Global Upper Expectations for Discrete-Time Stochastic Processes: In Practice, They Are All The Same!
aa r X i v : . [ m a t h . P R ] F e b Global Upper Expectations for Discrete-Time Stochastic Processes:In Practice, They Are All The Same!
Natan T’Joens
NATAN . TJOENS @ UGENT . BE Jasper De Bock
JASPER . DEBOCK @ UGENT . BE Foundations Lab for Imprecise Probabilities, ELIS, Ghent University, Belgium
Abstract
We consider three different types of global un-certainty models for discrete-time stochasticprocesses: measure-theoretic upper expecta-tions, game-theoretic upper expectations andaxiomatic upper expectations. The last two areknown to be identical. We show that they coin-cide with measure-theoretic upper expectationson two distinct domains: monotone pointwiselimits of finitary gambles, and bounded belowBorel-measurable variables. We argue that thesedomains cover most practical inferences, and thattherefore, in practice, it does not matter whichmodel is used.
Keywords: upper expectation, imprecise probabil-ities, monotone convergence, probability measure,supermartingale, capacitability
1. Introduction
To describe the dynamics of a discrete-time stochasticprocess, one may choose between a number of differ-ent mathematical approaches. There is of course themeasure-theoretic option [
2, 13, 14 ] —undoubtly themost popular one—but one can also use martingalesor game-theoretic principles to do so [
11, 12, 21 ] . Eachof these approaches has its own unique strengths andflaws, and each of them—rightly or not—has attracteda dedicated group of followers. Our aim here is not toargue for the use of one or the other though, but ratherto study the mathematical relation between the (global)uncertainty models that arise from these approachesin a general, imprecise-probabilistic context. As we willsee, they turn out to be surprisingly similar.All the global—imprecise—uncertainty models thatwe will consider take the form of an upper (or lower) ex-pectation [
18, 19 ] ; a non-linear operator that can—butneed not—be interpreted as a tight upper bound on aset of expectations. They are called global because theymodel beliefs about the entire, uncertain path taken bythe process. In that sense, they differ from—and aremore general than—local uncertainty models, whichonly give information about how the process is likelyto evolve from one time instant to the next. Such local models form the parameters of a stochastic process,whereas the global uncertainty model that follows fromit—in our case, a global upper expectation—extendsthe information incorporated in these local models. It isthe particular way in which this extension is done thatdistinguishes one type of global model from the other.We consider three global models. The first is a prob-abilistic model that is defined as an upper envelopeover a set of measure-theoretic global expectations [ ] . The second is based on game-theoretic principles,and defined as an infimum over hedging prices; seeRefs. [
11, 12 ] . The last is an abstract axiomatic model,whose defining axioms we have motivated in an earlierpaper [ ] on the basis of both a probabilistic anda behavioural interpretation. We have already shownthat the second and third of these three global upperexpectations are identical [ ] . In this paper, we re-late the first—measure-theoretic—one to this commonaxiomatic / game-theoretic upper expectation.Our contribution consists in showing that they areequal on two different domains: variables that aremonotone (upward or downward) limits of finitarygambles—bounded variables that only depend on theprocess’ state at a finite number of time instances—andbounded below Borel-measurable variables. Together,these two types of domains cover the vast majority ofinferences encountered in practice; hitting times, for in-stance, fall under the first category [ ] ; time averagesunder the second [ ] . Hence the title of this paper.That the three considered global upper expectations areequal on such a large domain is relevant in a numberof ways. First of all, it leaves no room for discussionwhen it comes to choosing a global model; it simplydoes not matter since all of them are equal. Philosoph-ically speaking, it is interesting that, whatever the inter-pretational point of view and associated system of lo-gical reasoning is, we always end up with exactly thesame object. Finally, and maybe most importantly, sucha relation provides us with a large number of additionalmathematical properties for the models at hand; prop-erties that were previously only known to hold for one ortwo of these models, suddenly hold for all three of them.We refer to Refs. [
4, 9, 10 ] for an illustration of how prop- © N. T’Joens & J.D. Bock. LOBAL U PPER E XPECTATIONS FOR D ISCRETE -T IME S TOCHASTIC P ROCESSES : I N P RACTICE , T
HEY A RE A LL T HE S AME ! erties acquired in this way have already led to importantconsequences.This paper is an extended version of a contributionthat is submitted for possible publication in the Pro-ceedings of ISIPTA 2021. Compared to the submittedversion, this extended version additionally includes anappendix containing proofs for the results in the maintext.
2. Local Uncertainty Models A discrete-time stochastic process is an infinite sequence X , X ,..., X k ,... of uncertain states, where the state X k at each discrete time point k ∈ N takes values in a fixednon-empty set X , called the state space . We will assumethat this state space X is finite . Typically, when mod-elling the dynamics of a stochastic process, one startsoff on a local level, by specifying how the process’ state X k is (likely) to evolve from one time instant to the next.In particular, we do this by attaching a so-called localuncertainty model to each possible situation ; a finite—possibly empty—sequence x k : = x x ··· x k of state val-ues that represents a possible history X = x , ··· , X k = x k up until some time point k ∈ N , with N : = N ∪{ } . Thelocal model associated with the situation x k then mod-els beliefs about the value of the next state X k + , condi-tional on the history represented by x k . We let X ∗ : = ∪ i ∈ N X i be the set of all situations and we denote the initial (empty) situation by ƒ : = x = X .Among the most popular types of local uncertaintymodels are (probability) mass functions p on X ; forany situation x k ∈ X ∗ , the mass function p ( ·| x k ) thenprovides, for each x k + ∈ X , the probability p ( x k + | x k ) that the value of the state X k + will be equal to x k + .Such a family of probability mass functions is represen-ted by a single function p : s ∈ X ∗ p ( ·| s ) , which we calla precise probability tree . What is equivalent, but less ofa popular habit, is to attach to each possible situation x k ∈ X ∗ an expectation E x k on the set L ( X ) of allreal-valued functions f on X . These expectations E x k may then be interpreted in a measure-theoretic sense,as coming from an underlying family of mass functions p ( ·| x k ) , but they can also be interpreted in a direct be-havioural way as a subject’s fair prices, as De Finettidoes [ ] .Unfortunately, irrespective of one’s preferencebetween mass functions and linear expectations,both of them are rather inadequate when modellingsituations where data is scarce, or when modellingthe beliefs of a conservative (risk-averse) subject. Insuch situations, one can reach for so-called ‘imprecise’
1. The reason why we call it a ‘tree’ is because it is a map on X ∗ ,which can naturally be visualised in terms of infinite (event) trees [
5, Figure 1 ] . probability models [
18, 19, 1 ] . These come in many dif-ferent shapes and forms (e.g. sets of desirable gambles,belief functions, credal sets,...), but, for our purpose ofmodelling the local dynamics of a process, we will onlyconsider two specific—yet wide-spread—ones; credalsets and coherent upper (and lower) expectations.The first, credal sets , are closed (under the topology ofpointwise convergence) convex sets of probability massfunctions; see e.g. [
1, Section 9.2 ] . If we attach to eachsituation s ∈ X ∗ a credal set P s on X , then we obtaina so-called imprecise probability tree P • : s ∈ X ∗
7→ P s ,which we will often simply denote by P . For any s ∈ X ∗ ,the associated credal set P s may then be interpreted asa set that contains all local mass functions p ( ·| s ) thatare deemed ‘possible’. Such an imprecise probabilitytree P parametrizes the stochastic process as a whole,and clearly does so in a more general manner than theprecise methods mentioned earlier; precise probabil-ity trees correspond to the special case where, for each s ∈ X ∗ , P s consists of a single mass function p ( ·| s ) . Wesay that a precise probability tree p is compatible withan imprecise probability tree P , and write p ∼ P , if p ( ·| s ) ∈ P s for all s ∈ X ∗ .Another—yet equivalent—approach consists in spe-cifying a local coherent upper (or lower) expectation Q s for each s ∈ X ∗ [ ] : a real-valued function on L ( X ) that satisfies, for all f , g ∈ L ( X ) and λ ∈ R ≥ ,C1. Q s ( f ) ≤ sup f [ upper bounds ] ;C2. Q s ( f + g ) ≤ Q s ( f )+ Q s ( g ) [ sub-additivity ] ;C3. Q s ( λ f ) = λ Q s ( f ) [ non-negative homogeneity ] .Any such family ( Q s ) s ∈X ∗ of local coherent upper ex-pectations will be gathered in a single upper expectationtree Q • : s ∈ X ∗ Q s , which we will also simply denoteby Q. For any x k ∈ X ∗ , the upper expectation Q x k can be interpreted as representing a subject’s minimumselling prices—a generalisation of De Finetti’s fair priceinterpretation for linear expectations. More concretely,this interpretation says that, given a situation x k ∈ X ∗ and any f ∈ L ( X ) , our subject is willing to sell theuncertain—possibly negative—payoff f ( X k + ) for anyprice α ≥ Q x k ( f ) . Axioms C1–C3 can then be seen asrationality criteria. We refer to Walley’s work [ ] for amore detailed motivation and justification for coherentupper (and lower) expectations.Mathematically speaking, it does not matter whetherwe use imprecise probability trees or upper expect-ation trees to characterise a stochastic process, be-cause credal sets and coherent upper expectations—
2. Traditionally, the behavioural interpretation of coherent upperexpectations says that they represent infimum selling prices,rather than minimum selling prices; see Ref. [ ] . We opt for min-imum selling prices here because they fit more naturally with thesupermartingales that we will introduce further on. LOBAL U PPER E XPECTATIONS FOR D ISCRETE -T IME S TOCHASTIC P ROCESSES : I N P RACTICE , T
HEY A RE A LL T HE S AME ! and therefore imprecise probability trees and upper ex-pectation trees—are in a one-to-one relation with eachother. In particular, with any imprecise probability tree P , we can associate an upper expectation tree Q • , P thatmaps each situation s ∈ X ∗ to the upper envelope Q s , P of the linear expectations corresponding to P s :Q s , P ( f ) : = sup ¦ X x ∈X f ( x ) p ( x | s ) : p ( ·| s ) ∈ P s © ,for all f ∈ L ( X ) . That each Q s , P is indeed a local coher-ent upper expectation follows from [
19, Theorem 3.6.1 ] .Conversely, with any upper expectation tree Q, we canassociate an imprecise probability tree P • ,Q ; for any s ∈X ∗ , its local credal set P s ,Q is the closed convex set of allmass functions p ( ·| s ) that are dominated by Q s , in thesense that X x ∈X f ( x ) p ( x | s ) ≤ Q s ( f ) for all f ∈ L ( X ) .It follows once more from [
19, Theorem 3.6.1 ] that thiscorrespondence between upper expectation trees andimprecise probability trees is one-to-one; that is, themap P 7→ Q • , P is bijective and Q
7→ P • ,Q is its inverse.We say that an imprecise probability tree P and an up-per expectation tree Q agree if they are related throughthese mappings.An important consequence of the one-to-one rela-tion described above is that imprecise probability treesand upper expectation trees can borrow each others in-terpretation; local credal sets can be interpreted as rep-resenting a subject’s infimum selling prices, whereaslocal upper expectations can be interpreted as upperenvelopes of the linear expectations associated with anunderlying local credal set.
