Infinite horizon utility maximisation from inter-temporal wealth
IINFINITE HORIZON UTILITY MAXIMISATION FROMINTER-TEMPORAL WEALTH
MICHAEL MONOYIOS
Abstract.
We develop a duality theory for the problem of maximising expected lifetimeutility from inter-temporal wealth over an infinite horizon, under the minimal no-arbitrageassumption of No Unbounded Profit with Bounded Risk (NUPBR). We use only deflators,with no arguments involving equivalent martingale measures, so do not require the strongercondition of No Free Lunch with Vanishing Risk (NFLVR). Our formalism also works withoutalteration for the finite horizon version of the problem. As well as extending work of Bouchardand Pham [2] to any horizon and to a weaker no-arbitrage setting, we obtain a strongerduality statement, because we do not assume by definition that the dual domain is the polarset of the primal space. Instead, we adopt a method akin to that used for inter-temporalconsumption problems, developing a supermartingale property of the deflated wealth andits path that yields an infinite horizon budget constraint and serves to define the correctdual variables. The structure of our dual space allows us to show that it is convex, withoutforcing this property by assumption. We proceed to enlarge the primal and dual domainsto confer solidity to them, and use supermartingale convergence results which exploit Fatouconvergence, to establish that the enlarged dual domain is the bipolar of the original dualspace. The resulting duality theorem shows that all the classical tenets of convex dualityhold. Moreover, at the optimum, the deflated wealth process is a potential converging tozero. We work out examples, including a case with a stock whose market price of risk is athree-dimensional Bessel process, so satisfying NUPBR but not NFLVR.
Contents
1. Introduction 22. Financial market and problem formulation 52.1. The financial market 52.2. The primal problem 72.3. The budget constraint 92.4. The dual problem 113. The main duality 124. Abstract bipolarity and duality 144.1. Abstract bipolarity 154.2. The abstract duality 165. Bipolarity relations 175.1. Sufficiency of the budget constraint 175.2. Convexity of the dual domain 205.3. On approaches to establishing bipolarity 246. Proofs of the duality theorems 267. Examples 36References 41
Date : October 13, 2020.Part of this work was carried out during a visit to the Laboratoire de Probabilit´es et Mod`eles Al´eatoires,Universit´e Paris Diderot. I am very grateful to Huyˆen Pham for generous hospitality. a r X i v : . [ q -f i n . P M ] O c t MICHAEL MONOYIOS Introduction
Let U : R + → R be a classical utility function and ( X t ) t ≥ a non-negative wealth processgenerated from self-financing investment in a semimartingale incomplete market on a completestochastic basis (Ω , F , F := ( F t ) t ∈ [0 , ∞ ) , P ), with the filtration F satisfying the usual hypothesesof right-continuity and augmentation with P -null sets of F . Under the minimal no-arbitrageassumption of No Unbounded Profit with Bounded Risk (NUPBR), we develop a dualitytheory for a problem in which utility is derived from inter-temporal wealth over the infinitehorizon:(1.1) E (cid:20)(cid:90) ∞ U ( X t ) d κ t (cid:21) → max!In (1.1), κ : [0 , ∞ ) → R + is a non-decreasing c`adl`ag adapted process that will act as a finitemeasure to assign a weight to utility of wealth at each time. We focus on the infinite horizoncase, but our approach also works without alteration for the finite horizon version of (1.1), aswe re-iterate in Remark 3.2.Problems of the type in (1.1) can arise when traditional utility of terminal wealth problemshave a random horizon date, as we shall illustrate by some examples in Section 2.2.1, butcan just as well be considered in their own right as one possible objective for a long-livedinvestment fund. A duality theory for such problems was developed by Bouchard and Pham[2] over a finite horizon, with a no-arbitrage assumption that allowed for the existence ofequivalent local martingale measures (ELMMs), so tantamount to assuming No Free Lunchwith Vanishing Risk (NFLVR) in the terminology of Delbaen and Schachermayer [6]. Here,the underlying assumptions as well as the approach and construction of the dual space aredifferent to those in [2], as we now describe.First, as indicated above, we relax the no-arbitrage assumption from NFLVR to NUPBR,so we do not rely on the existence of ELMMs, only on the existence of a class of deflators thatmultiply admissible wealth processes to create supermartingales. It was first made explicit byKaratzas and Kardaras [15] (though was implicit in the terminal wealth problem of Karatzaset al [17] in an incomplete Itˆo process market, in which which ELMMs were not invoked at all)that all one needs for well-posed utility maximisation problems is the existence of a suitableclass of deflators to act as dual variables. In particular, ELMMs are not needed. This is afirst reason for adopting NUPBR as our no-arbitrage condition.Aside from weakening the no-arbitrage assumption, there are other sound reasons for avoid-ing the use of ELMMs. It is well known that ELMMs will typically not exist over the infinitehorizon, because the candidate change of measure density process is not a uniformly integrablemartingale. This is the case for the Black-Scholes model for example, as discussed in Karatzasand Shreve [18, Section 1.7]. Moreover, even if ELMMs might exist when restricted to a finitehorizon, one needs to proceed with some care in invoking them in an infinite horizon model,by ensuring that events in the tail σ -algebra F ∞ := σ (cid:16)(cid:83) t ≥ F t (cid:17) have been excluded in a con-sistent way. We discuss this issue further in Section 2.1.1. Irrespective of such subtleties, sincedeflators are the key ingredient for establishing a duality for utility maximisation problems,it is natural to construct a theory which uses only deflators, and makes no use whatsoever ofconstructions involving ELMMs, and this is what we do. A key step in this approach will bethe use of the Stricker and Yan [33] version of the Optional Decomposition Theorem (ODT)to establish bipolarity relations between the primal and dual domains, as opposed to variantsof the ODT which state the result in terms of ELMMs.Second, our approach to establishing the duality between the primal problem in (1.1) andan appropriately defined dual problem differs quite markedly from that in Bouchard and Pham[2], and our basic duality statement is strengthened compared to that in [2], in essence because TILITY FROM INTER-TEMPORAL WEALTH 3 we are able to prove, as opposed to assume by definition, that the dual domain is the polar ofthe primal domain, as we now describe.The approach taken in [2], over a finite horizon time [0 , T ], is to define the dual domain(in the case where the initial value of the dual variables is unity) as the set of processes Y such that E (cid:104)(cid:82) T X t Y t d κ t (cid:105) ≤ X t , ( X s ) ≤ s ≤ t ) t ≥ , thatis, the value of an admissible wealth process at any time, as well as the wealth path up to thattime, as follows. Let S be any classical supermartingale deflator, so XS is a supermartingalefor all admissible wealth processes, and let β be a non-negative process such that (cid:82) · β s d κ s isalmost surely finite. We define associated supermartingales R and processes Y by(1.2) R := exp (cid:18) − (cid:90) · β s d κ s (cid:19) S, Y := βR = β exp (cid:18) − (cid:90) · β s d κ s (cid:19) S. With these processes in place, we show that M := XR + (cid:82) · X s Y s d κ s is a supermartingalefor all admissible wealth processes. The wealth-path deflators Y are then the appropriatedual variables for the problem in (1.1). They involve the auxiliary dual control β above andbeyond that implicit in the choice of supermartingale deflator, a typical feature of wealth pathdependent utility maximisation problems.This program yields an infinite horizon budget constraint satisfied by the wealth path,similar to that in Bouchard and Pham [2], over our infinite horizon: E (cid:2)(cid:82) ∞ X t Y t d κ t (cid:3) ≤ Y . The form of the dualproblem then emerges as E (cid:20)(cid:90) ∞ V ( Y t ) d κ t (cid:21) → min!over deflators with initial value Y = y >
0, where V : R + → R is the convex conjugate of theutility function.The particular structure in (1.2) of the inter-temporal wealth deflators, involving the super-martingale deflators and the auxiliary dual control β , is crucial, as it allows us to show thatthe dual space which emerges is convex. We then enlarge the primal domain to encompassprocesses dominated by admissible wealths (similar in spirit to the procedure used by Kramkovand Schachermayer [21, 22] for the terminal wealth utility maximisation problem), and showthat the budget constraint is also a sufficient condition for admissible primal processes, usingthe Stricker and Yan [33] version of the Optional Decomposition Theorem. Finally, we enlargethe dual domain in a similar manner, to encompass processes dominated by the deflators, showthat the resulting dual domain is closed in an appropriate topology (that of convergence inmeasure µ := κ × P ) by exploiting Fatou convergence of supermartingales, and obtain perfectbipolarity relations between the enlarged primal and dual domains. This bipolarity underliesthe subsequent duality results.We thus prove (as opposed to impose, by definition) that our dual domain has the requiredconvexity and closedness properties needed to establish bipolarity and hence duality, with asupermartingale constraint involving the admissible wealths as a starting point. Put anotherway, the procedure developed by Kramkov and Schachermayer [21, 22] for the terminal wealth MICHAEL MONOYIOS problem is adapted and made to work for an inter-temporal wealth problem under NUPBRand over the infinite (or indeed, finite) horizon.The main duality result (Theorem 3.1) shows that all the tenets of the theory hold in ourscenario: the marginal utility of optimal wealth is equal to the optimal deflator with initialvalue equal to the derivative of the primal value function, and the primal and dual valuefunctions are mutually conjugate. Moreover, at the optimum, the supermartingale M becomesa uniformly integrable martingale (cid:99) M , leading to an interesting additional representation ofthe optimal wealth process:(1.3) (cid:98) X t (cid:98) R t = E (cid:20) (cid:90) ∞ t (cid:98) X s (cid:98) Y s d κ s (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , t ≥ , where (cid:98) X, (cid:98) R, (cid:98) Y are the optimal manifestations of the processes X, R, Y . The supermartingale XR becomes a potential (satisfying lim t →∞ E [ (cid:98) X t (cid:98) R t ] = 0) at the optimum, and also convergesalmost surely to (cid:98) X ∞ (cid:98) R ∞ = 0.The philosophy of our approach can thus be summarised as: the use of a natural super-martingale property to derive an inter-termporal budget constraint as a means of identifyingthe dual space, then suitably enlarging that space and using supermartingale convergence tech-niques to get the bipolarity relations. A similar philosophy was applied recently by Monoyios[26] to the infinite horizon optimal consumption problem under NUPBR. There, using an ap-propriate class of consumption deflators (differing from the wealth-path deflators used here),deflated wealth plus cumulative deflated consumption at the optimum becomes a uniformlyintegrable martingale, while deflated wealth becomes a potential converging to zero. The finalresults in [26] thus have a similar flavour to those here (as they must, since classical tenetsof duality theory are shoiwn to hold in both cases), but the problem studied in [26] is quitedistinct from the one here, involving a different primal variable (consumption, as opposed tothe wealth path), and a different set of concomitant deflators. It turns out that the primaldomain in [26] is L ( µ )-bounded, while here it is the dual space that has this property. Wethus do not use any results from [26] here, as we need to formulate a separate, complete proof.Inter-temporal utility maximisation problems have a long history, usually treating the prob-lem of maximising utility of consumption, as opposed to the less standard objective here, whichinvolves aggregate expected utility from inter-temporal wealth, in the absence of a consump-tion stream. The consumption literature begins with Merton’s [25] dynamic programmingsolution of the problem in a constant coefficient complete Brownian model, extended to coverissues such as non-negativity constraints on consumption, and bankruptcy, by Karatzas etal [16] using similar methods. The infinite horizon problem for utility from consumptionin a complete Itˆo market was treated via duality methods by Huang and Pag`es [13], whileFoldes [9, 10] characterised optimal consumption plans in semimartingale markets with well-defined “shadow prices” (local