF^Θ, we allow a far richer class of functions, as is made clear by the following easy proposition. Theσ-algebraF^Θis also used in [17] and [13], where it is called the universal completion ofF.

Proposition 2.7. ForΘa family of probability measures on(Ω,F), whereN denotes theΘ-polar sets andF^θthe completion ofFunderθ,

F ⊆ F ∨ N ⊆ F^Θ⊆ F^θ for anyθ∈Θ.

Example 2.8. Let Ω = [0,1], F =B(Ω)and Θ = {δx}_x∈[0,1], the set of discrete point- mass measures on Ω. Then N = {∅}, so F ∨ N = B(Ω). However, F^θ = 2^Ω for all θ, so F^Θ = 2^Ω. This is perfectly reasonable, as one can take the expectation of any function underδx for anyx, so there is no need to insist on any stronger concepts of measurability.

Definition 2.9. LetΘbe a collection of probability measures on(Ω,F). We say that a functionX : Ω→R^is

• inmF^Θif it isF^Θ-measurable,

• inH^Θ_F if X ∈mF^Θ and at least one ofEθ[X_θ⁺]and Eθ[X_θ⁻] is finite for allθ ∈Θ, and

• inL¹(E;F)ifX ∈mF^Θandsup_θE_θ[|X|]is finite (and similarlyL^p(E;F)).

We can now extendEto the larger spaceH^Θ_F. Definition 2.10. We define the operator

E¯:H_F^Θ→R, X7→sup

θ∈Θ

E_θ[X],

It is easy to verify thatE¯satisfies properties (i-iv) and (vi) of Definition 2.2 withH replaced byH^Θ_F, as a mapH^Θ_F →R∪ {±∞}. It also satisfies property (v) provided all terms are well defined (in particular, this is satisfied onL¹(E)). Furthermore, comparing with Definition 2.2 and Theorem 2.3 we haveH ⊆ H^Θ_F andE|¯_H=E.

Hereafter, we shall takeΘas fixed, and simply writeH_F forH^Θ_F andEforE¯, when- ever this does not lead to confusion. However, we shall still distinguish betweenF and F^Θ.

Remark 2.11. We note at this point that we have defined our nonlinear expectation for allX in H^Θ_F, in line with the classical approach of measure theory when defining the integral for a wide class of functions. As mentioned above, this contrasts with many other works, for example [6], where the nonlinear expectation is only defined for quasi- continuous (as opposed to simply measurable) random variables.

Definition 2.12. We say that a statement holds quasi-surely (q.s.) if it holds except on a polar set.

2.1 Conditional sublinear expectations

Suppose now that we have a sub-σ-algebraG ⊆ F. In exactly the same way as before (Definition 2.9), we can define the spaceH_G^Θ, and it is easy to verify thatH^Θ_G ⊆ H^Θ_F and G^Θ⊆ F^Θ. As before, we shall simply writeHG forH^Θ_G.

We wish to consider the sublinear expectation conditional onG. This is an operator satisfying the following properties.

Definition 2.13. A pair of maps

E :HF→R

E_G :L¹(E;F)→L¹(E;G)

is called aG-consistent coherent sublinear expectation if for anyX, Y ∈L¹(E;F)

(i) Eis a coherent sublinear expectation

(ii) (Recursivity)E ◦ EG=E onL¹(E;F), that is,E(EG(X)) =E(X), (iii) (G-Regularity)E_G(IAY) =IAE_G(Y)q.s. for allA∈ G^Θ.

(iv) E_Gsatisfies the requirements of a coherent sublinear expectationG^Θ-conditionally, that is

(a) (G-monotonicity)X ≥Y impliesEG(X)≥ EG(Y)q.s.

(b) (G-triviality)EG(Y) =Y q.s. for allY ∈L¹(E;G).

(c) (G-cash additivity)E_G(X+Y) =E_G(X) +Y q.s. for allY ∈L¹(E;G). (d) (G-sublinearity)E_G(X+Y)≤ E_G(X) +E_G(Y)q.s.

(e) (G-coherence) EG(λY) = λ⁺EG(Y) +λ⁻EG(−Y) q.s. for all λ ∈ mG^Θ with (λY)∈L¹(E;G).