3. Three Types of Global Models
Imprecise probability trees and upper expectation treesdescribe the dynamics of a stochastic process on a locallevel—how it changes from one time instant to thenext—but they do not tell us anything, at least not dir-ectly, about more global features that relate to multipletime instances at once; e.g. the time it takes until theprocess is in a given state x ∈ X . We therefore face thefollowing question. How do we turn the local informa-tion captured by any of these trees into global informa-tion about the process as a whole? Three possible solu-tions are described in the current section, but we startby introducing some necessary terminology and nota-tion. A path ω = x x x ··· is an infinite sequence of state val-ues and represents a possible evolution of the process. The sample space Ω : = X N denotes the set of all paths.For any ω = x x x ··· ∈ Ω , we let ω k : = x k ∈ X k be thefinite sequence that consists of the initial k state values,and we let ω k : = x k ∈ X be the k -th state value. An eventA ⊆ Ω is a set of paths and, in particular, for any situation x k ∈ X ∗ , the cylinder event Γ ( x k ) : = { ω ∈ Ω : ω k = x k } is the set of all paths that go through the situation x k .We let R : = R ∪{ + ∞ , −∞} be the extended real num-bers, R ≥ be the subset of non-negative ones, and R ≥ be those that are moreover real. We extend the total or-der relation < on R to R by positing that −∞ < c < + ∞ for all c ∈ R and endow R with the associated order to-pology.Any extended real-valued function f : Y → R onsome non-empty set Y will be called a variable . Anybounded variable—that is, a variable f for which thereis a B ∈ R ≥ such that − B ≤ f ( y ) ≤ B for all y ∈ Y —willbe called a gamble . The set of all variables will be de-noted by L ( Y ) and the set of all gambles by L ( Y ) . Notethat this definition is in accordance with our earlier useof L ( X ) , where it denoted the real-valued functionson X —which are automatically bounded because X is finite. The elements of L ( X ) and L ( X ) are called local variables and gambles, respectively. On the otherhand, the variables in V : = L ( Ω ) and V : = L ( Ω ) are called global variables and gambles, respectively; they maydepend on the entire path ω ∈ Ω taken by the process.Variables that only depend on the process’ state at a fi-nite number of time instances are called finitary or n-measurable ; for such a finitary variable f ∈ V , there is an n ∈ N and some g ∈ L ( X n ) such that f ( ω ) = g ( ω n ) for all ω ∈ Ω . We often make this explicit by writing f = g ( X n ) ,where g ( X n ) : = g ◦ X n and where X n is the projectionof ω ∈ Ω on its first n state values ω n . Sometimes, wealso allow ourselves a slight abuse of notation by writing f ( x n ) to denote the constant value of f ( ω ) = g ( x n ) onall paths ω ∈ Ω such that ω n = x n . We collect all finitarygambles in the set F . A special type of global gamble isthe indicator I A of an event A , which assumes the value1 on A and 0 elsewhere. For any s ∈ X ∗ , the indicator I s : = I Γ ( s ) of the cylinder event Γ ( s ) is clearly a finitarygamble.A global upper expectation , finally, is a map E : V ×X ∗ → R ; it maps global variables f ∈ V and situations s ∈ X ∗ to a corresponding (conditional) upper expecta-tion E ( f | s ) . As we will see, such maps can play the roleof a global uncertainty model, in the sense that they canrepresent beliefs or knowledge about the path ω takenby the process, or about the value attained by a globalvariable f . Apart from global upper expectations, onecan also consider global lower expectations E: V ×X ∗ → R ; for each of the models that we will consider, these are
3. This choice of terminology is due to Walley [ ] . However, for us,the mathematical object of a gamble is not necessarily bound tothe interpretation as an uncertain payoff. LOBAL U PPER E XPECTATIONS FOR D ISCRETE -T IME S TOCHASTIC P ROCESSES : I N P RACTICE , T
HEY A RE A LL T HE S AME ! conjugate to the corresponding global upper expecta-tion E, in the sense that E ( f | s ) = − E ( − f | s ) for all f ∈ V and s ∈ X ∗ . It therefore suffices to focus on only one ofthem; our theoretical developments focus on E, leavingthe implications for E for Section 6. We start by presenting a traditional measure-theoreticapproach, where global upper expectations are definedas upper envelopes of sets of (linear) expectations, andwhere each of these (linear) expectations on its turn isderived from a different probability measure on Ω .Consider an imprecise probability tree P and let p ∼P be any precise probability tree that is compatiblewith P . With each x k ∈ X ∗ , we associate a probabilitymeasure P p ( ·| x k ) on the σ -algebra F generated by allcylinder events as follows. First, for any ℓ ∈ N and any C ⊆ X ℓ , letP p ( C | x k ) : = P p ( ∪ z ℓ ∈ C Γ ( z ℓ ) | x k ) : = X z ℓ ∈ C P p ( z ℓ | x k ) ,where P p ( z ℓ | x k ) : = (1) Q ℓ − i = k p ( z i + | z i ) if k < ℓ and z k = x k k ≥ ℓ and z ℓ = x ℓ p ( ·| x k ) forms a finitely addit-ive probability [
10, Chapter 3 ] . Hence, by [
2, Theorem2.3 ] , it is also a countably additive probability—thatis, a probability measure—on this algebra and so, byCarathéodory’s extension theorem [
21, Theorem 1.7 ] ,P p ( ·| x k ) can be uniquely extended to a probabilitymeasure on F .In accordance with standard practices, we then as-sociate with every probability measure P p ( ·| s ) an ex-pectation E p ( ·| s ) using Lebesgue integration. That is,we let E p ( f | s ) : = R Ω f dP p ( ·| s ) for all f ∈ V for which R Ω f dP p ( ·| s ) exists, which is guaranteed if f is F -measurable and bounded below (or bounded above).For general f ∈ V , we adopt an upper integral E p ( f | s ) defined byE p ( f | s ) : = inf ¦ E p ( g | s ) : g ∈ V σ ,b and g ≥ f © , (2)where V σ ,b is the set of all bounded below F -measurable variables in V . It follows from [
17, Propos-ition 12 ] that E p ( ·| s ) coincides with E p ( ·| s ) on the entiredomain where E p ( ·| s ) is well-defined—that is, wherethe Lebesgue integral with respect to P p ( ·| s ) exists—andhence, that E p ( ·| s ) is an extension of E p ( ·| s ) .Finally, the global upper expectation E P correspond-ing to the imprecise probability tree P is defined as the upper envelope of the upper integrals E p correspond-ing to each of the precise trees p ∼ P . That is, for each f ∈ V and s ∈ X ∗ ,E P ( f | s ) : = sup (cid:8) E p ( f | s ) : p ∼ P (cid:9) .This definition is in line with the sensitivity analysis in-terpretation for imprecise probability models [
19, Sec-tion 1.1.5 ] , which regards them as resulting from a lackof knowledge about a single ideal precise model.The approach set out above should look familiar toanyone with a measure-theoretic background, and wetherefore omit an in-depth conceptual discussion; weinstead refer to [
17, Section 9 ] for more details. Oneaspect, however, that we feel is worth pointing out isthe difference between our way of conditioning andwhat is usually done in measure-theory. Usually, condi-tional expectations (and probabilities) are derived froma single unconditional probability measure through theRadon-Nikodym derivative [
13, Section 2.7.2 ] . We, onthe other hand, associate with each situation s ∈ X ∗ a separate—in the traditional sense, unconditional—probability measure P p ( ·| s ) and use this probabilitymeasure P p ( ·| s ) to define the expectation E p ( ·| s ) . Thereason why we do so is because, unlike the traditionalapproach, it allows us to condition—in a meanigfulway—on (cylinder) events with probability zero; again,we refer to [
17, Section 9 ] for more details. The second global model that we will consider isthe game-theoretic upper expectation introduced and,for the most part, developed by Shafer and Vovk [ ] . This operator is defined in terms of infimumhedging prices; starting capitals that allow a gambler tocover—or hedge—the costs or gains of a given globalgamble. These hedging prices—and hence, these game-theoretic upper expectations—are determined usingthe notion of a supermartingale; a function that de-scribes the possible evolution of a gambler’s capital ashe gambles in a way that is in accordance with the localmodels Q s .Formally, for any upper expectation tree Q, a su-permartingale M is a real-valued function on X ∗ that satisfies Q s ( M ( s · )) ≤ M ( s ) for all s ∈ X ∗ , where M ( s · ) ∈ L ( X ) denotes the local gamble that takes thevalue M ( s x ) in x ∈ X . How can such a supermartin-gale be interpreted in the way described above? Con-sider any situation x k ∈ X ∗ and a gambler—called‘Skeptic’ in Shafer and Vovk’s framework—whose cur-rent capital equals M ( x k ) . Then, recalling our in-terpretation for the local model Q x k as represent-ing a subject’s minimum selling prices, the conditionthat Q x k ( M ( x k · )) ≤ M ( x k ) implies that Skeptic canuse his capital M ( x k ) to buy the uncertain reward LOBAL U PPER E XPECTATIONS FOR D ISCRETE -T IME S TOCHASTIC P ROCESSES : I N P RACTICE , T