martingales that characterise marginal utility of consumptionprocesses). These correspond to some class of deflators in the modern mathematical financeterminology. The set-up of [9, 10] is very much rooted in traditional economic phraseology,so certain contemporary notions such as the underlying no-arbitrage condition, and the com-pleteness or otherwise of the market, are not entirely transparent. Karatzas and ˇZitkovi´c[19] and ˇZitkovi´c [36] treated problems of optimal consumption with an additional randomendowment, in incomplete semimartingale markets on a finite horizon and under the classicalno-arbitrage assumption of the existence of ELMMs, equivalent to NFLVR. The consumptionproblem in an infinite horizon semimartingale incomplete market under NFLVR was treatedby Mostovyi [27], and later by Chau et al [4] under NUPBR, with the recent treatment in [26]establishing the duality without recourse to any arguments involving ELMMs.In contrast to the consumption problems analysed by these papers, the problem studiedhere, of maximising utility from inter-temporal wealth, has received much less attention. As TILITY FROM INTER-TEMPORAL WEALTH 5 indicated earlier, a dual theory for such problems, over a finite horizon and under NFLVR, wasdeveloped by Bouchard and Pham [2]. Aside from the duality developed in [2], wealth-path-dependent utility maximisation problems have arisen in models which consider investment andconsumption with a random horizon, such as Blanchet-Scalliet et al [1] (in complete Brow-nian markets with deterministic parameters), or Vellekoop and Davis [35] (who consider aMerton-type problem of optimal consumption in a Black-Scholes model, but with randomlyterminating income). Federico et al [8] analyse wealth-path-dependent problems from the dy-namic programming and Hamilton-Jacobi-Bellman (HJB) equation viewpoint, using viscositysolution methods to establish regularity of the value functions in Markovian market scenariosdriven by Brownian motions. As well as arising from a random termination date, the problemstudied here can also be viewed in its own right as describing an objective for a long-livedinvestment fund, looking to build, as oppposed to consume, wealth.The rest of the paper is structured as follows. In Section 2 we describe the financial market,introduce various classes of deflators and the primal problem, list some examples which fit intoour set-up, then derive the budget constraint and formulate the dual problem. In Section 3 wegive the main duality theorem (Theorem 3.1), and describe how the result may be re-cast inthe case when κ is absolutely continuous with respect to Lebesgue measure (Remark 3.3). InSection 4 we formulate the primal and dual problems in abstract notation on a finite measurespace with product measure µ := κ × P . We re-cast the optimisation problems over suitablyenlarged primal and dual domains, and present the bipolarity relations between these spaces(Proposition 4.4) as well as the abstract version of the duality theorem (Theorem 4.5). InSection 5 we prove Proposition 4.4. In many respects this is the heart of the paper. We usethe Stricker and Yan [33] optional decomposition results to show that the budget constraintis also a sufficient condition for primal admissibility, then show that the dual domain we haveconstructed is convex and closed, and make comparisons with the approach of Bouchard andPham [2]. In Section 6 we prove the abstract duality theorem in the classical manner ofKramkov and Schachermayer [21, 22], from which the concrete duality theorem follows, andalso prove the novel representation (1.3) of the optimal wealth process (Proposition 6.13). InSection 7 we work out two examples with power and logarithmic utility: a model whose marketprice of risk is a three-dimensional Bessel process (so satisfying NUPBR but not NFLVR) withstochastic volatility and correlation, and a Black-Scholes market.2. Financial market and problem formulation
The financial market.
We have an infinite horizon financial market on a completestochastic basis (Ω , F , F := ( F t ) t ∈ [0 , ∞ ) , P ), with the filtration F satisfying the usual hypothesesof right-continuity and augmentation with P -null sets of F . The market contains d stocks anda cash asset, the latter with strictly positive price process. We shall use the cash asset asnum´eraire, so without loss of generality (as we shall affirm in Remark 2.1) its price is normalisedto unity and we work with discounted quantities throughout. The (discounted) price processesof the stocks are given by a non-negative c`adl`ag vector semimartingale P = ( P , . . . , P d ).The σ -algebra F can contain more information than that generated by the asset prices, socan include, for example, a random time at which investment ceases, as this is one scenariowhere inter-temporal wealth utility maximisation can arise. Bouchard and Pham [2] had asimilar feature in a finite horizon version of our utility maximisation problem under NFLVR.Note that our formalism and results can be transferred with no alteration to the finite horizonsetting, as we re-iterate in Remark 3.2.A financial agent can trade a self-financing portfolio of the stocks and cash. The agent hasinitial capital x >
0, with the trading strategy represented by a d -dimensional predictable P -integrable process H = ( H , . . . , H d ), with H i , i = 1 , . . . , d the process for the number of MICHAEL MONOYIOS shares of the i th stock in the portfolio. The agent’s wealth process X is given by X t := x + ( H · P ) t , t ≥ , x > , where ( H · P ) := (cid:82) · H s d P s denotes the stochastic integral. Let X ( x ) denote the set of non-negative wealth processes with initial wealth x > X ( x ) := { X : X = x + ( H · P ) ≥ , almost surely } , x > . We write
X ≡ X (1) and we have X ( x ) = x X = { xX : X ∈ X } for x >
0. The set X is aconvex (and hence so is X ( x ) , x > y >
0, let S ( y ) denote the set of supermartingale deflators (SMDs), positive c`adl`agprocesses S with S = y such that the deflated wealth SX is a supermartingale for all X ∈ X :(2.1) S ( y ) := { S > , c`adl`ag , S = y : SX is a supermartingale for all X ∈ X } . We write
S ≡ S (1), and we have S ( y ) = y S for y >
0. The set S is clearly convex. Sincethe constant process X ≡ X , each S ∈ S is a supermartingale. The supermartingaledeflators are the processes used as dual variables by Kramkov and Schachermayer [21, 22] intheir treatment of the terminal wealth utility maximisation problem. The dual domain for theforthcoming inter-temporal wealth problem will be based on S ( y ) but will not coincide withthis space, as we shall see shortly.Let Z denote the set of local martingale deflators (LMDs), positive c`adl`ag local martingales Z with unit initial value such that deflated wealth XZ is a local martingale for all X ∈ X :(2.2) Z := { Z > , c`adl`ag , Z = 1 : XZ is a local martingale for all X ∈ X } . Since the local martingale XZ ≥ X ∈ X , it is also a supermartingale and, since X ≡ X , each Z ∈ Z is also a supermartingale, and we have the inclusion(2.3) S ⊇ Z . The set Z is convex, and contains the density processes of equivalent local martingale measures(ELMMs) in situations where such measures would exist. A feature of our approach is thatwe shall not be using any constructions involving ELMMs, even restricted to a finite horizon,as we discuss further below in Section 2.1.1.The standing no-arbitrage assumption we shall make is that the set of supermartingaledeflators is non-empty:(2.4) S ( y ) (cid:54) = ∅ . The condition (2.4) is equivalent to the condition of no unbounded profit with bounded risk(NUPBR) (also referred to as no arbitrage of the first kind, NA ), weaker than the no freelunch with vanishing risk (NFLVR) condition, the latter being equivalent to the existence ofequivalent local martingale measures (ELMMs), as established by Delbaen and Schachermayer[6] for the case of a locally bounded semimartingale stock price process. There are variouscharacterisations of NUPBR, including that the set Z of LMDs is non-empty: see Karatzasand Kardaras [15], Kardaras [20], Takaoka and Schweizer [34] and Chau et al [4], as well asthe recent overview by Kabanov, Kardaras and Song [14].2.1.1. Completion of the stochastic basis and equivalent measures.
As indicated earlier, weshall not use equivalent local martingale measures (ELMMs), even restricted to a finite hori-zon. This is partly for aesthetic reasons: since we work under NUPBR and assume only theexistence of various classes of deflators, which is the minimal requirement for well posed utilitymaximisation problems, it is natural to seek proofs which use only deflators.There is also a mathematical rationale for avoiding ELMMs. We are working on an infinitehorizon and have have assumed the usual conditions. Thus, each element of the filtration F = ( F t ) t ≥ includes all the P -null sets of F := σ ( (cid:83) t ≥ F t ) =: F ∞ , the tail σ -algebra. So, TILITY FROM INTER-TEMPORAL WEALTH 7 ultimate events (as time t ↑ ∞ ) of P -measure zero are included in any finite time σ -field F T , T < ∞ .It is well-known that in such a scenario many financial models will not admit an equivalentmartingale measure over the infinite horizon, because the candidate change of measure densityis not a uniformly integrable martingale. (This is true of the Black-Scholes model, see Karatzasand Shreve [18, Section 1.7].) One then has to proceed with caution when invoking argumentswhich utilise equivalent measures, by finding a consistent way to eliminate the tail σ -algebrafrom the picture when restricting to a finite horizon T < ∞ .One possible way forward is to not complete the space. This route was taken by Huang andPag`es [13] in an infinite horizon consumption model in a complete Brownian market. Thisis sound, though care is needed to ensure that no results are used which require the usualhypotheses to hold.Another way to proceed, if one wishes to consider equivalent measures restricted to a finitehorizon T < ∞ , is to augment the space with null events of a σ -field generated over a finitehorizon at least as big as T , that is by σ (cid:16)(cid:83) ≤ t ≤ T (cid:48) F t (cid:17) , for some 0 ≤ T ≤ T (cid:48) < ∞ . This canbe done in a consistent way, and relies on an application of Carath´eodory’s extension theorem(Rogers and Williams [31, Theorem II.5.1]). One can then obtain equivalent measures in aninfinite horizon model when restricting such measures to any finite horizon. This procedure iscarried out in a Brownian filtration in Karatzas and Shreve [18, Section 1.7], with a cautionaryexample [18, Example 1.7.6], showing that augmenting the σ -field generated by Brownianmotion over any finite horizon with null sets of the corresponding tail σ -algebra would renderinvalid the construction of equivalent measures, even over a finite horizon.The message is that one has to be careful in using any constructions involving equivalentmeasures, even restricted to a finite horizon, when working in an infinite horizon financialmodel.We avoid any such pitfalls, since we avoid ELMMs entirely. In particular, in Section 5we establish bipolarity results between the primal and dual domains using only the Strickerand Yan [33] version of the optional decomposition theorem, relying on deflators rather thanmartingale measures.We mention this issue because many papers appear to use a complete stochastic basis onan infinite horizon, and at the same time then use equivalent measures over a finite or infinitehorizon, without any statement about the elimination of the tail σ -field. This applies to someproofs in papers tackling the infinite horizon consumption problem (see Mostovyi [27, Lemma4.2] and Chau et al [4, Lemma 1]). In a similar vein, some celebrated papers working on aninfinite horizon, such as the seminal connection between ELMMs and NFLVR of Delbaen andSchachermayer [6], and the optional decomposition result of Kramkov [23], invoke ELMMsover an infinite horizon, without seeming to address the issue that these will not exist over aperpetual timeframe in even the simplest Brownian model such as the Black-Scholes model,and that care must sometimes be taken to eliminate the tail σ -algebra if invoking ELMMs(even restricted to a finite horizon) in an infinite horizon model.We would suggest that it was taken as implicit in the papers cited above that, when neces-sary, the tail σ -algebra was eliminated in a consistent way when invoking arguments involvingELMMs. But it should be said that no such qualifying statements were made. We conjecturethat all the arguments in these and other papers where such potential inconsistencies mayarise can be rendered sound by amendments as described above. This is an issue for possiblefuture investigation, though fortunately not one we need to address, as we bypass all theseproblems by arguments which avoid the use of ELMMs entirely.2.2. The primal problem.