The following simple lemma gives uniqueness of the conditional expectation.

Lemma 2.14. For a given coherent sublinear expectationE, a givenG ⊆ F, there exists at most one conditional coherent sublinear expectationE_G, up to equality q.s.

Proof. For a givenX, supposeE_GandE¯_Gare two versions of the conditional expectation.

By theG-triviality and cash additivity properties, we can see thatE¯_G(X −E¯_G(X)) = 0 q.s., and hence by regularity, for anyA∈ G^Θwe haveE(IA(X−E¯G(X))) = 0. Similarly we see that

E(IA(E_G(X)−E¯_G(X))) =E(E_G(IA(X−E¯_G(X)))) =E(IA(X−E¯_G(X))) = 0.

Therefore, takingAn ={ω:EG(X)>E¯G(X) +n⁻¹} ∈ G^Θ, we have 0≤ E(IA_nn⁻¹)≤ E(IA_n(E_G(X)−E¯_G(X))) = 0

and henceE(IA_n)is polar. Therefore∪nAn is polar, that is,EG(X)≤E¯G(X)q.s. Revers- ing the roles ofEG andE¯G yields the reverse inequality.

Note that, as in the classical case, we shall only require in the definition that E_G is well defined onL¹(E;F). However, it will often be the case (cf Remark 3.22) that the conditional expectation is well defined on a space of functions with significantly less integrability.

3 Representing the conditional expectation

For a givenG-consistent sublinear expectationE, we wish to have a representation of the conditional expectationE_G similar to that in Theorem 2.3. That is, we wish to write

“E_G(X) = sup

θ∈Θ

Eθ[X|G].” (3.1)

This statement has two key problems. First, the conditional expectationEθ[·|Ft]is only definedθ-a.s. rather thanE-q.s. WhenΘconsists of uncountably many possibly singular probability measures, this causes a significant problem. Second, ifΘ is uncountable, the pointwise supremum may be an inappropriate choice, as it is unclear whether it is even inmG^Θ.

To deal with these issues, we shall first assume that our set of measures satisfies a certain decomposition property, which is a generalisation of the separability assumed in Soner et al. [17]. Under this assumption, we shall be able to give a consistent definition

of the conditional expectation under θ, in a quasi-sure sense. We then follow Detlef- sen and Scandolo [7] in replacing the supremum in (3.1) with an essential supremum, which we construct quasi-surely. Hence, we show that the representation is valid. It is worth also noting the work of Bion-Nadal [1], where a similar representation is ob- tained (for the larger class of convex risk measures under uncertainty, that is, without the assumption of coherence) however no consideration is given to the construction of the conditional expectation in a quasi-sure sense.

Definition 3.1. For G ⊆ F, we shall write Θ|G for the set of measures θ ∈ Θ, all restricted toG.

3.1 Defining linear conditional expectations

Our key tool for the definition of the conditional expectation, in a sufficiently strong sense, will be the assumption that the following property holds.

Definition 3.2 (Hahn property). We shall say thatΘ has the Hahn property on G if there exists a ‘dominating’ set of probability measuresΦdefined on(Ω,G)such that

(i) ΦandΘ|_G generate the same polar sets andmG^Θ=mG^Φ,

(ii) for everyφ∈Φ, there is a setS_(φ,G)∈ G^Θthat supportsφ, that is, φ(S_(φ,G)) = 1,

such that the sets{S_(φ,G)}_φ∈Φare disjoint, and

(iii) for anyA∈ G, ifθ(A∩S_φ) = 0for allφ∈Φ, thenθ(A) = 0.

The collection{S_(φ,G)}_φ∈Φ, with the associated measuresΦ, will be called aΘ/G-dominating partition ofΩ. (Note that{S_(φ,G)}is aG^Θ-measurable partition ofΩminus a polar set.)

Note that the ‘dominating’ set Φ is not assumed to be countable. The reason for giving this name to the property will be outlined in Remark 3.15. The following example shows that the existence of a Hahn decomposition is not trivial in general.