HEY A RE A LL T HE S AME ! M ( x k X k + ) from this subject—called ‘Forecaster’ inShafer and Vovk’s framework. If Skeptic chooses tocommit to such a transaction, he is actually gambling against Forecaster, which explains why these playersare called Skeptic and Forecaster. So we see that a su-permartingale describes the evolution of Skeptic’s cap-ital if he chooses, in each situation, to buy a gamble thatForecaster is willing to sell.A hedging price α ∈ R for any f ∈ V is now a realnumber for which there is a bounded below super-martingale M that starts in M ( ƒ ) = α and such thatliminf M ( ω ) : = liminf k → + ∞ M ( ω k ) ≥ f ( ω ) for all ω ∈ Ω .A hedging price α for f is therefore worth more to Skep-tic than the global gamble f , because he is always ableto eventually turn the initial capital M ( ƒ ) = α into a cap-ital that is higher than the uncertain payoff correspond-ing to f , simply by choosing the right gambles from theones Forecaster is offering. That M should be boundedbelow, represents the condition that Skeptic can borrowat most a finite amount.For any f ∈ V , the infimum over all the hedging prices α is then what defines the (unconditional) global game-theoretic upper expectation E G,Q ( f ) of f . More gener-ally, the global game-theoretic upper expectation of any f ∈ V conditional on any s ∈ X ∗ , is defined asE G,Q ( f | s ) : = inf (cid:8) M ( s ) : M ∈ M b ( Q ) , ( ∀ ω ∈ Γ ( s )) liminf M ( ω ) ≥ f ( ω ) (cid:9) , (3)where M b ( Q ) denotes the set of all bounded below su-permartingales. The unconditional case corresponds to s = ƒ ; so, E G,Q ( f ) : = E G,Q ( f | ƒ ) .As the attentive reader may have noticed, the defini-tion above only applies to global gambles. So why notto general variables f ∈ V ? The reason is that, on thisextended domain, the formula presented above wouldyield an upper expectation with rather weak continu-ity properties [
16, section 8 ] . A simple solution is to usecontinuity with respect to so-called upper and lowercuts to extend the domain from V × X ∗ to V × X ∗ . Todo so, for any f ∈ V and any c ∈ R , let f ∧ c be definedby f ∧ c ( x ) : = min { f ( x ) , c } for all x ∈ X , and let f ∨ c bedefined analogously, as a pointwise maximum. Then wehenceforth let E G,Q : V × X ∗ → R be defined by Equa-tion (3) on V × X ∗ , and furthermore impose, for any s ∈ X ∗ , thatG1. E G,Q ( f | s ) = lim c → + ∞ E G,Q ( f ∧ c | s ) for all f ∈ V b ;G2. E G,Q ( f | s ) = lim c →−∞ E G,Q ( f ∨ c | s ) for all f ∈ V .Properties G1 and G2 together clearly imply that E G,Q is uniquely determined by its values on V × X ∗ . Hence,
4. This is similar to how [
18, Chapter 15 ] extends the notion of co-herence from gambles to unbounded real-valued variables. since E G,Q on this domain is described by Equation (3),it follows that E
G,Q is uniquely defined on all of V ×X ∗ .This way of extending a global game-theoretic up-per expectation is not that common, though. A tech-nique that is used more often consists in directly ap-plying Equation (3) to the entire domain V × X ∗ , butwith the real-valued supermartingales replaced by ex-tended real-valued ones [
12, 17, 16 ] . This of course firstrequires an extension Q ↑ s of the local models Q s to thedomain L ( X ) , which can be done in a way similar towhat we have done with E G,Q , by imposing continu-ity with respect to upper and lower cuts. An exten-ded real-valued supermartingale M : X ∗ → R is thencharacterised by the condition that Q ↑ s ( M ( s · )) ≤ M ( s ) for all s ∈ X ∗ . Remarkably enough, the global game-theoretic upper expectation that results from this ‘ex-tended supermartingale’-approach is identical to theoperator E G,Q we have defined above, using Proper-ties G1 and G2; see for example the end of [
16, Sec-tion 8 ] . We favor our approach, though, because the useof extended real-valued supermartingales undermineswhat we think is a key strength of the game-theoretic ap-proach: that supermartingales—and hence the result-ing game-theoretic upper expectations—can be givena clear behavioural meaning in terms of betting. Instead of relying on measure-theoretic or game-theoretic principles, one can also simply adopt an ab-stract global model E that is completely characterisedby a number of axioms. In particular, starting from anygiven upper expectation tree Q, we suggest to imposethe following list of axioms:P1. E ( f ( X n + ) | x n ) = Q x n ( f ) for all f ∈ L ( X ) andall x n ∈ X ∗ .P2. E ( f | s ) = E ( f I s | s ) for all f ∈ F and all s ∈ X ∗ .P3. E ( f | X k ) ≤ E ( E ( f | X k + ) | X k ) for all f ∈ F andall k ∈ N .P4. f ≤ g ⇒ E ( f | s ) ≤ E ( g | s ) for all f , g ∈ V and all s ∈ X ∗ .P5. For any sequence ( f n ) n ∈ N of finitary gambles that isuniformly bounded below and any s ∈ X ∗ :lim n → + ∞ f n = f ⇒ limsup n → + ∞ E ( f n | s ) ≥ E ( f | s ) .Here, as well as further on, we call a sequence ( f n ) n ∈ N of variables uniformly bounded below if there is some
5. The choice of extending the local models Q s in this partic-ular way, by imposing continuity with respect to upper andlower cuts, is motivated in [
16, Sections 2 and 8 ] and [
17, sec-tion 6 ] , and is, as far as the resulting global game-theoretic up-per expectation—with extended real-valued supermartingales—is concerned, completely equivalent with how Shafer and Vovkaxiomatise their local models in [
12, Part II ] . LOBAL U PPER E XPECTATIONS FOR D ISCRETE -T IME S TOCHASTIC P ROCESSES : I N P RACTICE , T
HEY A RE A LL T HE S AME ! B ∈ R such that f n ( ω ) ≥ B for all n ∈ N and ω ∈ Ω . Fur-thermore, the limit lim n → + ∞ f n , as well as all others inthis paper, are intended to be taken pointwise.Axioms P1–P5 are put forward here because, as weargue in [
17, Section 4 ] , they can be motivated on thebasis of two different interpretations for a global up-per expectation; a direct behavioural interpretation interms of minimum selling prices, or a probabilistic in-terpretation in terms of sets of linear expectations (orprobability measures). Basically, we find Axioms P1–P4 straightforward and believe them to be almost un-questionable, regardless of the adopted interpretation.Axiom P5, which imposes a form of continuity, is per-haps more disputable. Nonetheless, compared to otherwell-known continuity properties, such as dominatedconvergence or monotone convergence, Property P5 israther weak because it only applies to sequences of finit-ary gambles. Note that, in general, finitary gambles playa central role in our axiomatisation; with the exceptionof monotonicity (Axiom P4), all our axioms exclusivelyapply to finitary gambles (and their limits). We find thisimportant because, as explained in [
17, Section 4 ] , theyare the only global variables that we feel can be given adirect operational meaning, and hence, the only globalvariables for which axioms can be motivated directly.More general global variables in V , on the other hand,that depend on an infinite number of state values, or areunbounded or even infinite-valued, should be regardedas abstract idealisations.Of course, even if we agree upon Axioms P1–P5, itdoes not necessarily provide us with a global upper ex-pectation because there may be multiple—or, worse,no—global upper expectations satisfying these axioms.The following result shows that there is at least onemodel that satisfies P1–P5, and that among all the onesthat satisfy them, there is a unique most conservative—that is, largest—one. We denote this model by E A,Q . Theorem 1 ( [
17, Theorem 6 ] ) For any upper expecta-tion tree Q , there is a unique most conservative globalupper expectation E A,Q that satisfies P1 – P5 .