Let U : [0 , ∞ ) → R denote the agent’s utility function, assumedto be strictly increasing, strictly concave, continuously differentiable and satisfying the Inada MICHAEL MONOYIOS conditions(2.5) U (cid:48) (0) := lim x ↓ U (cid:48) ( x ) = + ∞ , U (cid:48) ( ∞ ) := lim x →∞ U (cid:48) ( x ) = 0 . Let κ : [0 , ∞ ) → R + be a non-negative, non-decreasing c`adl`ag adapted process, which willact as a finite measure that will discount utility from inter-temporal wealth. We assume that κ satisfies κ = 0 , P [ κ ∞ > > , κ ∞ ≤ K, for some finite constant K , so that E (cid:2)(cid:82) ∞ d κ t (cid:3) is bounded.The agent’s primal problem is to maximise expected utility from inter-temporal wealth overthe infinite horizon. The primal value function u ( · ) is defined by(2.6) u ( x ) := sup X ∈X ( x ) E (cid:20)(cid:90) ∞ U ( X t ) d κ t (cid:21) , x > . To exclude a trivial problem, we shall assume throughout that the primal value functionsatisfies u ( x ) > −∞ , ∀ x > . This is a mild condition, which can be guaranteed by assuming that for all wealth processes X ∈ X ( x ) we have E (cid:2)(cid:82) ∞ min(0 , U ( X t )) d κ t (cid:3) > −∞ . Remark . There is no loss of generality in working with discountedquantities (so in effect a zero interest rate). To see this, suppose instead that we have apositive interest rate process r = ( r t ) t ≥ , so the cash asset with initial value 1 has positiveprice process A t = e (cid:82) t r s d s , t ≥
0. If (cid:101) X is the un-discounted wealth process, then the problemin (2.6) is E (cid:104)(cid:82) ∞ U (cid:16) (cid:101) X t /A t (cid:17) d κ t (cid:105) → max! We can define another utility function (cid:101) U : R → R such that (cid:101) U ( A t , (cid:101) X t ) = U ( (cid:101) X t /A t ) , t ≥
0, and the problem in (2.6) can then be transportedto one in terms of the raw (un-discounted) wealth process. For example, if U ( · ) = log( · ) islogarithmic utility, we choose (cid:101) U ( A, (cid:101) X ) = log( (cid:101) X ) − log( A ). If U ( x ) = x p /p, p < , p (cid:54) = 0 ispower utility, then we choose (cid:101) U ( A, (cid:101) X ) = A − p (cid:101) X p /p . Remark . In the problem (2.6) we can allow U ( · ) to be stochastic, so toalso depend on ω ∈ Ω in an optional way. The analysis is unaffected, as the reader can easilyverify, so one can read the proofs with a stochastic utility in mind and with dependence on ω ∈ Ω suppressed throughout.2.2.1.
Some examples.
We list here some examples of inter-temporal wealth utility maximi-sation problems, illustrating how the measure κ manifests itself in various cases. Furtherexamples can be found in Bouchard and Pham [2, Section 2]. Example . Take d κ t = e − αt d t forsome positive discount rate α >
0, and an infinite horizon, so the objective is(2.7) E (cid:20)(cid:90) ∞ exp ( − αt ) U ( X t ) d t (cid:21) → max!This is the quintessential example we have in mind as our central problem, and can be thoughtof as an objective of a long-lived investment fund building up wealth. We shall treat thisexample under power and logarithmic utility in Section 7, to illustrate the application of theduality theorem of the paper, with different market environments: an incomplete market witha stock with whose market price of risk is a three-dimensional Bessel process (so will satisfyNUPBR but not NFLVR) and which has a stochastic volatility, and a Black-Scholes (thus,complete) market.There are no esoteric ingredients in (2.7) such as a random termination time which generatesthe wealth-path-dependent objective, but such modifications can be added. Indeed, suppose TILITY FROM INTER-TEMPORAL WEALTH 9 we have a random horizon given by T ∼ Exp( λ ), an exponentially distributed time withparameter λ >
0, independent of the stock price filtration. The objective can be re-cast withan additional integral over the probability density function of T , so we have E (cid:20)(cid:90) T exp ( − αt ) U ( X t ) d t (cid:21) = E (cid:20)(cid:90) ∞ λ exp( − λs ) (cid:90) s exp( − αt ) U ( X t ) d t d s (cid:21) . Integration by parts allows the objective to be re-written as E (cid:20)(cid:90) ∞ exp ( − ( α + λ ) t ) U ( X t ) d t (cid:21) → max!so we recover a problem of the same type as in (2.7) with a modified discount factor. Example . The other classical examplewhich yields an inter-temporal wealth objective is where we maximise expected utility ofterminal wealth E [ U ( X T )] at some random horizon T , an almost surely finite F -measurablenon-negative random variable.For instance, let T ∼ Exp( λ ) be an exponentially distributed random time with parameter λ >
0, independent of the stock price filtration. As in Example 2.3, we re-write the objectivewith an integral over the probability density function of T , so we have E [ U ( X T )] = E (cid:20)(cid:90) ∞ λ exp( − λt ) U ( X t ) d t (cid:21) , which again yields a problem of the type in Example 2.3. We observe that κ is given by κ t = (cid:90) t λ exp( − λs ) d s = 1 − exp( − λt ) = P [ T ≤ t ] , t ≥ . The obvious generalisation is to a general random time T which is independent of the assetprice filtration. In this case one has κ t = P [ T ≤ t ] , t ≥ T is a stopping time we have κ t = { t ≥ T } , t ≥
0, and this includes thecase where T is deterministic, so there is no time horizon uncertainty, and we revert to theclassical terminal wealth problem.Further similar examples are given in Bouchard and Pham [2, Examples 1–3], adapted tothe case of a finite horizon for the overall problem in (2.6).2.3. The budget constraint.
Our approach to establishing the form of the dual to the primalutility maximisation problem (2.6) is to determine an appropriate supermartingale constraintsatisfied by the pair ( X t , ( X s ) ≤ s ≤ t ) t ≥ , that is, the value of an admissible wealth process atany time as well as the wealth path up to that point. This gives an infinite horizon budgetconstraint on the wealth path. Using a supermartingale constraint in this way is analogous tothe procedure followed in consumption problems, where one considers the wealth process atany time as well the consumption plan up to that time.Let B denote the set of all non-negative c`adl`ag adapted processes β satisfying (cid:82) t β s d κ s < ∞ almost surely for all t ≥ B := (cid:8) β ≥ (cid:82) · β s d κ s < ∞ almost surely (cid:9) . The processes in B will act as an additional dual control, above and beyond that implied inthe classical supermartingale deflators, as we shall see in due course.For any β ∈ B and for any supermartingale deflator S ∈ S ( y ), define a process R by(2.9) R t := exp (cid:18) − (cid:90) t β s d κ s (cid:19) S t , t ≥ , β ∈ B , S ∈ S ( y ) , y > . Denote the set of such processes with initial value y > R ( y ):(2.10) R ( y ) := { R : R is defined by (2.9) } , y > . We write
R ≡ R (1) and we have R ( y ) = y R for y >
0. We shall prove in Section 5.2 that theset R is convex (see Lemma 5.5), which will lead to the corresponding property for the dualdomain to the primal problem (2.6), to be defined shortly.Since β ∈ B is almost surely non-negative, the supermartingale property of the deflatedwealth SX in (2.1) also holds for RX , for any R ∈ R ( y ), so we have the inclusion S ( y ) ⊇R ( y ) , y >
0, and each R ∈ R ( y ) is also a supermartingale.For each R ∈ R ( y ) and for the same β ∈ B appearing in the definition (2.9), define a process Y by(2.11) Y t := β t R t = β t exp (cid:18) − (cid:90) t β s d κ s (cid:19) S t , t ≥ , β ∈ B , S ∈ S ( y ) , y > . Denote the set of such processes by Y ( y ):(2.12) Y ( y ) := { Y : Y is defined by (2.11) } , y > . The set Y ( y ) will form the domain of the dual problem to the inter-temporal wealth problem(2.6), as we shall see shortly. We shall refer to processes Y ∈ Y ( y ) as wealth-path deflators or inter-temporal wealth deflators or, simply, as deflators, when no confusion can arise. Wewrite Y ≡ Y (1), with Y ( y ) = y Y for y >
0. The set Y turns out to be convex, as we shallshow in Section 5.2. This is an important ingredient in our approach to establishing certainbipolarity relations between the primal and dual domains, which underlie the duality resultsof the paper. As we shall see, the convexity of Y will stem from the particular structure of thedual variables as given in (2.11). This structure seems to have eluded some previous studiesof inter-temporal wealth utility maximisation problems. We shall say more on this structureand make comparisons with the approach of Bouchard and Pham [2] in Section 5.3, after weprove the bipolarity relations.The following lemma gives the supermartingale constraint and the resultant infinite horizonbudget constraint on admissible wealth processes, which will lead to the form of the dualproblem. Lemma 2.5 (Supermartingale and budget constraints) . Let β ∈ B be any non-negative c`adl`agadapted process satisfying (cid:82) t β s d κ s < ∞ almost surely for all t ≥ . Define the processes R ∈ R ( y ) and the wealth-path deflators Y ∈ Y ( y ) by (2.9) and (2.11) , respectively. We thenhave that (2.13) M := RX + (cid:90) · X s Y s d κ s is a supermartingale . As a consequence, we have the infinite horizon budget constraint(2.14) E (cid:20)(cid:90) ∞ X t Y t d κ t (cid:21) ≤ xy, x, y > , ∀ X ∈ X ( x ) , Y ∈ Y ( y ) . Proof.