Example 3.3. Consider the spaceΩ = [0,1]² with its Borelσ-algebra. For simplicity, we take G = B(ω1), the Borel σ-algebra generated by the first component of Ω. Let Θ ={δ(x,y) : (x, y)∈[0,1]²}, the family of single-point measures onΩ. ThenΘhas the Hahn property, withΘ = ΦandS_(δ(x,y),G) ={x} ×[0,1]. The set of measures obtained by taking all countable mixtures of elements ofΘwill also have the Hahn property, a Θ/G-dominating partition being the sets{{x} ×[0,1]}_x∈[0,1].

Conversely, if Θ⁰ = Θ∪ {λ}, whereλis Lebesgue measure on [0,1]², thenΘ⁰ does not have the Hahn property. This is because any dominating setΦ must generate no non-empty polar sets, and for every pointxthere is a measureφ∈Φsuch thatφ(x)>0. As the supports of the measures inΦare disjoint,Φmust be built up only of measures supported by countably many points. This implies, however, that all functions areG^φ- measurable for eachφ∈Φ, so all functions are inmG^Φ. On the other hand, mG^Θonly contains Lebesgue measurable functions, so we see thatmG^Φ6=mG^Θ.

Example 3.4. Suppose there exists a G-dominating measure φ ∈ Θ, that is, θ|_G is absolutely continuous with respect toφ|G for allθ ∈Θ. ThenΘhas the Hahn property withΦ ={φ}.

As pointed out by a referee, if θ|_G is absolutely continuous with respect toφ|_G for all θ ∈ Θ but φ 6∈ Θ, then we cannot guarantee that G^Θ ⊆ G^φ. Hence there may exist functions in mG^Θ for whichEθ[X] is well defined for everyθ, but which are not φ-measurable. In such a case, working only withG^φ-measurable functions provides a distinct approach to ours, but depends more heavily on the choice ofφ.

The usefulness of the Hahn property is due to the following simple lemma.

Lemma 3.5. Let Θhave the Hahn property onG and let A ∈ G^Θ withA ⊆S_(φ,G) for someφ∈Φ. ThenAis polar if and only ifAisφ-null.

Hence for every θ ∈ Θ, everyφ ∈ Φ, we know θ|G is absolutely continuous with respect toφonS_(φ,G).

Proof. By assumption (i) of the Hahn property, all polar sets must be φ-null for every φ ∈Φ. Conversely, as A ⊆S_(φ,G) and the supports{S_(ψ,G)}ψ∈Φare disjoint, ψ(A) = 0 for everyψ 6= φ, ψ ∈ Φ. As φ(A) = 0 also, we know thatA is Φ-polar, and hence is Θ-polar.

In some cases, the Hahn property may be most easily verified using the following lemma, which generalizes Example 3.4 above.

Lemma 3.6. Suppose there exists a subsetΦ⊆ Θwith disjoint supports {S_(φ,G)}_φ∈Φ, such that for anyθ∈Θthere exists a countable set{φ^θ_n} ⊆Φwith

• S

nS_(φθ

n,G)supportsθ, and

• θ|G is absolutely continuous with respect toφ^θ_n|GonS_(φθ n,G). ThenΘhas the Hahn property (andΦ|_G is aΘ/G-dominating partition).

Proof. Condition (ii) of the Hahn property is direct. We need to show condition (i) holds, that is, thatΦandΘgenerate the same polar sets inGandmG^Θ=mG^Φ. AsΦ⊆Θ, any Θ-polar set is clearlyΦ-polar andmG^Φ⊇mG^Θ.

To showmG^Φ⊆mG^Θ, for anyθ∈Θ, by assumption there is a countable set{φ^θ_n}in Φsuch thatS

nS_(φθ

n,G)supportsθ. For anyΦ-polarA∈ G^Φ, we then have θ(A) =X

n

θ(A∩S_(φθ n,G)).

However,θ|_G is absolutely continuous with respect toφ^θ_n|_G onS_(φθ

n,G), so ifφ^θ_n(A) = 0we haveθ(A∩S_(φθ

n,G)) = 0. Henceθ(A) = 0, and asθwas arbitrary we knowAisΘ-polar.