4. An Equality for Monotone Limits ofFinitary Gambles
Having introduced all three global upper expectations,we can finally turn to the central problem of this paper:how are these upper expectations related to each other?More specifically, we ask ourselves the following. If theparameters of a stochastic process are equivalent—thatis, if the trees P and Q agree—are the global models E P ,E G,Q and E
A,Q then equal? In a recent paper [ ] , we haveshown that the answer is affirmative for the latter twomodels. Theorem 2 ( [
17, Theorem 6 ] ) The global upper expect-ations E A,Q and E G,Q are equal.
So it only remains to study the relationship betweenthe measure-theoretic upper expectation E P and thecommon upper expectation E Q : = E A,Q = E G,Q . To do so,we will build on two earlier results, gathered from thatsame paper [ ] ; the first one [
17, Theorem 14 ] says thatE P coincides with E Q if the tree P is a precise probab-ility tree p (and Q is the agreeing (upper) expectationtree); the second one [
17, Proposition 21 ] says that theyare also equal for general imprecise probability trees P , provided that we limit ourselves to finitary gambles.Our main results extend this equality for general impre-cise probability trees in two ways: to variables that aremonotone limits of finitary gambles and to bounded be-low F -measurable variables. In the current section, wework towards establishing the first extension. Our ap-proach is straightforward; we will prove that E P andE Q are both continuous with respect to monotone se-quences of finitary gambles. Since they coincide on fi-nitary gambles, this directly implies the desired equal-ity.We start by showing that E P and E Q are both continu-ous with respect to non-decreasing sequences in V b —and hence definitely in F . Proposition 3
For any P and Q , any s ∈ X ∗ and anynon-decreasing sequence ( f n ) n ∈ N in V b , we have that lim n → + ∞ E P ( f n | s ) = E P ( f | s ) , with f = sup n ∈ N f n = lim n → + ∞ f n , and similarly for E Q . Proof
That the statement holds for E Q follows imme-diately from [
17, Theorem 9(i) ] . To prove the statementfor E P , recall [
17, Theorem 14 ] , which says that, forany precise probability tree p and the agreeing (upper)expectation tree Q p , we have that E p ( g | s ) = E Q p ( g | s ) for all g ∈ V . Then, since E Q p is continuous with re-spect to non-decreasing sequences in V b [
17, Theorem9(i) ] , we have, for any precise probability tree p , thatlim n → + ∞ E p ( f n | s ) = E p ( f | s ) . Hence, it follows thatsup p ∼P lim n → + ∞ E p ( f n | s ) = sup p ∼P E p ( f | s ) = E P ( f | s ) .On the other hand, we also have thatsup p ∼P lim n → + ∞ E p ( f n | s ) ≤ lim n → + ∞ sup p ∼P E p ( f n | s )= lim n → + ∞ E P ( f n | s ) ,where the two last limits exist because ( f n ) n ∈ N is non-decreasing and E p —and therefore also E P —is mono-tone; see e.g. Lemma 18 in Appendix A.2. So we ob-tain that E P ( f | s ) ≤ lim n → + ∞ E P ( f n | s ) . The converse in-equality follows from the fact that f n ≤ f for all n ∈ N and the monotonicity of E P . LOBAL U PPER E XPECTATIONS FOR D ISCRETE -T IME S TOCHASTIC P ROCESSES : I N P RACTICE , T
HEY A RE A LL T HE S AME ! Next, we prove that E P is also continuous with respectto non-increasing sequences in F —that E Q satisfies thistype of continuity was already established in [
17, The-orem 9(ii) ] . The proof is less straightforward, though,and first requires us to establish the following two topo-logical lemmas concerning probability trees. We will saythat a sequence ( p i ) i ∈ N of precise probability trees con-verges if there is some limit tree p such that, for each s ∈ X ∗ , the mass functions ( p i ( ·| s )) i ∈ N converge (point-wise) to the mass function p ( ·| s ) . Lemma 4
Consider any imprecise probability tree P .Then any sequence ( p i ) i ∈ N of precise probability treesthat are compatible with P has a converging sub-sequence whose limit is compatible with P . Lemma 5
Consider any sequence ( p i ) i ∈ N of preciseprobability trees that converges to some limit tree p.Then, for any g ∈ F and any s ∈ X ∗ , lim i → + ∞ E p i ( g | s ) = E p ( g | s ) . Combined, the two lemmas above suffice to provethe continuity of E P with respect to non-increasing se-quences in F . Proposition 6
For any P and Q , any s ∈ X ∗ and anynon-increasing sequence ( f n ) n ∈ N of finitary gambles, wehave that lim n → + ∞ E P ( f n | s ) = E P ( f | s ) , with f = inf n ∈ N f n = lim n → + ∞ f n , and similarly for E Q . Proof
The statement for E Q follows from [
17, The-orem 9(ii) ] . To prove the statement for E P first note that,since ( f n ) n ∈ N is non-increasing and all f n are gambles,the variable f is bounded above. Moreover, f is F -measurable because it is a pointwise limit of finitary—and therefore certainly F -measurable—gambles [ ] . Taking both facts into account, we de-duce that, for any p ∼ P , the expectation E p ( f | s ) ex-ists and hence, because E p is an extension of E p (seeSection 3.2), that E p ( f | s ) = E p ( f | s ) . Since this obviouslyalso holds for each f n —because they are finitary andbounded—the desired statement follows if we manageto show thatlim n → + ∞ sup ¦ E p ( f n | s ) : p ∼ P © = sup ¦ E p ( f | s ) : p ∼ P © .The ‘ ≥ ’-inequality follows immediately from the factthat f m ≥ inf n ∈ N f n = f for all m ∈ N and the monoton-icity of E p . It remains to prove the converse inequality.Fix any ε > ( p i ) i ∈ N be a sequence of preciseprobability trees such that p i ∼ P andE p i ( f i | s )+ ε ≥ sup (cid:8) E p ( f i | s ) : p ∼ P (cid:9) for all i ∈ N . Note that this is indeed possible because, for all i ∈ N ,sup (cid:8) E p ( f i | s ) : p ∼ P (cid:9) ≤ sup f i and, since f i is a gamble,sup f i ∈ R . Then Lemma 4 guarantees that ( p i ) i ∈ N hasa convergent subsequence ( p i ( k ) ) k ∈ N whose limit p ∗ is compatible with P . Since E p ∗ satisfies continuitywith respect to non-increasing sequences [
17, Prop-erty M9 ] (the required conditions are obviously satis-fied because f n is finitary and f n ≤ f ≤ sup f ∈ R for all n ∈ N ), there is, for any real a > E p ∗ ( f | s ) , some n ∗ ∈ N such that a ≥ E p ∗ ( f n ∗ | s ) . Furthermore, since f n ∗ is finit-ary, and since E p ∗ ( f n ∗ | s ) ∈ R because f n ∗ is a gamble,Lemma 5 implies that there is some k ∗ ∈ N such thatE p i ( k ) ( f n ∗ | s ) − ε ≤ E p ∗ ( f n ∗ | s ) for all k ≥ k ∗ . We thereforeget that a ≥ E p i ( k ) ( f n ∗ | s ) − ε for all k ≥ k ∗ . (4)Now consider any k ≥ k ∗ such that i ( k ) ≥ n ∗ , whichis possible because ( i ( k )) k ∈ N is increasing. Then,since ( f n ) n ∈ N is non-increasing, and E p i ( k ) is mono-tone, Equation (4) implies that a ≥ E p i ( k ) ( f i ( k ) | s ) − ε .Since the tree p i ( k ) was chosen in such a way thatE p i ( k ) ( f i ( k ) | s )+ ε ≥ sup (cid:8) E p ( f i ( k ) | s ) : p ∼ P (cid:9) , this impliesthat a ≥ sup (cid:8) E p ( f i ( k ) | s ) : p ∼ P (cid:9) − ε . Because this holdsfor any k ≥ k ∗ such that i ( k ) ≥ n , we find that a ≥ lim k → + ∞ sup (cid:8) E p ( f i ( k ) | s ) : p ∼ P (cid:9) − ε = lim n → + ∞ sup (cid:8) E p ( f n | s ) : p ∼ P (cid:9) − ε ,where the equality follows from the fact that ( f n ) n ∈ N is non-increasing and the monotonicity of E p . Sincethis holds for any real a > E p ∗ ( f | s ) , it follows thatE p ∗ ( f | s ) ≥ lim n → + ∞ sup (cid:8) E p ( f n | s ) : p ∼ P (cid:9) − ε . Finally,it suffices to recall that p ∗ ∼ P , to see thatsup ¦ E p ( f | s ) : p ∼ P © ≥ lim n → + ∞ sup (cid:8) E p ( f n | s ) : p ∼ P (cid:9) − ε ,which, since ε > P and E Q coincide on F ×X ∗ [ ] to arrive at our first main result. Theorem 7
Consider any P and Q that agree, anys ∈ X ∗ and any f ∈ V that is the pointwise limit of anon-decreasing or non-increasing sequence of finitarygambles. Then we have that E P ( f | s ) = E Q ( f | s ) .