For x, y > X ∈ X ( x ) be an admissible wealth process and let S ∈ S ( y ) be anysupermartingale deflator. The Itˆo product rule applied to XR = XS exp (cid:0) − (cid:82) · β s d κ s (cid:1) gives(2.15) M t := X t R t + (cid:90) t X s Y s d κ s = xy + (cid:90) t exp (cid:18) − (cid:90) s β u d κ u (cid:19) d( X s S s ) , t ≥ , where we have used the definition (2.11) of the inter-temporal wealth deflators. Since XS is a supermartingale, it has a Doob-Meyer decomposition XS = xy + L − A for some localmartingale L and a non-decreasing process A , with L = A = 0. Using this Doob-Meyer de-composition, the integral on the right-hand-side of (2.15) is also seen to be a supermartingale,so we obtain the supermartingale property of M := XR + (cid:82) · X s Y s d κ s as stated in the lemma. TILITY FROM INTER-TEMPORAL WEALTH 11
The supermartingale property gives E (cid:20) X t R t + (cid:90) t X s Y s d κ s (cid:21) ≤ xy, t ≥ . Since XR is non-negative, we thus also have E (cid:20)(cid:90) t X s Y s d κ s (cid:21) ≤ xy, t ≥ . Letting t ↑ ∞ and using monotone convergence we obtain the infinite horizon budget constraint(2.14). (cid:3) Note that for β ≡ XS is a supermartingale for all X ∈ X and S ∈ S . This is the basic sense in which weare extending the starting point of the methodology of Kramkov and Schachermayer [21, 22]towards duality: begin with a supermartingale constraint to build a budget constraint. Thepresence of the supermartingales S ∈ S , R ∈ R in these arguments will ultimately be exploitedto invoke supermartingale convergence results involving Fatou convergence of processes, inproving that an abstract dual domain D (an enlargement of the domain Y to encompassprocesses dominated by some Y ∈ Y ) is closed (see Lemmata 5.6 and 5.9).2.4. The dual problem.
Let V : R + → R denote the convex conjugate of the utility function,defined by V ( y ) := sup x> [ U ( x ) − xy ] , y > . The map y (cid:55)→ V ( y ) , y >
0, is strictly convex, strictly decreasing, continuously differentiableon R + , − V ( · ) satisfies the Inada conditions, and we have the bi-dual relation U ( x ) := inf y> [ V ( y ) + xy ] , x > , as well as V (cid:48) ( · ) = − I ( · ) = − ( U (cid:48) ) − ( · ), where I ( · ) denotes the inverse of marginal utility. Inparticular, we have the inequality(2.16) V ( y ) ≥ U ( x ) − xy, ∀ x, y > , with equality iff U (cid:48) ( x ) = y. From the budget constraint (2.14) we can motivate the form of the dual problem to (2.6)by bounding the achievable utility in the familiar way. For any X ∈ X ( x ) and Y ∈ Y ( y ) wehave E (cid:20)(cid:90) ∞ U ( X t ) d κ t (cid:21) ≤ E (cid:20)(cid:90) ∞ U ( X t ) d κ t (cid:21) + xy − E (cid:20)(cid:90) ∞ X t Y t d κ t (cid:21) = E (cid:20)(cid:90) ∞ ( U ( X t ) − X t Y t ) d κ t (cid:21) + xy ≤ E (cid:20)(cid:90) ∞ V ( Y t ) d κ t (cid:21) + xy, x, y > , (2.17)the last inequality a consequence of (2.16). This motivates the definition of the dual problemassociated with the primal problem (2.6), with dual value function v : R + → R defined by(2.18) v ( y ) := inf Y ∈Y ( y ) E (cid:20)(cid:90) ∞ V ( Y t ) d κ t (cid:21) , y > . We shall assume that the dual problem is finitely valued:(2.19) v ( y ) < ∞ , for all y > . Remark . As is known from Kramkov and Schacher-mayer [22], (2.19) is a mild condition that will guarantee a well-posed primal problem. It isalso well known that one can alternatively impose the reasonable asymptotic elasticity condi-tion of Kramkov and Schachermayer [21] on the utility function:(2.20) AE( U ) := lim sup x →∞ xU (cid:48) ( x ) U ( x ) < , along with the assumption that u ( x ) < ∞ for some x >
0. Then, as in Kramkov andSchachermayer [22, Note 2] or Bouchard and Pham [2, Remark 5.1], these conditions can beshown to yield (2.19). 3.
The main duality
Here is the central duality statement of the paper.
Theorem 3.1 (Perpetual inter-temporal wealth duality under NUPBR) . Define the primalinter-temporal wealth utility maximisation problem by (2.6) and the corresponding dual problemby (2.18) . Assume (2.4) , (2.5) and that u ( x ) > −∞ , ∀ x > , v ( y ) < ∞ , ∀ y > . Then: (i) u ( · ) and v ( · ) are conjugate: v ( y ) = sup x> [ u ( x ) − xy ] , u ( x ) = inf y> [ v ( y ) + xy ] , x, y > . (ii) The primal and dual optimisers (cid:98) X ( x ) ∈ X ( x ) and (cid:98) Y ( y ) ∈ Y ( y ) exist and are unique,so that u ( x ) = E (cid:20)(cid:90) ∞ U ( (cid:98) X t ( x )) d κ t (cid:21) , v ( y ) = E (cid:20)(cid:90) ∞ V ( (cid:98) Y t ( y )) d κ t (cid:21) , x, y > , with (cid:98) Y ( y ) = (cid:98) β (cid:98) R ( y ) = (cid:98) β exp (cid:16) − (cid:82) · (cid:98) β s d κ s (cid:17) (cid:98) S ( y ) , for an optimal (cid:98) β ∈ B and optimalsupermartingales (cid:98) R ( y ) ∈ R ( y ) and (cid:98) S ( y ) ∈ S ( y ) . (iii) With y = u (cid:48) ( x ) (equivalently, x = − v (cid:48) ( y ) ), the primal and dual optimisers are relatedby U (cid:48) ( (cid:98) X t ( x )) = (cid:98) Y t ( y ) , equivalently , (cid:98) X t ( x ) = − V (cid:48) ( (cid:98) Y t ( y )) , t ≥ , and satisfy (3.1) E (cid:20)(cid:90) ∞ (cid:98) X t ( x ) (cid:98) Y t ( y ) d κ t (cid:21) = xy. Moreover, the associated optimal wealth process (cid:98) X ( x ) satisfies (3.2) (cid:98) X t ( x ) (cid:98) R t ( y ) = E (cid:20) (cid:90) ∞ t (cid:98) X s ( x ) (cid:98) Y s ( y ) d κ s (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , t ≥ , and the process (cid:99) M := (cid:98) X ( x ) (cid:98) R ( y ) + (cid:82) · (cid:98) X s ( x ) (cid:98) Y s ( y ) d κ s is a uniformly integrable martin-gale. (iv) The functions u ( · ) and − v ( · ) are strictly increasing, strictly concave, satisfy the Inadaconditions, and for all x, y > their derivatives satisfy xu (cid:48) ( x ) = E (cid:20)(cid:90) ∞ U (cid:48) ( (cid:98) X t ( x )) (cid:98) X t ( x ) d κ t (cid:21) , yv (cid:48) ( y ) = E (cid:20)(cid:90) ∞ V (cid:48) ( (cid:98) Y t ( y )) (cid:98) Y t ( y ) d κ t (cid:21) . TILITY FROM INTER-TEMPORAL WEALTH 13
Remark . As the analysis in the sequel will show, it is easy toverify that all our methodology works without alteration for the finite horizon version of (2.6),with some terminal time
T < ∞ . The budget constraint is altered to have an upper limit of T as are all the results of Theorem 3.1. We thus extend the problem studied in Bouchardand Pham [2] to the NUPBR scenario, in addition to the strengthening of the basic dualitystatement as described below, where we do not have to assume a priori that the dual domainis the polar of the primal domain.The proof of Theorem 3.1 will be given in Section 6, and will rely on bipolarity results andan abstract version of the duality theorem in Section 4, with the bipolarity results proven inSection 5. Duality results akin to items (i)–(iii) of the theorem (but not the additional novelcharacterisation (3.2) of the optimal wealth process) were obtained by Bouchard and Pham[2] over a finite horizon and under NFLVR. Compared to [2], Theorem 3.1 makes a strongerstatement in other ways. We describe this strengthening briefly here, and will give furtherdetails in Section 5.3 after we prove bipolarity relations between the primal and dual domains,as some of the features are directly concerned with such polarity results.First, we strengthen the duality for inter-temporal wealth utility maximisation to the weakerno-arbitrage assumption of NUPBR, compared to the NFLVR assumption in [2]. Second, weavoid having to define the dual domain as the polar of the primal domain. As indicated inthe Introduction, the dual domain in [2] was directly defined as the set of deflators for whicha finite horizon version of the budget constraint holds. In the language of the polar of aset (defined in Section 4, see Definition 4.1) the dual space is set equal to the polar of theprimal space, by definition. This automatically renders the dual domain convex and closed,but the statement of the duality result is then somewhat weaker, because one half of theperfect bipolarity between the primal and dual domains (as given in Proposition 4.4) has beenachieved by definition.In our approach, the dual space arises from the budget constraint (2.14), itself derived fromthe supermartingale property (2.13) of the process M . This renders the budget constraint anecessary condition for admissibility. On enlarging the primal domain to include processesdominated by some admissible wealth, we show in Lemma 5.2 that the budget constraint isalso a sufficient condition for admissibility. This uses the Stricker and Yan [33] version of theoptional decomposition theorem, avoiding martingale measures in favour of local martingaledeflators. This equivalence between primal admissibility and the budget constraint establishesthat the enlarged primal set C is the polar of the dual space Y .We then show that our dual space is convex, relying on the particular structure of thewealth path deflators in (2.11). An enlargement of the dual domain (in a similar vein tothe primal enlargement), combined with supermartingale convergence results which exploitFatou convergence of processes, culminates in Lemma 5.6, which shows that the enlarged dualdomain D is closed (in an appropriate topology). This, along with convexity and solidity,yields that the enlarged dual domain D is the bipolar of the original domain Y . Thus gives usthe perfect bipolarity we need between C and D .The above procedure is in essence the Kramkov and Schachermayer [21, 22] program forbipolarity and duality, adapted to an inter-temporal wealth framework. We shall describethese features of the bipolarity derivations in more detail in Section 5.3, and compare theprogram to that of Bouchard and Pham [2], after we have proven the bipolarity relations. Remark κ (cid:28) Leb) . Theorem 3.1 holds true regardless of whether themeasure κ admits a density with respect to Lebesgue measure. However, when κ (cid:28) Leb thereis a natural change of variable which one would use in computations, as we shall see in thecourse of some examples in Section 7, so we highlight here how the Theorem 3.1 is slightlyre-cast