Similarly, if X ∈mG^Φ, then for any θ ∈Θwe have the countable set{φ^θ_n}, and for eachn, we see thatX differs from aG-measurable function on aφ^θ_n-null set. OnS_(φθ

n,G), we know θ is absolutely continuous with respect to φ^θ_n, so there is a G-measurable functionX˜ such that{X 6= ˜X} ∩S_(φθ

n,G)isθ-null. From the representation X =X

n

IS_(φθ

n ,G)X θ−a.s., we see thatX ∈ G^θfor allθ, soX ∈mG^Θ.

Finally, we show condition (iii). We know that for any measurableA,θ(A) =θ(A∩ (∪nS_(φθ

n,G))) =P

nθ(A∩S_(φθ

n,G))and so ifθ(A∩S_(φ,G)) = 0for allφ∈Φ, thenθ(A) = 0. Remark 3.7. As pointed out by a referee, ifΘhas the Hahn property, then it follows that Θ∪Φ satisfies the above Lemma. However, it seems preferable to maintain a distinction between the setsΘ, which defines our nonlinear expectation, andΦ, which provides the necessary technical machinery for our analysis. This is particularly the case as the setΦwill typically not be unique.

We can now see that the setting of Soner et al. [17] has the Hahn property.

Example 3.8. Let Ωbe the classical Wiener space, with canonical processB starting at zero. LetF_t=σ{B_s}_0≤s≤t, andG=F_tfor somet. LethBibe the quadratic variation, which is a progressively measurable continuous function and can be universally defined for all local martingale measures onB, as in Karandikar [10] (this is a scalar version of the setting of [17], see also Nutz [13]).

Consider the set of orthogonal measures θ parameterized by some subset of the F-predictable absolutely continuous nonnegative functions, where underθv,B is a lo- cal martingale with quadratic variation v. Then we can take S(θ_v,G) = {ω : hBis = v_sfor alls≤t}, which is aG-measurable set. Soner et al. [17] takevof the form

dv dt =

∞

X

n=0

∞

X

i=0

aⁱ_nIEⁿ_iI_[τn,τn+1[,

where the (aⁱ_n) come from a generating class (for example, the class of deterministic processes), the(τ_n)is an increasing sequence of stopping times taking countably many values and q.s. reaching∞for finiten, and{E_iⁿ} ⊂ Fτ_n is a family of partitions ofΩ. Such processesvare said to satisfy the separability condition.

We claim the measures associated with the generating class, restricted toG, form aΘ/G-dominating partition ofΩ(up to repeated sets in the partition). Under the sep- arability condition, the measures associated with the generating class, restricted toG, have either identical or disjoint supports and are included inΘ. As every measure inΘ is generated by a countable collection of elements of the generating class, the first re- quirement of our Lemma 3.6 is satisfied. Lemma 5.2 of [17] then proves the equivalence (in fact, the equality) of any two measures in Θon the intersection of their supports, yielding the second condition of our Lemma 3.6.

3.2 The essential supremum

It is useful to be able to combine families of random variables in a quasi-surely consistent manner. A key tool for doing this is the essential supremum, which we now construct in a quasi-sure sense. To begin, we cite the following result on the existence of the essential supremum in a classical setting.

Theorem 3.9(Föllmer and Schied [8] (Theorem A.18)).LetXbe any set ofG-measurable random variables on a (complete) probability space(Ω,G, θ).

(i) Then there exists a random variableX^∗ such thatX^∗ ≥X θ-a.s. for allX ∈ X. MoreoverX^∗isθ-a.s. unique in the sense that any other random variableY with this property satisfiesY ≥X θ-a.s. We callX^∗theθ-essential supremum ofX, and writeX^∗=θ-ess supX.

(ii) Suppose thatX is upward directed, that is, forX, X⁰ ∈ X there is X⁰⁰ ∈ X with X⁰⁰≥X∨X⁰. Then there exists an increasing sequenceX1≤X2...inX such that X^∗= limnXn θ-a.s.

We can extend the first half of this result to our setting, using the Hahn property.

Theorem 3.10. Suppose Θis a collection of measures with the Hahn property onG. Then for any setX ⊂mG^Θ, the result of Theorem 3.9(i) holds, where all random vari- ables are taken to be in mG^Θ, and inequalities are taken to hold q.s. For clarity, we denote theΘ-q.s. essential supremum byΘ-ess sup.