5. An Equality for F -Measurable Variables In order to prove our second main result—that E P coincides with E Q on bounded below F -measurablevariables—we require the notions of upper and lowersemicontinuity. LOBAL U PPER E XPECTATIONS FOR D ISCRETE -T IME S TOCHASTIC P ROCESSES : I N P RACTICE , T
HEY A RE A LL T HE S AME ! Let Ω be endowed with the topology generated bythe cylinder events { Γ ( s ) : s ∈ X ∗ } . As we show in Ap-pendix A.2, this topology is metrizable and compact,and coincides with the product topology on Ω = X N .For any topological space Y —and hence for Ω inparticular—a function f : Y → R is called upper semi-continuous (u.s.c.) if { y ∈ Y : f ( y ) < a } is an open sub-set of Y for each a ∈ R ; see [
8, Section 11.C and 23.F ] or [
20, Section 3.7.K ] . A function f : Y → R is called lowersemicontinuous (l.s.c.) if − f is u.s.c. and it is called con-tinuous if it is both u.s.c. and l.s.c. In general, semi-continuous functions can always be written as point-wise limits of monotone sequences of continuous real-valued functions (see e.g. [
8, Theorem 23.19 ] ). In ourcase, though, where Y = Ω , a stronger property holds. Lemma 8
Any f ∈ V is u.s.c. (l.s.c.) if and only if itis the pointwise limit of a non-increasing (resp. non-decreasing) sequence ( f n ) n ∈ N of extended real variables,each of which is n-measurable and bounded below (resp.bounded above). Moreover, f is both u.s.c. (l.s.c.) andbounded above (resp. bounded below) if and only if itis the pointwise limit of a non-increasing (resp. non-decreasing) sequence ( f n ) n ∈ N of n-measurable gambles. Lemma 8 leads us to two important intermediate res-ults, the first of which being that E P and E Q coincideon the domain of all u.s.c. variables that are boundedabove and all l.s.c. variables that are bounded below.The result can simply be seen as a restatement of The-orem 7 and is therefore stated without proof. Corollary 9
For any P and Q that agree, any s ∈ X ∗ and any variable f ∈ V that is u.s.c. and bounded above,or l.s.c. and bounded below, we have that E P ( f | s ) = E Q ( f | s ) . On the other hand, Lemma 8 also implies that con-tinuity with respect to non-increasing sequences of(bounded above) u.s.c. variables is actually not strongerthan continuity with respect to non-increasing se-quences of finitary gambles; see Lemma 17 in Ap-pendix A.2. Since both E P and E Q satisfy the latter typeof continuity, we immediately obtain the following res-ult. Proposition 10
Consider any P and Q , any s ∈ X ∗ and any non-increasing sequence ( f n ) n ∈ N of u.s.c. vari-ables that are bounded above. Then we have that lim n → + ∞ E P ( f n | s ) = E P ( f | s ) for f = lim n → + ∞ f n , andsimilarly for E Q . Proof
This follows from Lemma 17 in Appendix A.2, Pro-position 6 and the fact that E P and E Q are clearly bothmonotonous (see Lemma 18 in Appendix A.2). Note that, conversely, E P and E Q are also continuouswith respect to non-decreasing sequences of l.s.c. vari-ables that are bounded below, simply because, due toProposition 3, they satisfy continuity with respect to anynon-decreasing (bounded below) sequence.As a final step towards establishing our desired res-ult, we will use what is called Choquet’s capacitabilitytheorem. This theorem can be found in many differenttextbooks, but we will make use of the specific versionof Dellacherie [ ] . We do this because Dellacherie’s no-tion of a capacity can directly be applied to an extendedreal-valued functional—such as E P and E Q —whereasmost other sources restrict capacities to take the formof set-functions. Let us start by introducing some keyconcepts and terminology regarding capacitability andanalytic functions.Let V ≥ be the set of all variables taking values in R ≥ and V u ≥ the set of all (possibly unbounded) variablestaking values in R ≥ . A functional F : V ≥ → R ≥ is calleda Ω -capacity if it satisfies the following three properties [
7, Section II.1.1 ] :CA1. f ≤ g ⇒ F ( f ) ≤ F ( g ) for all f , g ∈ V ≥ ;CA2. lim n → + ∞ F ( f n ) = F (cid:0) lim n → + ∞ f n (cid:1) for any non-decreasing sequence ( f n ) n ∈ N in V ≥ ;CA3. lim n → + ∞ F ( f n ) = F (cid:0) lim n → + ∞ f n (cid:1) for any non-increasing sequence ( f n ) n ∈ N of u.s.c. variablesin V u ≥ .Recall from the beginning of this section that Ω is com-pact and metrizable, which is in line with Dellacherie’sassumption about the set ‘E’ in [
7, Section II.1.1 ] ; see [
7, Introduction, Paragraph 2 ] . Furthermore, observethat CA3 only applies to sequences in V u ≥ instead ofsequences in V ≥ ; this too corresponds to the defini-tion given in [
7, Section II.1.1 ] because Dellacherie al-ways considers u.s.c. functions to be real-valued [
7, In-troduction, Paragraph 2 ] . In fact, one could restate CA3so as to only apply to sequences that are uniformlybounded above; this follows immediately from the non-increasing character and the following lemma. Lemma 11
Any u.s.c. variable f ∈ V u ≥ is boundedabove. For any Ω -capacity F, we say that a variable f ∈ V ≥ isF -capacitable ifF ( f ) = sup (cid:8) F ( g ) : g ∈ V u ≥ , g is u.s.c. and f ≥ g (cid:9) . (5)A variable f ∈ V ≥ is called universally capacitable ifit is F-capacitable for all Ω -capacities F. Now, Cho-quet’s capacitability theorem [
7, Theorem II.2.5 ] statesthat any analytic variable is universally capacitable.The definition of an analytic variable can be found in [
7, 8 ] ; we do not explicitly give it here, because it is a LOBAL U PPER E XPECTATIONS FOR D ISCRETE -T IME S TOCHASTIC P ROCESSES : I N P RACTICE , T
HEY A RE A LL T HE S AME ! rather abstract concept that, in practice, can often bereplaced by the simpler and better-known notion ofa Borel-measurable variable. Indeed, according to [ ] , each Borel-measurable variable in V ≥ is analytic. Moreover, by Corollary 16 in Appendix A.2,the Borel σ -algebra on Ω coincides with the σ -algebra F generated by all cylinder events, so the notions ofBorel-measurability and F -measurability are equival-ent. Combined with [
7, Theorem II.2.5 ] , this allows usto state the following weaker version of Choquet’s capa-citability theorem: Theorem 12 (Choquet’s capacitability light)
Any F -measurable variable f ∈ V ≥ is universally capacitable. As an almost immediate consequence of Proposi-tion 3, Proposition 6 and Lemma 11, it can be shownthat, for any s ∈ X ∗ , the restrictions of both E P ( ·| s ) and E Q ( ·| s ) to V ≥ are Ω -capacities; see Appendix A.2.Therefore, and because these upper expectations co-incide on the u.s.c. variables in V u ≥ —due to Corol-lary 9 and Lemma 11 above—the desired equality for F -measurable variables in V ≥ follows from Equation (5)and Theorem 12. We can moreover replace V ≥ by V b ,simply because E P and E Q are linear with respect toadding constants (see Lemma 18 in Appendix A.2). Thisleads to our second main result. Theorem 13
For any P and Q that agree, any s ∈ X ∗ and any F -measurable variable f ∈ V b that is boundedbelow, we have that E P ( f | s ) = E Q ( f | s ) .
6. Relation with Shafer and Vovk’s Work
Before we conclude this paper, it seems appropriate tosay a few words about how our work here comparesto that of Shafer and Vovk. As readers that are famil-iar with their work may have noticed, the idea to useChoquet’s capacitability theorem to extend the domainof the equality to F -measurable (or analytic) variablesalready appears in [
12, Section 9 ] . Another part thatstrongly builds on ideas from [
12, Section 9 ] is the proofof Proposition 6; some key steps there were inspired bythe proof of [
12, Lemma 9.10 ] . So it is fair to say that [
12, Section 9 ] served as an important inspiration forour work. In fact, to the untrained eye, it might per-haps even seem as if our results do not differ much fromthose in [
12, Section 9 ] ; but take a closer look.First of all—and most importantly—the setting inwhich we define game-theoretic upper expectations dif-fers considerably from theirs. More specifically, theyconsider supermartingales under the prequential prin-ciple, which says that Forecaster’s moves—the specific-ation of the local models Q s (or Q ↑ s )—are not necessar-ily known beforehand for each situation s ∈ X ∗ , but in-stead are allowed to also depend on previous moves by Skeptic; see [
12, Theorem 7.5 ] for more details. Whilethis assumption allows them to remain more general—though, in many practical cases, it does not make muchof a difference—the benefit that we gain from droppingit is remarkable; it allows us to replace [
12, Lemma 9.10 ] and [
12, Theorem 9.7 ] , which require strong topologicalconditions on the parametrization of the local models,with respectively Theorem 7 and Theorem 13, whichare similar, but do not need any topological conditionsat all.A second notable difference is that our results involvea larger domain; Theorem 7, or equivalently, Corollary 9,applies to both u.s.c. variables that are bounded aboveand l.s.c. variables that are bounded below, whereas [ ] only applies to bounded u.s.c. variables;Theorem 13 applies to bounded below ( F -measurable)variables, whereas [
12, Theorem 9.7 ] only applies tobounded (analytic) variables. Our results also allowconditioning on situations; theirs only apply to uncon-ditional upper expectations. The fact that this extensionin domain is relevant in practice becomes clear whenwe also take a look at lower expectations. Indeed, in(more) practical situations, we are usually not only in-terested in the upper expectation of a variable, but also,and simultaneously, in its lower expectation [
9, 10 ] . Ourresults can be easily extended to this two-sided setting,by combining the conjugacy relation between globalupper and lower expectations with our two main res-ults. Corollary 14
Consider any P and Q that agree, anys ∈ X ∗ and any f ∈ V that is (a) the pointwise limit ofa monotone sequence of finitary gambles or (b) an F -measurable gamble. Then we have that E P ( f | s ) = E Q ( f | s ) and E P ( f | s ) = E Q ( f | s ) .Note that many practically relevant inferences—e.g. hit-ting times [ ] —fall under category (a) but not under cat-egory (b), simply because they are not bounded. Yet, itis exactly this class of variables that is missing in Shaferand Vovk’s main result [
12, Theorem 9.7 ] .Finally, recall that our results relate E P to E Q , wherethe latter represents, apart from the game-theoretic up-per expectation E G,Q , also the axiomatic upper expect-ation E
A,Q . Shafer and Vovk, on the other hand, only re-late E P to the game-theoretic upper expectation E G,Q .