in that case. The scenario to keep in mind is the case where d κ t = e − αt d t for a positiveimpatience rate α > In the definition (2.8) of the set B , one replaces κ by Lebesgue measure. With an abuse ofnotation, to use the same symbol for this set of auxiliary dual controls, B now denotes the setof non-negative c`adl`ag processes β such that (cid:82) · β s d s < ∞ almost surely. With similar abuseof notation, the set R ( y ) is composed of processes R := exp (cid:0) − (cid:82) · β s d s (cid:1) S , for supermartingaledeflators S ∈ S ( y ). The wealth-path deflators are then given by Y := βR , and once againwe denote the set of such processes by Y ( y ). The supermartingale property (2.13) convertsto the statement that the process M := XR + (cid:82) · X s Y s d s is a supermartingale. The budgetconstraint (2.14) becomes E (cid:2)(cid:82) ∞ X t Y t d t (cid:3) ≤ xy .With this notation, define the positive process γ = ( γ t ) t ≥ as the reciprocal of ( d κ t / d t ) t ≥ : γ t := (cid:18) d κ t d t (cid:19) − , t ≥ . The dual problem then takes the form(3.3) v ( y ) := inf Y ∈Y ( y ) E (cid:20)(cid:90) ∞ V ( γ t Y t ) d κ t (cid:21) , y > , as can be confirmed by repeating the computation that led to (2.17) in this altered set-up.With these changes, items (ii)–(iv) of Theorem 3.1 are altered to:(ii) (cid:48) The primal and dual optimisers (cid:98) X ( x ) ∈ X ( x ) and (cid:98) Y ( y ) ∈ Y ( y ) exist and are unique,so that u ( x ) = E (cid:20)(cid:90) ∞ U ( (cid:98) X t ( x )) d κ t (cid:21) , v ( y ) = E (cid:20)(cid:90) ∞ V ( γ t (cid:98) Y t ( y )) d κ t (cid:21) , x, y > , with (cid:98) Y ( y ) = (cid:98) β (cid:98) R ( y ) = (cid:98) β exp (cid:16) − (cid:82) · (cid:98) β s d s (cid:17) (cid:98) S ( y ), for an optimal (cid:98) β ∈ B and optimalsupermartingales (cid:98) R ( y ) ∈ R ( y ) and (cid:98) S ( y ) ∈ S ( y ).(iii) (cid:48) With y = u (cid:48) ( x ) (equivalently, x = − v (cid:48) ( y )), the primal and dual optimisers are relatedby(3.4) U (cid:48) ( (cid:98) X t ( x )) = γ t (cid:98) Y t ( y ) , equivalently , (cid:98) X t ( x ) = − V (cid:48) ( γ t (cid:98) Y t ( y )) , t ≥ , and satisfy(3.5) E (cid:20)(cid:90) ∞ (cid:98) X t ( x ) (cid:98) Y t ( y ) d t (cid:21) = xy. Moreover, the associated optimal wealth process (cid:98) X ( x ) satisfies(3.6) (cid:98) X t ( x ) (cid:98) R t ( y ) = E (cid:20) (cid:90) ∞ t (cid:98) X s ( x ) (cid:98) Y s ( y ) d s (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , t ≥ , and the process (cid:99) M := (cid:98) X ( x ) (cid:98) R ( y ) + (cid:82) · (cid:98) X s ( x ) (cid:98) Y s ( y ) d s is a uniformly integrable martin-gale.(iv) (cid:48) The functions u ( · ) and − v ( · ) are strictly increasing, strictly concave, satisfy the Inadaconditions, and for all x, y > xu (cid:48) ( x ) = E (cid:20)(cid:90) ∞ U (cid:48) ( (cid:98) X t ( x )) (cid:98) X t ( x ) d κ t (cid:21) , yv (cid:48) ( y ) = E (cid:20)(cid:90) ∞ V (cid:48) ( γ t (cid:98) Y t ( y )) (cid:98) Y t ( y ) d t (cid:21) . Abstract bipolarity and duality
In this section we specify a finite measure space which allows us to write the primal anddual problems in abstract notation, over suitably enlarged primal and dual domains. We thenstate the bipolarity relations between the abstract primal and dual domains in Proposition4.4, which forms the basis for the subsequent abstract duality of Theorem 4.5.
TILITY FROM INTER-TEMPORAL WEALTH 15
Set Ω := [0 , ∞ ) × Ω . Let G denote the optional σ -algebra on Ω , that is, the sub- σ -algebra of B ([0 , ∞ )) ⊗F generatedby evanescent sets and stochastic intervals of the form (cid:74) T, ∞ (cid:74) for arbitrary stopping times T .Define the measure(4.1) µ := κ × P on ( Ω , G ). On the resulting finite measure space ( Ω , G , µ ), denote by L ( µ ) the space of non-negative µ -measurable functions, corresponding to non-negative infinite horizon processes.The primal and dual domains for our optimisation problems (2.6) and (2.18) are now con-sidered as subsets of L ( µ ). The abstract primal and dual domains will be enlargements of X ( x ) and Y ( y ) to accommodate processes dominated by some element of the original domainin question.The abstract primal domain is C ( x ), defined by(4.2) C ( x ) := { g ∈ L ( µ ) : g ≤ X, µ -a.e., for some X ∈ X ( x ) } , x > . We write
C ≡ C (1), with C ( x ) = x C for x >
0, and the set C is convex. Since U ( · ) is increasing,the primal value function of (2.6) is now written in the abstract notation as an optimisationover g ∈ C ( x ):(4.3) u ( x ) := sup g ∈C ( x ) (cid:90) Ω U ( g ) d µ, x > . The abstract dual domain is obtained by a similar enlargement of the original dual domain.Define the set D ( y ) by(4.4) D ( y ) := { h ∈ L ( µ ) : h ≤ Y, µ -a.e., for some Y ∈ Y ( y ) } , y > . We write
D ≡ D (1), we have D ( y ) = y D for y >
0, and the set D is convex, inheriting thisproperty from Y . This is a crucial feature, and relies on our demonstration of the convexity of Y in Section 5.2 (see Lemma 5.5), which in turn relies on the inter-temporal wealth deflatorshaving the particular structure in (2.11).With this notation, and since V ( · ) is decreasing, the dual problem (2.18) takes the form(4.5) v ( y ) := inf h ∈D ( y ) (cid:90) Ω V ( h ) d µ, y > . Abstract bipolarity.
The abstract duality theorem relies on the abstract bipolarityresult in Proposition 4.4 below which connects the sets C and D . The result is of course inthe spirit of Kramkov and Schachermayer [21, Proposition 3.1].We shall sometimes employ the notation(4.6) (cid:104) g, h (cid:105) := (cid:90) Ω gh d µ, g, h ∈ L ( µ ) . Let us recall the concepts of set solidity and the polar of a set.
Definition 4.1 (Solid set, closed set) . A subset A ⊆ L ( µ ) is called solid if f ∈ A and0 ≤ g ≤ f, µ -a.e. implies that g ∈ A .A set is closed in µ -measure , or simply closed , if it is closed with respect to the topology ofconvergence in measure µ . Definition 4.2 (Polar of a set) . The polar , A ◦ , of a set A ⊆ L ( µ ), is defined by A ◦ := (cid:8) h ∈ L ( µ ) : (cid:104) g, h (cid:105) ≤ , for each g ∈ A (cid:9) . For clarity and for later use, we state here the bipolar theorem of Brannath and Schacher-mayer [3, Theorem 1.3], originally proven in a probability space, and adapted here to themeasure space ( Ω , G , µ ). Theorem 4.3 (Bipolar theorem, Brannath and Schachermayer [3], Theorem 1.3) . On thefinite measure space ( Ω , G , µ ) : (i) For a set A ⊆ L ( µ ) , its polar A ◦ is a closed, convex, solid subset of L ( µ ) . (ii) The bipolar A ◦◦ , defined by A ◦◦ := (cid:8) g ∈ L ( µ ) : (cid:104) g, h (cid:105) ≤ , for each h ∈ A ◦ (cid:9) , is the smallest closed, convex, solid set in L ( µ ) containing A . Proposition 4.4 (Abstract bipolarity) . Under the condition (2.4) , the abstract primal anddual sets C and D satisfy the following properties: (i) C and D are both closed with respect to convergence in measure µ , convex and solid; (ii) C and D satisfy the bipolarity relations g ∈ C ⇐⇒ (cid:104) g, h (cid:105) ≤ , ∀ h ∈ D , that is, C = D ◦ , (4.7) h ∈ D ⇐⇒ (cid:104) g, h (cid:105) ≤ , ∀ g ∈ C , that is, D = C ◦ ;(4.8)(iii) C and D are bounded in L ( µ ) , and D is also bounded in L ( µ ) . The proof of Proposition 4.4 will be given in Section 5, where we shall establish thatthe infinite horizon budget constraint is also a sufficient condition for admissibility, once theprimal domain is enlarged to accommodate processes dominated by admissible wealths. Thisculminates in the full bipolarity relations once we enlarge dual domain in a similar manner.The derivations in Section 5 are quite distinct from previous approaches, and are the bedrock ofthe mathematical results. As indicated earlier, we shall establish the bipolarity results withoutany recourse whatsoever to constructions involving ELMMs, by exploiting ramifications of theStricker and Yan [33] version of the optional decomposition theorem.4.2.
The abstract duality.
Armed with the abstract bipolarity in Proposition 4.4, we havethe following abstract version of the convex duality relations between the primal problem(4.3) and its dual (4.5). The theorem shows that all the natural tenets of utility maximisationtheory, as established by Kramkov and Schachermayer [21, 22] in the terminal wealth problemunder NFLVR, extend to the infinite horizon inter-temporal wealth problem under NUPBR,with weak underlying assumptions on the primal and dual domains.
Theorem 4.5 (Abstract duality theorem) . Define the primal value function u ( · ) by (4.3) andthe dual value function by (4.5) . Assume that the utility function satisfies the Inada conditions (2.5) and that (4.9) u ( x ) > −∞ , ∀ x > , v ( y ) < ∞ , ∀ y > . Then, with Proposition 4.4 in place, we have: (i) u ( · ) and v ( · ) are conjugate: (4.10) v ( y ) = sup x> [ u ( x ) − xy ] , u ( x ) = inf y> [ v ( y ) + xy ] , x, y > . (ii) The primal and dual optimisers (cid:98) g ( x ) ∈ C ( x ) and (cid:98) h ( y ) ∈ D ( y ) exist and are unique, sothat u ( x ) = (cid:90) Ω U ( (cid:98) g ( x )) d µ, v ( y ) = (cid:90) Ω V ( (cid:98) h ( y )) d µ, x, y > . TILITY FROM INTER-TEMPORAL WEALTH 17 (iii)
With y = u (cid:48) ( x ) (equivalently, x = − v (cid:48) ( y ) ), the primal and dual optimisers are relatedby U (cid:48) ( (cid:98) g ( x )) = (cid:98) h ( y ) , equivalently , (cid:98) g ( x ) = − V (cid:48) ( (cid:98) h ( y )) , and satisfy (cid:104) (cid:98) g ( x ) , (cid:98) h ( y ) (cid:105) = xy. (iv) u ( · ) and − v ( · ) are strictly increasing, strictly concave, satisfy the Inada conditions,and their derivatives satisfy xu (cid:48) ( x ) = (cid:90) Ω U (cid:48) ( (cid:98) g ( x )) (cid:98) g ( x ) d µ, yv (cid:48) ( y ) = (cid:90) Ω V (cid:48) ( (cid:98) h ( y )) (cid:98) h ( y ) d µ, x, y > . The proof of Theorem 4.5 will be given in Section 6, and uses as its starting point thebipolarity result in Proposition 4.4.The duality proof itself follows some of the classical steps (with adaptations) of Kramkovand Schachermayer [21, 22]. For completeness and clarity we shall give a full, self-containedtreatment. 5.