Proof. Let {S_(φ,G)} be a Θ/G-dominating partition of Ω. As mG^Θ = mG^Φ, we know thatX ∈ X isG^φ-measurable for allφ. Hence we can use Theorem 3.9(i) to construct

the essential supremumX_φ^∗ = φ-ess sup{X }, and then define the ‘universal’ essential supremum by the disjoint sum

X^∗:=X

φ∈Φ

IS_(φ,G)X_φ^∗.

Clearly for anyX ∈ X we haveX^∗ ≥X q.s. onS_(φ,G) for allφ, which directly implies X^∗ ≥X φ-a.s. for allφ. AsΦandΘgenerate the same polar sets, we see thatX^∗≥X q.s. onΩ. It is easy to verify thatX^∗is unique q.s., asX_φ^∗ is unique q.s. for eachφ. To show measurability, note thatX^∗∈mG^φ for allφ, soX^∗∈mG^Φ. AsmG^Φ=mG^Θ by the Hahn property, the result is proven.

We can now construct, in a q.s. unique way, the supports of the measuresθ∈Θ. Definition 3.11. LetΘhave the Hahn property. Forθ∈Θ,φ∈Φ, define

λ_θ|φ:= dθ|_G dφ IS_(φ,G)

where by Lemma 3.5 the Radon–Nikodym derivative is well definedφ-a.s. onS_(φ,G), and henceλ_θ|φis defined up to a polar set. Then define theG^Θ-measurable support ofθ,

S_(θ,G):={ω: Θ-ess sup_φ∈Φ(λ_θ|φ)>0} ∈ G^Θ.

Lemma 3.12. S_(θ,G)supportsθ, and anyG^Θ-measurableθ-null subset ofS_(θ,G)is polar.

Proof. To showS_(θ,G)supportsθ, note that by Lemma 3.5, for anyφ∈Φwe have θ((S_(θ,G))^c∩S_(φ,G)) =

Z

(S_(θ,G))^c

λ_θ|φdφ= 0,

as0≤λ_θ|φ ≤Θ-ess sup_φ∈Φ(λ_θ|φ) = 0on(S_(θ,G))^c. By part (iii) of the Hahn property, this implies that(S_(θ,G))^c isθ-null, henceS_(θ,G)is a support ofθ.

To show that anyG^Θ-measurableθ-null subset ofS_(θ,G)is polar, letA∈ G^Θbe aθ-null subset ofS_(θ,G). IfAis not polar, then there exists φ ∈Φsuch thatφ(A) >0. By the definition ofS_(θ,G)and the essential supremum, we knowλ_θ|φ >0φ-a.s. onS_(φ,G)∩S_(θ,G), so

θ(A)≥ Z

A

λ_θ|φdφ >0, which impliesAis notθ-null, giving a contradiction.

Remark 3.13. Note that this lemma implies thatS_(θ,G)is the ‘smallest’G^Θ-measurable support of θ, in a q.s. sense. That is, if there was another G^Θ-measurable support R ⊂S_(θ,G), then we knowS_(θ,G)\R ∈ G^Θ would beθ-null, hence from the lemma it is polar.

Lemma 3.14. For any two measuresθ, θ⁰∈Θ, their restrictionsθ|Gandθ⁰|Gare equiva- lent on the intersection of their supports. That is, ifA⊂S_(θ,G)∩S_(θ0,G),A∈ G^Θisθ-null, it is alsoθ⁰-null.

Proof. IfAisθ-null it is polar, by Lemma 3.12, and hence is alsoθ⁰-null.

Remark 3.15. This lemma is the reason why we have used the name ‘Hahn property’.

From this lemma, we see that our assumption allows us to decompose our space into supports for our restricted measuresΘ|_Gsuch that they are equivalent on the intersec- tion of their supports. When we consider only two measures, this can be done using

a combination of the Lebesgue decomposition theorem and the Hahn decomposition theorem. Here we assume enough that we cansimultaneouslyfind supports for our un- countable family of measures such that the decomposition holds for all pairs, keeping the supports fixed.

We can also reproduce an aggregation result similar to that of Touzi, Soner and Zhang [17].