7. Conclusion
Our main results, Theorem 7 and Theorem 13, showthat measure-theoretic, game-theoretic and axiomatic
6. Recall that we could just as well have stated Theorem 13 for ana-lytic variables instead of F -measurable variables. LOBAL U PPER E XPECTATIONS FOR D ISCRETE -T IME S TOCHASTIC P ROCESSES : I N P RACTICE , T
HEY A RE A LL T HE S AME ! upper expectations are equal on a large domain of vari-ables; it contains all variables that are the limit of amonotone sequence of finitary gambles, and all vari-ables that are bounded below and F -measurable. It re-mains to be seen whether we can extend this equival-ence even further, to all variables; so far, we have yet tofind a counterexample showing that this is not possible. References [ ] Thomas Augustin, Frank P.A. Coolen, Gertde Cooman, and Matthias C.M. Troffaes.
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Communications inComputer and Information Science , pages 224–238,2020. [ ] Natan T’Joens, Jasper De Bock, and Gertde Cooman. Game-theoretic upper expectationsfor discrete-time finite-state uncertain processes.Submitted for publication; see arXiv:2008.03133for a preliminary online version, 2020. [ ] Natan T’Joens, Jasper De Bock, and Gert deCooman. A particular upper expectation as globalbelief model for discrete-time finite-state uncer-tain processes.
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Appendix A. Proofs
A.1. Proofs for the Results in Section 4
Let P X be the set of all probability mass functions on X and let d be the total variation distance [
3, Section 7.1 ] LOBAL U PPER E XPECTATIONS FOR D ISCRETE -T IME S TOCHASTIC P ROCESSES : I N P RACTICE , T
HEY A RE A LL T HE S AME ! defined, for any two mass functions π , π ∈ P X , by d ( π , π ) : = max A ⊆X | π ( A ) − π ( A ) | = X x ∈X | π ( x ) − π ( x ) | ,(6)where we allowed ourselves a slight abuse of notationby writing π i ( A ) to mean P x ∈ A π i ( x ) for i ∈ { } . Let P X be endowed with the topology induced by d , whichis equivalent—see [
3, Appendix A ] —to the topology ofpointwise convergence that we have implicitly adoptedin the main text. So P X is metrizable and, by [
3, Sec-tion 7 ] , compact. Also, note that any precise probabil-ity tree p : s ∈ X ∗ p ( ·| s ) ∈ P X can be regarded as anelement of the product space × s ∈X ∗ P X , and any im-precise probability tree P can be seen as a subset of × s ∈X ∗ P X . Saying that a precise probability tree p iscompatible with an imprecise probability tree P is thenthe same as saying that p ∈ P . We will moreover en-dow the space × s ∈X ∗ P X with the product topology. It isclear that a sequence of precise probability trees ( p i ) i ∈ N then converges if, for each situation s ∈ X ∗ , the massfunctions ( p i ( ·| s )) i ∈ N converge pointwise, which is in ac-cordance with our assumptions in the main text. Proof of Lemma 4
For any s ∈ X ∗ , since the credal set P s is a closed and convex subset of the compact space P X , it follows that P s is compact. Therefore, by Tychon-off’s theorem [
20, Theorem 17.8 ] , the tree P is com-pact too (as a subset of × s ∈X ∗ P X ). Moreover, note that,due to [
20, Theorem 22.3 ] and the metrizability of P X (and the fact that X ∗ is countable), the space × s ∈X ∗ P X is also metrizable. So, by [
20, 17G.3. ] , the compactnessof P implies its sequential compactness. Hence, bydefinition, each sequence in P has a convergent sub-sequence whose limit belongs to P . Proof of Lemma 5
First of all, observe that, for all i ∈ N ,the expectations E p ( g | s ) and E p i ( g | s ) are indeed well-defined—the corresponding Lebesgue integrals exist—because g is bounded and finitary (and therefore cer-tainly F -measurable). In fact, because we can write g = P z ℓ ∈X ℓ g ( z ℓ ) I z ℓ for some ℓ ∈ N , these expectationssimply reduce—by definition of the Lebesgue integral;see [
13, Section 2.6.1 ] —to the finite weighted sumsE p ( g | s ) = X z ℓ ∈X ℓ g ( z ℓ ) P p ( z ℓ | s ) (7)and E p i ( g | s ) = X z ℓ ∈X ℓ g ( z ℓ ) P p i ( z ℓ | s ) , (8)where P p ( ·| s ) and P p i ( ·| s ) are defined according to Equa-tion (1). Let x k ∈ X ∗ be such that s = x k and fix any z ℓ ∈ X ℓ . We will now show that P p i ( z ℓ | s ) converges toP p ( z ℓ | s ) as a function of i ∈ N .If k ≥ ℓ and z ℓ = x ℓ , then P p i ( z ℓ | s ) = i ∈ N and also P p ( z ℓ | s ) =
1, so P p i ( z ℓ | s ) surely converges toP p ( z ℓ | s ) . Similar observations lead us to conclude thatthis is also true for the cases where, either, k ≥ ℓ and z ℓ = x ℓ , or, k < ℓ and z k = x k . So it remains to checkwhether it is true for the case where k < ℓ and z k = x k . In that case, P p i ( z ℓ | x k ) = Q ℓ − n = k p i ( z n + | z n ) con-verges to P p ( z ℓ | x k ) = Q ℓ − n = k p ( z n + | z n ) if, for all n ∈{ k , ··· , ℓ − } , p i ( z n + | z n ) converges to p ( z n + | z n ) . Thelatter is implied by the convergence of p i to p . Indeed,since × s ∈X ∗ P X is equipped with the product topo-logy, the convergence of p i to p implies that, for any n ∈ { k , ··· , ℓ − } , the mass function p i ( ·| z n ) convergesto p ( ·| z n ) . Since the set P X on its turn is equippedwith the topology of pointwise convergence, this im-plies that p i ( z n + | z n ) converges to p ( z n + | z n ) .Now, to conclude the proof, note that the sums inEquations (7) and (8) are over a finite set X ℓ —because X is finite—and the coefficients g ( z ℓ ) are real be-cause g is a gamble. Since we have just shown that,for any z ℓ ∈ X ℓ , the probability P p i ( z ℓ | s ) convergesto P p ( z ℓ | s ) , it is therefore clear that the expectationE p i ( g | s ) converges to E p ( g | s ) . Proof of Theorem 7
Suppose that f is the pointwiselimit of a non-decreasing sequence of finitary gambles.Then we have that E P ( f | s ) = E Q ( f | s ) because, on theone hand, E P coincides with E Q for all finitary gambles [
17, Proposition 21 ] , and on the other hand, due to Pro-position 3, both E P and E Q are continuous with respectto non-decreasing sequences of gambles. Suppose nowthat f is the pointwise limit of a non-increasing se-quence ( f n ) n ∈ N of finitary gambles. Then similarly, thedesired equality follows from [
17, Proposition 21 ] andProposition 6. A.2. Proofs for the Results in Section 5
Consider the distance function δ on Ω defined by δ ( ω , ω ′ ) : = − n with n : = inf { k ∈ N : ω k = ω ′ k } , (9)for all ω , ω ′ ∈ Ω . Then it can easily be checked that δ isa metric on Ω . Furthermore, as is shown by the lemmabelow, the topology on Ω corresponding to this metric δ is the same as the topology that we have adoptedthroughout the main text—that is, the topology gener-ated by the cylinder events { Γ ( s ) : s ∈ X ∗ } . This confirmsour claim that Ω is metrizable. Moreover, the lemma be-low also shows that this metric topology coincides withthe product topology and therefore, by Tychonoff’s the-orem [
20, Theorem 17.8 ] and the finiteness of X (andtherefore the compactness of X ), that Ω is compact. LOBAL U PPER E XPECTATIONS FOR D ISCRETE -T IME S TOCHASTIC P ROCESSES : I N P RACTICE , T
HEY A RE A LL T HE S AME ! Lemma 15
The set { Γ ( s ) : s ∈ X ∗ } of all cylinder eventsis a subbase for the metric topology on Ω correspondingto δ . The same holds for the product topology on Ω , andhence, the metric topology and product topology coin-cide. Moreover, a set in this topology is open if and onlyif it is a countable union of cylinder events. Proof
Recall that the set of all open ε -disks form asubbase for the metric topology; see e.g. [