Bipolarity relations
In this section we prove Proposition 4.4, which establishes in particular the bipolarityrelations (4.7) and (4.8) between the enlarged primal and dual domains C and D in (4.2) and(4.4).5.1. Sufficiency of the budget constraint.
The budget constraint (2.14), as derived inLemma 2.5, constitutes a necessary condition for admissible inter-temporal wealth processes.Setting x = y = 1 in (2.14), we thus have the implications(5.1) X ∈ X = ⇒ E (cid:20)(cid:90) ∞ X t Y t d κ t (cid:21) ≤ , ∀ Y ∈ Y , and(5.2) Y ∈ Y = ⇒ E (cid:20)(cid:90) ∞ X t Y t d κ t (cid:21) ≤ , ∀ X ∈ X . We wish to establish the reverse implications in some form, if need be by enlarging the primaland dual domains.Recall the enlarged primal domain
C ≡ C (1) in (4.2) of processes dominated by admissiblewealths with initial capital 1. The budget constraint (2.14) clearly holds with g ∈ C in placeof X ∈ X , so the implication (5.1) extends from X to C :(5.3) g ∈ C = ⇒ E (cid:20)(cid:90) ∞ g t Y t d κ t (cid:21) ≤ , ∀ Y ∈ Y . We establish the reverse implication to (5.3) in Lemma 5.2 below. This requires some ver-sion of the Optional Decomposition Theorem (ODT), originally formulated by El Karoui andQuenez [7] in a Brownian setting. This was generalised to markets with locally boundedsemimartingale stock prices by Kramkov [23], extended to the non-locally bounded case byF¨ollmer and Kabanov [11], and to models with constraints by F¨ollmer and Kramkov [12]. Therelevant version of the ODT for us is the one due to Stricker and Yan [33], which uses localmartingale deflators, rather then ELMMs. We shall use a result from [33] which applies tothe super-hedging of American claims, so is designed to construct a process which can super-replicate a payoff at an arbitrary time. The salient observation is that this result can also beused to dominate a process over all times, and this is how we shall employ it.For clarity we state here the ODT results we need, and specify afterwards precisely whichresults from [33] we have taken.
For t ≥
0, let T ( t ) denote the set of F -stopping times with values in [ t, ∞ ). For t = 0, write T ≡ T (0), and recall the set Z of local martingale deflators in (2.2). Theorem 5.1 (Stricker and Yan [33] ODT) . (i) Let W be an adapted non-negative pro-cess. The process ZW is a supermartingale for each Z ∈ Z if and only if W admits adecomposition of the form W = W + ( φ · P ) − A, where φ is a predictable P -integrable process such that Z ( φ · P ) is a local martingale foreach Z ∈ Z , A is an adapted increasing process with A = 0 , and for all Z ∈ Z and T ∈ T , E [ Z T A T ] < ∞ . In this case, moreover, we have sup Z ∈Z ,T ∈T E [ Z T A T ] ≤ W . (ii) Let b = ( b t ) t ≥ be a non-negative c`adl`ag process such that sup Z ∈Z ,T ∈T E [ Z T b T ] < ∞ .Then there exists an adapted c`adl`ag process W that dominates b : W t ≥ b t almostsurely for all t ≥ , ZW is a supermartingale for each Z ∈ Z , and the smallest suchprocess W is given by (5.4) W t = ess sup Z ∈Z ,T ∈T ( t ) Z t E [ Z T b T |F t ] , t ≥ . Part (i) of Theorem 5.1 is taken from [33, Theorem 2.1]. Part (ii) is a combination of [33,Lemma 2.4 and Remark 2].The following lemma establishes the reverse implication to (5.3).
Lemma 5.2.
Suppose g is a non-negative c`adl`ag process satisfying (5.5) E (cid:20)(cid:90) ∞ g t Y t d κ t (cid:21) ≤ , ∀ Y ∈ Y . Then, g ∈ C .Proof. Since g is assumed to satisfy (5.5) for all Y ∈ Y and because we have the inclusion(2.3), we see that (5.5) is satisfied for Y = β exp (cid:0) − (cid:82) · β s d κ s (cid:1) Z , for any non-negative c`adl`ag β ∈ B and for any local martingale deflator Z ∈ Z .Fix a stopping time T ∈ T , and for each n ∈ N choose β according to β t = { T ≤ t Letting n → ∞ , and using Fatou’s lemma and the right-continuity of Zg , we obtain E [ Z T g T ] ≤ , ∀ Z ∈ Z , T ∈ T . Since Z ∈ Z and T ∈ T were arbitrary, we havesup Z ∈Z ,T ∈T E [ Z T g T ] ≤ < ∞ . Thus, from part (ii) of the Stricker-Yan version of optional decomposition, Theorem 5.1,there exists a c`adl`ag process W that dominates g , so W t ≥ g t , a . s ., ∀ t ≥ 0, and ZW is asupermartingale for each Z ∈ Z . From (5.4), the smallest such W given by W t = ess sup Z ∈Z ,T ∈T ( t ) Z t E [ Z T g T |F t ] , t ≥ , so that W ≤ 1. Further, by part (i) of Theorem (5.1), there exists a predictable P -integrableprocess H and an adapted increasing process A , with A = 0, such that W has decomposition W = W + ( H · P ) − A , with Z ( H · P ) a local martingale for each Z ∈ Z , and E [ Z T A T ] < ∞ for all Z ∈ Z and T ∈ T .Since W dominates g , we can define a process X by X t := 1 + ( H · P ) t , t ≥ , which also dominates g , since its initial value is no smaller than W and we have dispensedwith the increasing process A . We observe that X corresponds to the value of a self-financingwealth process with initial capital 1 which dominates g , so that g ∈ C . (cid:3) We can now assemble consequences of the budget constraint and of Lemma 5.2 which,combined with the bipolar theorem, gives the following polarity properties of the set C . Lemma 5.3 (Polarity properties of C ) . The set C ≡ C (1) of admissible wealth processes withinitial capital x = 1 is a closed, convex and solid subset of L ( µ ) . It is equal to the polar ofthe set Y ≡ Y (1) of (2.12) with respect to measure µ : (5.6) C = Y ◦ , so that (5.7) C ◦ = Y ◦◦ , and C is equal to its bipolar: (5.8) C ◦◦ = C . Proof. Lemma 5.2, combined with the implication in (5.3), gives the equivalence g ∈ C ⇐⇒ E (cid:20)(cid:90) ∞ g t Y t d κ t (cid:21) ≤ , ∀ Y ∈ Y . Equivalently, in terms of the measure µ of (4.1), we have(5.9) g ∈ C ⇐⇒ (cid:90) Ω gY d µ ≤ , ∀ Y ∈ Y . The characterisation (5.9) is the dual representation of C : C = (cid:8) g ∈ L ( µ ) : (cid:104) g, Y (cid:105) ≤ , for each Y ∈ Y (cid:9) . This says that C is the polar of Y , establishing (5.6) and thus (5.7).Part (i) of the bipolar theorem, Theorem 4.3, along with (5.6), imply that C is a closed,convex and solid subset of L ( µ ) (since it is equal to the polar of a set) as claimed. Part (ii)of Theorem 4.3 gives C ◦◦ ⊇ C with C ◦◦ the smallest closed, convex, solid set containing C . Butsince C is itself closed, convex and solid, we have (5.8). (cid:3) Remark . There are other ways to obtain the closed, convex and solid properties of C . First,the equivalence (5.9) along with Fatou’s lemma yields that the set C is closed with respect tothe topology of convergence in measure µ . To see this, let ( g n ) n ∈ N be a sequence in C whichconverges µ -a.e. to an element g ∈ L ( µ ). For arbitrary Y ∈ Y we obtain, via Fatou’s lemmaand the fact that g n ∈ C for each n ∈ N , (cid:90) Ω gY d µ ≤ lim inf n →∞ (cid:90) Ω g n Y d µ ≤ , so by (5.9), g ∈ C , and thus C is closed. Further, it is straightforward to establish the convexityof C (inherited from the convexity of X ) from its definition. Finally, solidity of C is also clear:if one can dominate an element g ∈ C with a self-financing wealth process, then one can alsodominate any smaller process with the same portfolio.5.2. Convexity of the dual domain. We now turn to the dual side of the analysis. The firststep is to establish convexity properties of the sets R and Y . Here, the particular structureof the dual variables in (2.9) and (2.11) comes into play. Lemma 5.5. The sets R and Y of (2.10) and (2.12) are convex.Proof. Take two elements S , S ∈ S and two elements β , β ∈ B , and define R , R ∈ R and Y , Y ∈ Y by R i := exp (cid:18) − (cid:90) · β is d κ s (cid:19) S i , Y i := β i R i , i = 1 , . For two constants λ , λ ≥ λ + λ = 1, define the convex combinations S := λ S + λ S , R := λ R + λ R , Y := λ Y + λ Y . Observe that S ∈ S because the set S of supermartingale deflators is convex.Since β i ≥ , i = 1 , S is convex, we have R ≤ λ S + λ S = S ∈ S . We can therefore define a non-negative process (cid:101) β ∈ B by the relation(5.10) R = exp (cid:18) − (cid:90) · (cid:101) β s d κ s (cid:19) S, This shows that R ∈ R , so that R is convex, as claimed.Define a non-negative process (cid:98) β ∈ B by(5.11) Y = (cid:98) βR. To establish that Y is convex, we need to show the existence of a process ¯ β ∈ B such that(5.12) Y = ¯ β exp (cid:18) − (cid:90) · ¯ β s d κ s (cid:19) S. From (5.10), (5.11) and (5.12) we thus require ¯ β to satisfy the relation¯ β exp (cid:18) − (cid:90) · ¯ β s d κ s (cid:19) = (cid:98) β exp (cid:18) − (cid:90) · (cid:101) β s d κ s (cid:19) , which, given processes (cid:101) β and (cid:98) β , does have a unique solution for ¯ β , due to the monotonicity ofthe exponential function. Thus Y is convex. (cid:3) TILITY FROM INTER-TEMPORAL WEALTH 21 The next step is to attempt to reach some form of reverse polarity result to (5.6). It is herethat the enlargement of the dual domain from Y to the set D of (4.4) comes into play.To see why this enlargement is needed, we first observe that the implication (5.2) extendsfrom X to C , so we have(5.13) Y ∈ Y = ⇒ (cid:104) g, Y (cid:105) ≤ , ∀ g ∈ C , which implies that(5.14) Y ⊆ C ◦ . We do not have the reverse inclusion, because we do not have the reverse implication to (5.13),so cannot write a full bipolarity relation between sets C and Y . The enlargement from Y tothe set D resolves the issue, yielding the inter-temporal wealth bipolarity of Lemma 5.7 below.This procedure, in the spirit of Kramkov and Schachermayer [21], requires us to establish thatthe enlarged domain is closed in an appropriate topology. Here is the relevant result. Lemma 5.6. The enlarged dual domain D ≡ D (1) of (4.4) is closed with respect to thetopology of convergence in measure µ . The proof of Lemma 5.6 will be given