Theorem 3.16. SupposeΘhas the Hahn property onG. Let{X^θ}_θ∈Θbe any family of functions such that for allθ, ψ∈Θ

• X^θisG^θ-measurable (whereG^θis the completion ofGunderθ)and

• X^θ=X^ψ(θ-a.s.) onS_(θ,G)∩S_(ψ,G).

Then there exists anaggregation functionY which isG^Θ-measurable, such thatY =X^θ θ-a.s. for allθ.

Proof. Simply take

Y = Θ-ess sup_θ∈ΘX^θ.

For anyθ ∈ Θ, by our second assumption we see thatY =X^θ θ-a.s. onS_(θ,G), and as S_(θ,G)supportsθ,Y =X^θθ-a.s.

As shown in [17], many of the results of stochastic analysis can be obtained as soon as we have a result of this kind.

3.3 A dual representation

We now seek to prove that a modified version of the representation (3.1) is valid.

Lemma 3.17. LetEbe aG-consistent sublinear expectation, with representationE(·) = sup_θ∈ΘEθ[X]. Then for anyθ∈Θ, anyX such that all terms areθ-a.s. finite, anyt <∞,

−E(−X|G)≤Eθ[X|G]≤ E(X|G) θ−a.s.

Proof. For anyA∈ G^Θ, anyX we have

EG[IA(X− Et(X))] =EG(IAX)−IAEG(X) = 0 and so by time consistencyEG[IA(X− Et(X))] = 0.Hence

Eθ[IA(X− EG(X))]≤ E[IA(X− EG(X))] = 0

and rearrangement gives Eθ[IAX] ≤ Eθ[IAEG(X))], which is equivalent to the upper boundE_θ[X|G]≤ EG(X). For the lower bound, applying this result to−X gives

E_θ[X|G] =−E_θ[−X|G]≥ −E_G(−X) θ−a.s.

Using the Hahn property, we can consistently define our conditional expectations Eθ[·|G]up to equalityE-q.s.

Definition 3.18. SupposeΘhas the Hahn property onG. For eachθ∈Θ, we define

E_θ|Θ[X|G] =

(Y ω∈S_(θ,G)

−∞ ω6∈S_(θ,G)

where Y ∈ mG^Θ is any version of the classical conditional expectation Eθ[X|G]. By Lemma 3.12, this definition is unique up to a polar set (as it is unique up to aθ-null subset ofS_(θ,G)).

Remark 3.19. Note thatE_θ|Θ[X|G]is a version of the usual conditional expectation, but is definedE-q.s. rather thanθ-a.s. Furthermore,E_θ|Θ[X|G]satisfies the usual properties of the conditional expectation, when considered only onS_(θ,G), i.e. linearity, recursivity, monotonicity, etc., again E-q.s. rather than simply θ-a.s. The reason for setting the expectation to−∞off S_(θ,G) is simply so that we can take the supremum in a simple manner. It also gives the following lemma.

Lemma 3.20. E_θ|Θ[X|G]is the q.s. minimal version of the θ-conditional expectation.

That is, ifY ∈mG^Θis another version of the conditional expectation andY ≤E_θ|Θ[X|G], then{ω:Y < E_θ|Θ[X|G]}is polar.

Proof. By definition,Y =E_θ|Θ[X|G] =−∞except onS_(θ,G). Hence{ω :Y < E_θ|Θ[X|G]}

is aθ-null subset ofS_(θ,G). By Lemma 3.12, this set is polar.

We can now prove our general representation.

Theorem 3.21. LetE be aG-consistent sublinear expectation, with a representationΘ having the Hahn property onG. Then the conditional expectation has a representation

EG(X) = Θ-ess sup_θ∈Θ{E_θ|Θ[X|G]}

up to equality q.s.

Proof. First note that for anyA∈ G^Θ, E(IAE_G(X)) =E(IAX) = sup

θ∈Θ

Eθ[IAX] = sup

θ∈Θ

Eθ[IAE_θ|Θ[X|G]]

≤sup

θ∈Θ

E_θ[I_A(Θ-ess sup_ψ∈Θ{E_ψ|Θ[X|G]})]

=E(I_A(Θ-ess sup_ψ∈Θ{E_ψ|Θ[X|G]})) from which we see

E_G(X)≤Θ-ess sup_ψ∈θ{E_ψ|Θ[X|G]} q.s.