20, Example3.2(a) ] . Consider any such open ε -disk; that is, for any ε > ω ∈ Ω , consider the set { ω ′ ∈ Ω : δ ( ω , ω ′ ) <ε } . If ε >
1, let ℓ : =
0; otherwise, let ℓ ∈ N be the uniquenatural number such that 2 − ℓ − < ε ≤ − ℓ . Then, for all ω ′ ∈ Γ ( ω ℓ ) , since inf { k ∈ N : ω ′ k = ω k } ≥ ℓ +
1, we haveby Equation (9) that δ ( ω , ω ′ ) ≤ − ℓ − < ε . On the otherhand, for any ω ′ Γ ( ω ℓ ) , we infer in a similar way that δ ( ω , ω ′ ) ≥ − ℓ ≥ ε . Hence, both facts taken together, weobtain that Γ ( ω ℓ ) = { ω ′ ∈ Ω : δ ( ω , ω ′ ) < ε } is the open ε -disk around ω . Conversely, one can see that any cylin-der event Γ ( x ℓ ) with x ℓ ∈ X ∗ , is an open ε -disk aroundany ω ∈ Γ ( x ℓ ) if ε > − ℓ − < ε ≤ − ℓ . Asa consequence, the family of open ε -disks in Ω is thesame as the set { Γ ( s ) : s ∈ X ∗ } of all cylinder events andtherefore, since the former is a subbase of the metric to-pology, the set { Γ ( s ) : s ∈ X ∗ } is a subbase of the metrictopology. This establishes the first statement.Let us show that the same holds for the product topo-logy on Ω = X N . Since X has the discrete topology, thesets U n , y : = { ω ∈ Ω : ω n = y } with n ∈ N and y ∈ X forma subbase of this topology [
20, Definition 8.3 ] . Clearly,any such set U n , y is the union of the cylinder events Γ ( x n − y ) with x n − ∈ X n − , so the topology gener-ated by the cylinder events { Γ ( s ) : s ∈ X ∗ } is finer than(includes) the product topology. On the other hand,any cylinder event Γ ( x n ) with x n ∈ X ∗ is the finiteintersection of the sets U i , x i with i ∈ { ··· , n } , so wealso have that the product topology is finer than theone generated by { Γ ( s ) : s ∈ X ∗ } . All together, we con-clude that the topology generated by the cylinder events { Γ ( s ) : s ∈ X ∗ } coincides with the product topology—and hence { Γ ( s ) : s ∈ X ∗ } is a subbase—which estab-lishes the second statement.It remains to prove the last statement, which saysthat a set in this common topology is open if and onlyif it is a countable union of cylinder events. In otherwords, we have to prove that τ : = {∪ i ∈ N Γ ( s i ) : ( ∀ i ∈ N ) s i ∈X ∗ } is the topology generated by the subbase { Γ ( s ) : s ∈X ∗ } . That τ is closed under arbitrary unions followsfrom the fact that the set X ∗ of all situations is count-able. Indeed, any union of elements of τ is a unionof cylinder events, and since X ∗ —and therefore also { Γ ( s ) : s ∈ X ∗ } —is countable, this union can always bewritten as a countable union, therefore implying that itis an element of τ . Now, consider any finite intersection ∩ j ∈{ ··· , n } ∪ i ∈ N Γ ( s i , j ) of elements of τ and let us check that this too is an element of τ . Using distributivity, thefinite intersection ∩ j ∈{ ··· , n } ∪ i ∈ N Γ ( s i , j ) can be rewrittenas a countable union of finite intersections of cylinderevents Γ ( s i , j ) . So we can conclude that this countableunion is an element of τ if we manage to show thatany finite intersection of cylinder events is itself a cyl-inder event. In order to do so, consider the intersec-tion of any two cylinder events Γ ( x n ) and Γ ( y m ) with x n ∈ X ∗ and y m ∈ X ∗ . Note that this intersection isnon-empty if and only if, either, n ≤ m and x n = y n ,or, if n > m and x m = y m . In the first case, we havethat Γ ( x n ) ∩ Γ ( y m ) = Γ ( y m ) and, in the second case,we have that Γ ( x n ) ∩ Γ ( y m ) = Γ ( x n ) . Hence, the inter-section of any two cylinder events is itself a cylinderevent and therefore, any finite intersection of cylinderevents is also a cylinder event. By our previous consid-erations, this implies that τ is indeed closed under finiteintersections. Together with the fact that τ is closed un-der arbitrary unions—and trivially includes Ω and theempty set ; —we may conclude that τ is a topology on Ω . Since { Γ ( s ) : s ∈ X ∗ } is clearly a subbase of this topo-logy τ , this finalises the proof.The last statement in the lemma above immediatelyimplies the following corollary, in which the Borel setsare the open sets with respect to the common topologyfrom Lemma 15. Corollary 16
The Borel σ -algebra on Ω coincides withthe σ -algebra F generated by the cylinder events. Proof
By Lemma 15, any open set in Ω is the countableunion of cylinder events. As a result, all open sets areincluded in the σ -algebra F and therefore, F includesthe Borel σ -algebra. On the other hand, it is clear that F is not larger than the Borel σ -algebra because each cyl-inder event is itself open (because it is a—trivial—unionof cylinder events). Proof of Lemma 8
Since − f ∈ V is l.s.c. if and only if f is u.s.c., it clearly suffices to prove the statement foru.s.c. variables. We start by proving the two direct im-plications. Let f ∈ V be u.s.c. and let ( f n ) n ∈ N be definedby f n ( ω ) : = sup ( {− n }∪{ f ( ω ′ ) : ω ′ ∈ Γ ( ω n ) } ) ,for all ω ∈ Ω and all n ∈ N . Then ( f n ) n ∈ N is clearly a non-increasing sequence of variables that are n -measurableand bounded below (since f n ≥ − n ). If f is boundedabove, then each f n is clearly also bounded above, so inthat case ( f n ) n ∈ N is a sequence of gambles. So it only re-mains to show that lim n → + ∞ f n ( ω ) = f ( ω ) for any ω ∈ Ω .That lim n → + ∞ f n ( ω ) ≥ f ( ω ) holds, follows from the factthat, due to the definition of the variables f n , f n ( ω ) ≥ LOBAL U PPER E XPECTATIONS FOR D ISCRETE -T IME S TOCHASTIC P ROCESSES : I N P RACTICE , T
HEY A RE A LL T HE S AME ! f ( ω ) for all n ∈ N . Hence, if f ( ω ) = + ∞ , we automat-ically have that lim n → + ∞ f n ( ω ) = f ( ω ) , so we may as-sume that f ( ω ) < + ∞ . Fix any real a > f ( ω ) . Since f isu.s.c., the set { ω ′ ∈ Ω : f ( ω ′ ) < a } is an open neighboor-hood of ω . According to Lemma 15, any open set in Ω isa countable union of cylinder events. Since ω belongsto { ω ′ ∈ Ω : f ( ω ′ ) < a } , one of these cylinder events con-tains ω . This implies that there is some n ∈ N such that f ( ω ′ ) < a for all ω ′ ∈ Γ ( ω n ) . Then, for any k ≥ n , since Γ ( ω k ) ⊆ Γ ( ω n ) , we obviously also have that f ( ω ′ ) < a forall ω ′ ∈ Γ ( ω k ) . Hence, f k ( ω ) ≤ a for all k ≥ max {| a | , n } ,which implies that lim k → + ∞ f k ( ω ) ≤ a . This holds forany real a > f ( ω ) , so we obtain that lim k → + ∞ f k ( ω ) ≤ f ( ω ) as desired.To prove the two converse implications, consider any f ∈ V that is the pointwise limit of a non-increasing se-quence ( f n ) n ∈ N of n -measurable bounded below vari-ables. We show that, for any a ∈ R , the set A : = { ω ∈ Ω : f ( ω ) < a } is open, and therefore that f is a u.s.c. vari-able. It is then clear that f is moreover bounded aboveif ( f n ) n ∈ N is a sequence of gambles, because in thatcase f ≤ f ≤ sup f ∈ R . So fix any a ∈ R and note thatthe sequence ( A n ) n ∈ N of events defined by A n : = { ω ∈ Ω : f n ( ω ) < a } for all n ∈ N , is non-decreasing and con-verges to A because ( f n ) n ∈ N converges non-increasinglyto f . So we have that A = ∪ n ∈ N A n . Moreover, for any n ∈ N , because f n is n -measurable, the set A n is a fi-nite union of cylinder events of the form Γ ( x n ) with x n ∈ X n . So, by Lemma 15, each set A n is open. Sinceany union of open sets is open again, we obtain that A = ∪ n ∈ N A n is open, therefore concluding the proof. Lemma 17
Any operator
F : V → R that is monotoneand that is continuous with respect to non-increasing(or non-decreasing) sequences of finitary gambles, is alsocontinuous with respect to non-increasing (resp. non-decreasing) sequences ( f n ) n ∈ N of u.s.c. (resp. l.s.c.) vari-ables that are bounded above (resp. bounded below); i.e. lim n → + ∞ F ( f n ) = F ( f ) , with f = inf n ∈ N f n = lim n → + ∞ f n . Proof