Conversely, by Lemma 3.12, we know that for everyψ ∈Θ, E_ψ|Θ[X|G] ≤ EG(X)ψ-a.s.

By definition,E_ψ|Θ[X|G] =−∞except onS_(ψ,G), so by Lemma 3.5 we know E_ψ|Θ[X|G]≤ EG(X) q.s.

Therefore, by Theorem 3.10,

Θ-ess sup_ψ∈θ{E_ψ|Θ[X|G]} ≤ E_G(X) q.s.

giving the desired equality.

As mentioned earlier, Bion-Nadal [1] gives a similar result to this, however without a quasi-sure construction of the conditional expectation. Therefore, her result presents only the θ-a.s. equality of the conditional sublinear expectation and the θ-essential supremum. Our result is strictly stronger, as both the equality and the essential supre- mum are taken in a quasi sure sense.

Remark 3.22. We note that this result immediately allows us to consistently extendE_G to the larger spaceH_F, using a generalized conditional expectation, as in [9, p2]. That is, we no longer require substantial integrability conditions onX to defineE_G(X). This will, however, lead to somewhat different statements of the properties of the conditional expectation (as finiteness is no longer guaranteed).

3.4 G-consistency and pasting of measures

Using this result, we can give a type of ‘pasting stability’ of the measures related to G-consistency. This is closely related to them-stability of Delbaen [5], see also a similar result in Nutz and Soner [14, Proposition 3.6].

Definition 3.23. ForΘwith the Hahn property, we sayΘis stable underG-pastingif for anyθ, θ⁰∈Θ, anyA⊆S_(θ,G)∩S(θ⁰,G),A∈ G^Θwe haveψ∈Θ, whereψis the measure onΩwith

ψ(B) :=E_θ[I_AE_θ⁰[I_B|G] +I_AcI_B].

For a setΘ, we can defineΘ^G, the finiteG-stabilization ofΘ, as the set of all measures obtained fromΘthrough finitely many combinations of this form, that is,

Θ^G =n

ψ:Eψ[·] =Eθ₀

hX^k

n=0

IA_nEθ_n[·|G]i

, k∈No

(3.2)

for measures θn ∈ Θ and An ∈ G^Θ with An ⊆ S_(θn,G)∩S_(θ0,G) and A0 = (∪^k_n=1An)^c. Clearly ifΘhas the Hahn property onGthen so willΘ^G, and note that∪nAn⊆S_(ψ,G).

Note that this pasting only needs to hold for A in the intersection of the minimal supports of the two measures. By Lemma 3.14, θ|G and θ⁰|G are equivalent on the intersection of their minimal supports, and hence the (classical) conditional expectation can be used without difficulty.

In some applications the analogous stabilization where countably many combina- tions are permitted may be of interest (particularly if we wish for the supremum to be attained), however the finite case will be sufficient for our result.

Theorem 3.24. Let E be a sublinear expectation with representationΘ. SupposeΘ has the Hahn property onG. Then

(i) IfEisG-consistent, thenE has an equivalent representation E(X) = sup

θ∈Θ^G

Eθ[X].

(ii) IfΘ = Θ^G thenEisG-consistent.

Proof. (i) SupposeE isG-consistent. ClearlyΘ⊆Θ^G, and so E(X)≤ sup

θ∈Θ^G

E_θ[X].

Conversely, for anyψ∈Θ^G we knowψis of the form indicated by (3.2). Then Eψ[X] =Eθ₀

"

X

n

IA_nEθ_n[X|G]

#

≤sup

θ

Eθ[Θ-ess sup_θ0E_θ⁰_|Θ[X|G]] =E(X) and so

E(X)≥ sup

θ∈Θ^G

Eθ[X].

(ii) AsΘ = Θ^G has the Hahn property, for each fixedX ∈L¹(E,F)we can define the putative sublinear conditional expectation

E˜_G(X) := Θ-ess sup_ψ∈ΘE_ψ|Θ[X|G].

All the properties of a G-consistent sublinear expectation are trivial to verify except recursivity.

To show recursivity, first select someθ∈Θ. Consider the family Y={I_AE_ψ|Θ[X|G] +I_AcE_θ[X|G] :ψ∈Θ, A⊆S_(θ,G)∩S_(ψ,G)}.

Now suppose we have

Y =IAE_ψ|Θ[X|G] +IA^cEθ[X|G]

Y⁰ =IBE_ψ⁰_|Θ[X|G] +IB^cEθ[X|G].

Define, (with the convention−∞ 6>−∞)

A˜={ω:E_ψ|Θ[X|G]> Eθ[X|G]} ∩S_(θ,G)⊆S_(θ,G)∩S_(ψ,G) B˜ ={ω:E_ψ⁰_|Θ[X|G]> Eθ[X|G]} ∩S_(θ,G)⊆S_(θ,G)∩S_(ψ⁰_,G) Y⁰⁰=IA˜E_ψ|Θ[X|G] +IB˜E_ψ0|Θ[X|G] +I_{( ˜A∪}B)˜^cE_θ|Θ[X|G].

Then,A,˜ B˜ ∈ G^Θ, soY⁰⁰ ∈ Y andY⁰⁰ ≥ Y ∨Y⁰ θ-a.s., so Y is upward directed (up to equalityθ-a.s.).

The quasi-sure essential supremum given by Theorem 3.10 must also be a version of theθ-a.s. essential supremum given by Theorem 3.9. AsE_ψ|Θ[X|G] =−∞except on S_(ψ,G), andΘ = Θ^G, we see that

E˜_G(X) =θ-ess sup{IAE_ψ|Θ[X|G] +IA^cEθ[X|G]} θ−a.s.

By Theorem 3.9(ii), we can then find appropriate sequencesψ_n^θ,A^θ_n such that {I_Aθ

nE_ψθ

n|Θ[X|G] +I_(Aθ

n)^cEθ[X|G]} ↑E˜G(X) θ−a.s.

We now relax our selection ofθ, and consider the equation E( ˜EG(X)) = sup

θ

E_θ[ ˜EG(X)]

= sup

θ

Eθ[lim

n {I_Aθ nE_ψθ

n|Θ[X|G] +I_(Aθ

n)^cEθ[X|G]}]

= sup

θ

sup

n

Eθ[I_Aθ nE_ψθ

n|Θ[X|G] +I_(Aθ

n)^cEθ[X|G]]

= sup

θ

sup

n

E_θn[X].

where

θn(B) :=Eθ[I_Aθ nE_ψθ

n|Θ[IB|G] +I_(Aθ

n)^cEθ[IB|G]].

As we knowΘ = Θ^G, all the induced measuresθn are inΘ. Therefore we have E( ˜E_G(X)) = sup

θ

Eθ[X] =E(X)

and soE˜G(X)satisfies the recursivity assumption.

4 Conclusion

We have considered sublinear expectations on general probability spaces, where the set of measures in the dual representation of the expectation are not necessarily abso- lutely continuous with respect to any dominating measure. In this context, we have shown that the assumption of a Hahn property provides a simple means to aggregate processes defined with respect to each measure, thereby giving a straightforward ap- proach to quasi-sure analysis in this context. This approach also removes reliance on

quasi-continuity of the random variables, instead defining the expectations for all mea- surable random variables.

Our methods generalize the approach of [17], as the Hahn property has a natural interpretation in a general setting. Consequently, this paper provides a quasi-sure con- struction of the conditional expectation under each test measure, and shows that a dual representation then holds for the conditional sublinear expectation. We have given a version of the aggregation result of [17].

For any specific problem, determining whether the Hahn property holds may be a difficult task, (as is made clear by the analysis in [17]). However, our approach shows that for any given problem, once the Hahn property has been shown, many of the results of stochastic analysis transfer simply into a quasi-sure setting.

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Acknowledgments. Thanks to Terry Lyons and Freddy Delbaen for useful conversa- tions during the preparation of this paper, and to two anonymous referees for their careful reading and helpful suggestions. In particular, thanks to F. Delbaen for the proof of Lemma 3.17.