Consider any non-increasing sequence ( f n ) n ∈ N of u.s.c. variables that are bounded above. Then itfollows from Lemma 8 that, for all n ∈ N , there is anon-increasing sequence ( g n , m ) m ∈ N of m -measurablegambles such that lim m → + ∞ g n , m = f n . Now let ( h m ) m ∈ N be the sequence of variables defined by h m ( ω ) : = min { g n , m ( ω ) : 0 ≤ n ≤ m } for all ω ∈ Ω .Because each ( g n , m ) m ∈ N is non-increasing, ( h m ) m ∈ N isalso non-increasing. The variables h m for all m ∈ N are clearly bounded—and hence, they are gambles—and they are also m -measurable because, on the onehand, g n , m is m -measurable for all n ∈ N , and on the other hand, the minimum over a finite num-ber of m -measurable variables is trivially also m -measurable. So ( h m ) m ∈ N is a non-increasing sequenceof m -measurable gambles. Furthermore, note that h m ≥ f because g n , m ≥ f n ≥ f for all n , m ∈ N , andtherefore lim m → + ∞ h m ≥ f . To see that lim m → + ∞ h m ≤ f , fix any ω ∈ Ω and any a ∈ R such that a > f ( ω ) .Since lim n → + ∞ f n = f , there is some n ′ ∈ N such that a > f n ′ ( ω ) and since also lim m → + ∞ g n ′ , m = f n ′ , there issome m ′ ≥ n ′ such that a > g n ′ , m ′ ( ω ) . Then certainly a > h m ′ ( ω ) , and since ( h m ) m ∈ N is non-increasing, wehave that a > lim m → + ∞ h m ( ω ) . This holds for any a ∈ R such that a > f ( ω ) , so we have that lim m → + ∞ h m ( ω ) ≤ f ( ω ) , which in turn implies that lim m → + ∞ h m ≤ f because ω ∈ Ω was chosen arbitrarily. So we havethat lim m → + ∞ h m = inf m ∈ N h m = f . Now, recalling that ( h m ) m ∈ N is moreover a non-increasing sequence of m -measurable gambles, it follows from the assumptionsabout F that lim m → + ∞ F ( h m ) = F ( f ) . Furthermore, notethat, due to the non-increasing character of ( g n , m ) m ∈ N and ( f n ) n ∈ N , h m ( ω ) = min { g n , m ( ω ) : 0 ≤ n ≤ m }≥ min { f n ( ω ) : 0 ≤ n ≤ m } = f m ( ω ) ,for all m ∈ N and all ω ∈ Ω . So, f m ≤ h m for all m ∈ N ,which by the monotonicity of F implies thatlim m → + ∞ F ( f m ) ≤ lim m → + ∞ F ( h m ) = F ( f ) .The converse inequality—that lim m → + ∞ F ( f m ) ≥ F ( f ) —follows from the non-increasing character of ( f n ) n ∈ N and the monotonicity of F.Finally, that the complementary statement holdsfor any F : V → R that is (monotone and) continuouswith respect to non-decreasing sequences of finitarygambles, can easily be deduced from what we have justproved above, and the fact that f ∈ V is l.s.c. if andonly if − f is an u.s.c. variable Indeed, the operator F ′ defined by F ′ ( f ) : = − F ( − f ) for all f ∈ V satisfies mono-tonicity and continuity with respect to non-increasingsequences of finitary gambles, so it follows that F ′ is alsocontinuous with respect to non-increasing sequencesof u.s.c. variables that are bounded above. As a res-ult, F is continuous with respect to non-decreasing se-quences of l.s.c. variables that are bounded below. Proof of Lemma 11
Recall from Lemma 8 that f is thepointwise limit of a non-increasing sequence ( f n ) n ∈ N of n -measurable (bounded below) variables. Assume exabsurdo that f is not bounded above. Then, for each n ∈ N , since f n ≥ inf m ∈ N f m = f , it follows that f n isalso not bounded above. Since each f n can only takea finite number of different values—because it is n -measurable and X is finite—we must have that f n ( ω ) = LOBAL U PPER E XPECTATIONS FOR D ISCRETE -T IME S TOCHASTIC P ROCESSES : I N P RACTICE , T
HEY A RE A LL T HE S AME ! + ∞ for at least one ω ∈ Ω . So, for each n ∈ N , theset A n : = { ω ∈ Ω : f n ( ω ) = + ∞} is non-empty. Moreover,since ( f n ) n ∈ N is non-increasing, ( A n ) n ∈ N is also non-increasing and therefore, ∩ ni = A i = A n = ; for all n ∈ N .So ( A n ) n ∈ N has the finite intersection property. Then,since Ω is compact, it follows from [
20, Theorem 17.4 ] that the sets ( A n ) n ∈ N have a non-empty intersectionif each of the A n is closed. We proceed to show thatthe sets A n are closed. Note that because each f n is n -measurable, the set A n is a finite union of cylinderevents. In particular, there is some S n ⊆ X n such that A n = ∪ x n ∈ S n Γ ( x n ) . Since ∪ x n ∈X n Γ ( x n ) = Ω , this im-plies that A cn = ∪ x n ∈X n \ S n Γ ( x n ) is a finite union of cyl-inder events and therefore, by Lemma 15, it is open.So A n is closed and we can therefore apply [
20, The-orem 17.4 ] to find that ∩ n ∈ N A n = ; . Then, for any ω ∈∩ n ∈ N A n , since ω ∈ A n for all n ∈ N , it follows from thedefinition of the sets A n that f n ( ω ) = + ∞ for all n ∈ N .As a consequence, f ( ω ) = lim n → + ∞ f n ( ω ) = + ∞ , whichis in contradiction with the fact that f is real-valued. Lemma 18
Consider any P , any Q and any s ∈ X ∗ .Then, for all f , g ∈ V and µ ∈ R , we have that E1 . f ≤ g ⇒ E P ( f | s ) ≤ E P ( g | s ) [ monotonicity ] ;E2 . E P ( f + µ | s ) ≤ E P ( f | s )+ µ [ constant additivity ] , and similarly for E Q . Proof
To prove both properties for E P , consider anycompatible p ∼ P . That Properties E1 and E2 hold forE p follows from the fact that they are satisfied by the ex-pectation E p (if it exists; see [
17, Properties M1 and M2 ] )together with Equation (2). Since E P is then simply theupper envelope of all E p with p ∼ P , it follows that bothproperties are also satisfied by E P . Furthermore, thatE Q satisfies monotonicity is immediate from P4 andTheorems 1 and 2. That it is also constant additive fol-lows from [
17, Proposition 7 (V4) ] and, again, Theor-ems 1 and 2. Proposition 19
For any s ∈ X ∗ , the restrictions of E P ( ·| s ) and E Q ( ·| s ) to V ≥ are Ω -capacities. Proof
Property CA1 follows for both E P ( ·| s ) and E Q ( ·| s ) from Lemma 18 (E1) above. That E P ( ·| s ) and E Q ( ·| s ) sat-isfy Property CA2 follows from Proposition 3 and thefact that V ≥ ⊆ V b . Finally, that they satisfy Property CA3follows from Proposition 10, together with the fact that,as a consequence of Lemma 11, u.s.c. variables in V u ≥ are always bounded above. Proof of Theorem 13
Let f ∈ V b be bounded below and F -measurable. Since f is bounded below, and both E P and E Q are constant additive (see Lemma 18 (E2)), wemay assume without loss of generality that f is non-negative—and therefore, that f ∈ V ≥ . Then, accordingto Theorem 12, the variable f is universally capacitable.Since E P ( ·| s ) and E Q ( ·| s ) are both Ω -capacities by Pro-position 19, this implies thatE P ( f | s ) = sup ¦ E P ( g | s ) : g ∈ V u ≥ , g is u.s.c. and f ≥ g © andE Q ( f | s ) = sup ¦ E Q ( g | s ) : g ∈ V u ≥ , g is u.s.c. and f ≥ g © .Now recall Corollary 9, which says that E P ( h | s ) = E Q ( h | s ) for all u.s.c. variables h ∈ V that are boundedabove. Since all u.s.c. variables g ∈ V u ≥ are automatic-ally bounded above due to Lemma 11, we obtain thatE P ( f | s ) = E Q ( f | s ) . Proof of Corollary 14
The statement for the upper ex-pectations follows immediately from Theorem 7 andTheorem 13. To prove the statement for the lowerexpectations, we distinguish two cases. If f is thepointwise limit of a monotone sequence of finitarygambles, then the same holds for − f , and hence, byTheorem 7, E P ( − f | s ) = E Q ( − f | s ) . This implies by con-jugacy that − E P ( f | s ) = − E Q ( f | s ) and therefore thatE P ( f | s ) = E Q ( f | s ) . On the other hand, if f is an F -measurable gamble, then so is − f , and therefore, byTheorem 13, we have that E P ( − f | s ) = E Q ( − f | s ) . Con-jugacy then again implies that E P ( f | s ) = E Q ( f | s ) ..