further below. First, we use the result of the lemmato establish the bipolarity result below. Lemma 5.7 (Inter-temporal wealth bipolarity) . Given Lemma 5.6, the set D is a closed,convex and solid subset of L ( µ ) , and the the sets C and D satisfy the bipolarity relations (5.15) C = D ◦ , D = C ◦ . Proof. For any h ∈ D there will exist an element Y ∈ Y such that h ≤ Y, µ -almost everywhere.Hence, the implication (5.13) holds true with D in place of Y : h ∈ D = ⇒ (cid:104) g, h (cid:105) ≤ , ∀ g ∈ C , which yields the analogue of (5.14):(5.16) D ⊆ C ◦ . Combining (5.7) and (5.16) we have(5.17) D ⊆ Y ◦◦ . Part (ii) of the bipolar theorem, Theorem 4.3, says that Y ◦◦ ⊇ Y and that Y ◦◦ is thesmallest closed, convex, solid set which contains Y . But D is also closed, convex and solid(closed due to Lemma 5.6, convexity following easily from the convexity of Y , and solidity isobvious), and by definition D ⊇ Y , so we also have(5.18) D ⊇ Y ◦◦ . Thus, (5.17) and (5.18) give(5.19) D = Y ◦◦ . In other words, in enlarging from Y to D we have succeeded in reaching the bipolar of theformer.Combining (5.19) and (5.7) we see that D is the polar of C ,(5.20) D = C ◦ , so we have the second equality in (5.15). From (5.20) we get D ◦ = C ◦◦ which, combined with(5.8), yields the first equality in (5.15), and the proof is complete. (cid:3) It remains to prove Lemma 5.6, which we used above. We recall the concept of Fatouconvergence of stochastic processes from F¨ollmer and Kramkov [12], that will be needed. Definition 5.8 (Fatou convergence) . Let ( Y n ) n ∈ N be a sequence of processes on a stochasticbasis (Ω , F , F := ( F t ) t ≥ , P ), uniformly bounded from below, and let τ be a dense subset of R + . The sequence ( Y n ) n ∈ N is said to be Fatou convergent on τ to a process Y if Y t = lim sup s ↓ t, s ∈ τ lim sup n →∞ Y ns = lim inf s ↓ t, s ∈ τ lim inf n →∞ Y ns , a.s ∀ t ≥ . If τ = R + , the sequence is simply called Fatou convergent .The relevant consequence for our purposes is F¨ollmer and Kramkov [12, Lemma 5.2], thatfor a sequence ( S n ) n ∈ N of supermartingales, uniformly bounded from below, with S n = 0 , n ∈ N , there is a sequence ( Y n ) n ∈ N of supermartingales, with Y n ∈ conv( S n , S n +1 , . . . ), and asupermartingale Y with Y ≤ 0, such that ( Y n ) n ∈ N is Fatou convergent on a dense subset τ of R + to Y . Here, conv( S n , S n +1 , . . . ) denotes a convex combination (cid:80) N ( n ) k = n λ k S k for λ k ∈ [0 , (cid:80) N ( n ) k = n λ k = 1. The requirement that S n = 0 is of course no restriction, since for asupermartingale with (say) S n = 1 (as we shall have when we apply these results below forsupermartingales in Y ), we can always subtract the initial value 1 to reach a process whichstarts at zero.To prove Lemma 5.6 we shall need the following lemma on Fatou convergence of convexcombinations of elements in R , S and, as a consequence, Y . This result could instead havebeen developed in the course of proving Lemma 5.6, but it simplifies the proof of the latter agreat deal to establish it separately. Lemma 5.9. Let τ be a dense subset of R + . Let ( (cid:101) R n ) n ∈ N be a sequence in R , so given by (cid:101) R n = exp (cid:18) − (cid:90) · (cid:101) β ns d κ s (cid:19) (cid:101) S n , n ∈ N , for a sequence ( (cid:101) β n ) n ∈ N in B and a sequence of supermartingale deflators ( (cid:101) S n ) n ∈ N in S .Then for each n ∈ N there exist convex combinations R n ∈ conv( (cid:101) R n , (cid:101) R n +1 , . . . ) ∈ R , S n ∈ conv( (cid:101) S n , (cid:101) S n +1 , . . . ) ∈ S , and a process β n ∈ B such that (5.21) R n = exp (cid:18) − (cid:90) · β ns d κ s (cid:19) S n , n ∈ N , and such that the sequence ( R n ) n ∈ N (respectively, ( S n ) n ∈ N ) is Fatou convergent on τ to to asupermartingale R ∈ R (respectively, S ∈ S ), with (5.22) R = exp (cid:18) − (cid:90) · β s d κ s (cid:19) S, for a process β ∈ B . As a consequence, the sequence of inter-temporal wealth deflators ( Y n ) n ∈ N ∈ Y given by Y n = β n R n is Fatou convergent on τ to the element Y = βR ∈ Y .Proof. Since R and S are both convex sets, the convex combinations R n , S n of the lemma lie in R , S , respectively. Indeed, by similar reasoning as in the proof of Lemma 5.5, for non-negativeconstants ( λ k ) N ( n ) k = n such that (cid:80) N ( n ) k = n λ k = 1, we have(5.23) R n := N ( n ) (cid:88) k = n λ k (cid:101) R k = N ( n ) (cid:88) k = n λ k exp (cid:18) − (cid:90) · (cid:101) β ks d κ s (cid:19) (cid:101) S k ≤ N ( n ) (cid:88) k = n λ k (cid:101) S k =: S n , which shows that R n ≤ S n , implying R n ∈ R and S n ∈ S , and implying the existence of β n ∈ B such that (5.21) holds. From F¨ollmer and Kramkov [12, Lemma 5.2] there existsupermartingales R and S such that the sequences ( R n ) n ∈ N and ( S n ) n ∈ N Fatou converge on τ to R and S respectively.Define a supermartingale sequence ( (cid:101) V n ) n ∈ N by (cid:101) V n := X (cid:101) S n , for X ∈ X . Once againfrom [12, Lemma 5.2] there exists a sequence ( V n ) n ∈ N of supermartingales with each V n ∈ TILITY FROM INTER-TEMPORAL WEALTH 23 conv( (cid:101) V n , (cid:101) V n +1 , . . . ) = X conv( (cid:101) S n , (cid:101) S n +1 , . . . ), and a supermartingale V , such that ( V n ) n ∈ N isFatou convergent on τ to V . Since V n ∈ X conv( (cid:101) S n , (cid:101) S n +1 , . . . ) for each n ∈ N , we have V n = XS n , for S n ∈ conv( (cid:101) S n , (cid:101) S n +1 , . . . ). Because the sequence ( S n ) n ∈ N is Fatou convergenton τ to the supermartingale S , the sequence ( V n ) n ∈ N = ( XS n ) n ∈ N is Fatou convergent on τ to the supermartingale V = XS . Since XS is a supermartingale and X ∈ X , we have S ∈ S .The same argument as in the last paragraph, now applied to the supermartingale sequence( (cid:102) W n ) n ∈ N defined by (cid:102) W n := X (cid:101) R n , establishes that R ∈ S . But because R n ≤ S n , µ -a.e., wehave R ≤ S, µ -a.e., so there exists a process β ∈ B such that (5.22) holds, and thus in fact wehave R ∈ R ⊆ S . We have thus established that the sequence in (5.21) Fatou converges tothe process R in (5.22), and this implies that the sequence ( Y n ) n ∈ N defined by Y n := β n R n must Fatou converge to a process βR =: Y ∈ Y , since the same process β n ∈ B appears in thesequence in (5.21) as well as in the sequence ( Y n ) n ∈ N , and the proof is complete. (cid:3) With this preparation, we can now prove Lemma 5.6. Proof of Lemma 5.6. Let ( h n ) n ∈ N be a sequence in D , converging µ -a.e. to some h ∈ L ( µ ).We want to show that h ∈ D .Since h n ∈ D , for each n ∈ N we have h n ≤ (cid:98) Y n , µ -a.e for some element (cid:98) Y n ∈ Y givenby (cid:98) Y n = (cid:98) β n (cid:98) R n , for a non-negative process (cid:98) β n ∈ B and a supermartingale (cid:98) R n ∈ R given by (cid:98) R n = exp (cid:16) − (cid:82) · (cid:98) β ns d κ s (cid:17) (cid:98) S n , for a supermartingale deflator (cid:98) S n ∈ S .Consider a convex combination(5.24) Y n = N ( n ) (cid:88) k = n λ k (cid:98) Y k ≥ N ( n ) (cid:88) k = n λ k h k , n ∈ N , for non-negative constants ( λ k ) N ( n ) k = n such that (cid:80) N ( n ) k = n λ k = 1.By convexity of the set Y , we have Y n ∈ Y for each n ∈ N , so there exist processes β n ∈ B , R n ∈ R , S n ∈ S such that Y n = β n R n = β n exp (cid:18) − (cid:90) · β ns d κ s (cid:19) S n , n ∈ N . By convexity of the sets R and S there will exist sequences ( (cid:101) R n ) n ∈ N in R and ( (cid:101) S n ) n ∈ N in S ,such that R n ∈ conv( (cid:101) R n , (cid:101) R n +1 , . . . ) ∈ R , and S n ∈ conv( (cid:101) S n , (cid:101) S n +1 , . . . ) ∈ S , and these convexcombinations will in general differ from that in (5.24). We thus have the analogue of (5.23): R n = (cid:101) N ( n ) (cid:88) k = n (cid:101) λ k (cid:101) R k = (cid:101) N ( n ) (cid:88) k = n (cid:101) λ k exp (cid:18) − (cid:90) · (cid:101) β ks d κ s (cid:19) (cid:101) S k ≤ (cid:101) N ( n ) (cid:88) k = n (cid:101) λ k (cid:101) S k = S n , n ∈ N , for some sequence ( (cid:101) β n ) n ∈ N in B . and non-negative constants ( (cid:101) λ k ) (cid:101) N ( n ) k = n such that (cid:80) (cid:101) N ( n ) k = n (cid:101) λ k = 1.By Lemma 5.9, the sequences ( R n ) n ∈ N in R and ( S n ) n ∈ N in S Fatou converge on a dense subset τ of R + to supermartingales R ∈ R and S ∈ S , respectively, and such that (5.22) holds forsome process β ∈ B . Then, again by Lemma 5.9, the sequence ( Y n ) n ∈ N Fatou converges on τ to Y = βR ∈ Y . So the first term in (5.24) converges to Y ∈ Y while the last term convergesto h as n → ∞ , so the inequality in (5.24) gives h ≤ Y , and thus h ∈ D . (cid:3) With the inter-temporal wealth bipolarity of Lemma 5.7, we can establish Proposition 4.4. Proof of Proposition 4.4. From the properties of C established in Lemma 5.3, we have all theclaimed properties of C in items (i) and (ii). The corresponding assertions for D follow fromLemma 5.7. For item (iii), consider first the set D . Since the wealth process X ≡ ∈ X , the constantfunction g ≡ ∈ C , and the budget constraint (equivalently, the polar relation (4.7)) in thiscase gives (cid:82) Ω h d µ ≤ 1, so D is bounded in L ( µ ) and hence in L ( µ ).For the L -boundedness of C , we shall find a positive element h ∈ D and show that C is bounded in L ( h d µ ), and hence bounded in L ( µ ). Since the constant supermartingale S ≡ ∈ S and since the constant process β ≡ α > α , lies in B ,we can take Y (cid:51) Y t := α exp( − ακ t ) , t ≥ 0, and then choose D (cid:51) h ≡ Y . We see that h ∈ D isstrictly positive except on a set of µ -measure zero. Then, the budget constraint (equivalently,the polar relation (4.8)) gives (cid:82) Ω gh d µ ≤ g ∈ C . Thus, C is bounded in L ( h d µ )and hence bounded in L ( µ ). (cid:3) On approaches to establishing bipolarity. In this section we compare the approachwe have taken to establishing the polar relations (4.7) and (4.8) in Proposition 4.4, between theenlarged primal and dual domains C and D , with the approach taken by Bouchard and Pham[2]. This is instructive and will indicate how we have been able to strengthen the statementof the final duality result, in essence by proving, as opposed to partially assuming, the polarrelations, which is what Bouchard and Pham [2] were compelled to do.5.3.1. The Kramkov-Schachermayer approach. Our approach is in the spirit of the recipe cre-ated by Kramkov and Schachermayer [21, 22] for the terminal wealth utility maximisationproblem, adapted to an inter-temporal framework. One begins with a supermartingale prop-erty linking the elements of the primal and dual domains. (In the terminal wealth problem onehas the admissible wealth processes X ∈ X and the supermartingale deflators S ∈ S , with XS a supermartingale for each X ∈ X and S ∈ S .) Here, we invoke the additional dual controls β ∈ B , and from these and the supermartingale deflators we construct the supermartingales R ∈ R and the inter-temporal wealth deflators Y ∈ Y according to the relations in (2.9) and(2.11), repeated below for the case y = 1, so for S ∈ S :(5.25) R := exp (cid:18) − (cid:90) · β s d κ s (cid:19) S, Y := βR, β ∈ B , S ∈ S . Observe that the deflators Y ∈ Y are given by Y = νS, S ∈ S , with the process ν given by(5.26) ν t := β t exp (cid:18) − (cid:90) t β s d κ s (cid:19) , t ≥ , β ∈ B . We see that ν satisfies (cid:90) ∞ ν t d κ t = 1 − exp (cid:18) − (cid:90) ∞ β t d κ t (cid:19) ≤ , almost surely , and hence also E (cid:2)(cid:82) ∞ ν t d κ t (cid:3) ≤ (cid:104) ν, (cid:105) ≤ . This structure of dual variables for wealth-path-dependent utility maximisation problems,namely a multiplicative auxiliary control which augments the classical deflators and whichsatisfies a constraint of the form in (5.27), is not uncommon, and we shall see a similar featureshortly when we describe the Bouchard and Pham [2] approach. The key insight that arisesin our approach is that this auxiliary control must have the very specific structure in (5.26),which confers convexity to the dual domain.From (5.25) and the properties of S ∈ S , we get that the process M in (2.13) is a super-martingale, and in turn this gives the budget constraint (2.14), repeated below for the case TILITY FROM INTER-TEMPORAL WEALTH 25 x = y = 1, as a necessary condition for admissibility of a wealth process: E (cid:20)(cid:90) ∞ X t Y t d κ t (cid:21) ≤ , ∀ X ∈ X , Y ∈ Y . Then, enlarging the primal domain from X to C , Lemma 5.2 establishes that the budgetconstraint is also a sufficient condition for admissibility, so we obtain the polar properties ofLemma 5.3 for C C = Y ◦ , C ◦ = Y ◦◦ , C ◦◦ = C , which imply that C is a closed, convex and solid (CCS) subset of L ( µ ).Now to the dual side of the story. Using the particular form of the dual variables in (5.25)we established in Lemma 5.5 that the dual domain Y is convex. This convexity is passed onto the enlarged dual domain D . Then, again using the structure in (5.25), and in particularthat the deflators Y ∈ Y contain the supermartingales R ∈ R , S ∈ S , we are able to exploitFatou convergence of supermartingales to show that D is closed with respect to the topology ofconvergence in µ -measure. This, along with the convexity and (obvious) solidity of D , showsthat D is also a CCS subset of L ( µ ), matching the property we obtained for C . In particular,we obtain the key result that the enlargement from Y to D has taken as to the bipolar of theoriginal dual domain: D = Y ◦◦ . This result then readily combines with the earlier polarity properties of C to establish theperfect bipolarity relations (4.7) and (4.8).The message is that we have made the Kramkov and Schachermayer [21, 22] prescriptionfor obtaining bipolarity work: begin with a supermartingale property to arrive at the correctdefinition of the dual variables, make no assumptions regarding convexity and closed propertiesof either the primal or dual domains, show that with a natural enlargement of these domainsto obtain solid sets, all the required CCS properties of the domains, and hence bipolarity,follows. This bipolarity is then the bedrock of the subsequent program for the proof of theduality theorem, as we shall see in Section 6.This methodology is to be contrasted with the approach in [2], which we now describe.5.3.2. The Bouchard-Pham approach. The first difference between our methodology and thatof Bouchard and Pham [2] is that in [2], the dual domain (let us call in D BP ) is defined as thepolar of the primal domain. Over a finite horizon T < ∞ , the dual variables Y BP and dualdomain are thus defined according to D BP := (cid:26) Y BP ≥ E (cid:20)(cid:90) T X t Y BP t d κ t (cid:21) ≤ , ∀ X ∈ X (cid:27) , (see the definition of the set D ( y ) in [2, Page 584]). In other words,(5.28) D BP := X , by assumption. This automatically confers the CCS property to the dual domain, but thestatement of the result is weakened, having been obtained by definition. The reason that thisapproach had to be adopted, we conjecture, is that the authors of [2] did not have to handthe specific structure of the dual variables in (5.25) that emerges in our approach.This conjecture is reinforced by the reasoning which now follows. In a subsequent refine-ment Bouchard and Pham [2] show that, under an assumption called (Hf ) (namely, that κ decomposes into a continuous density plus a linear combination of indicator functions of theform { τ ≤ t } , t ∈ [0 , T ], for any F -stopping time τ ), processes of the form ν BP Z M lie in theirdual domain, where Z M is the density process of an ELMM, and ν BP is any process satisfying (cid:104) ν BP , (cid:105) T := E (cid:104)(cid:82) T ν BP t d κ t (cid:105) ≤ 1. The similarity with the structure we have in (5.27) is clear.If we denote the set of processes ν BP Z M by Z BP , then under their additional assumption (Hf ) , Bouchard and Pham [2] are able to re-cast their dual problem as a minimisation overthe convex hull of Z BP . This, therefore, is the analogue, under NFLVR and over a finitehorizon, of the dual structure we have used, but with two caveats. First, they have to use theconvex hull of Z BP , because the set Z BP is not known to be convex in general. Second, thislack of convexity is due to the fact that the authors of [2] do not have the particular structureof the auxiliary dual control ν BP that we have found in (5.26), a structure that was crucialin our establishing the convexity of our dual domain. All that is known about the processes ν BP is that they satisfy (cid:104) ν BP , (cid:105) T ≤ 1, and this is not enough to afford a proof of convexity of Z BP .Finally, the discussion above also explains why the bulk of the analysis in [2] is carried outon the primal side of the problem. Since the definition in (5.28) confers the CCS propertyto the dual domain by assumption, the remaining work in [2] is concerned with enlarging theprimal domain to confer solidity and proving the remaining polarity relation, as can be verifiedby examining [2, Section 5].In summary, we are able to strengthen the duality statement in [2] to any horizon andunder NUPBR, by making the broad pattern of the Kramkov and Schachermayer [21, 22]program for bipolarity work, without having to assume the associated properties of either theprimal or dual domain. Instead, we begin with a natural supermartingale property linking theprimal and dual elements, thus identifying the natural dual space for the problem, along withits particular structure, so that the closed and convex features of the domains, from whichthe existence and uniqueness of the optimisers are ultimately deduced, are demonstrated, asopposed to being assumed.6. Proofs of the duality theorems In this section we prove the abstract duality of Theorem 4.5, from which the concreteduality of Theorem 3.1 is then deduced. Throughout this section, we have in place the resultof Proposition 4.4, as this bipolarity is the starting point of the duality proof. The proofof Theorem 4.5 proceeds via a series of lemmas. The procedure has a similar flavour tothat of Kramkov and Schachermayer [21, 22] for an abstract duality proof in the context ofthe terminal wealth utility maximisation problem, with variations where appropriate, andwith an additional result, Proposition 6.13, which gives the additional characterisation (3.2)of the optimal wealth process as well as the uniformly integrable martingale property of theprocess (cid:99) M := (cid:98) X ( x ) (cid:98) R ( y )+ (cid:82) · (cid:98) X s ( x ) (cid:98) Y s ( y ) d κ s . This proposition also establishes that the process (cid:98) X ( x ) (cid:98) R ( y ) is a potential, and that its limiting value is lim t →∞ (cid:98) X t ( x ) (cid:98) R t ( y ) = 0 almost surely.Let us state the basic properties that are taken as given throughout this section. Fact 6.1. Throughout this section, assume that the utility function satisfies the Inada con-ditions (2.5), that the sets C and D satisfy all the properties in Proposition 4.4, and that theabstract primal and dual value functions in (4.3) and (4.5) satisfy the minimal conditions in(4.9). All subsequent lemmata and propositions in this section implicitly take Fact 6.1 as given. The first step is to establish weak duality. Lemma 6.2 (Weak duality) . The primal and dual value functions u ( · ) and v ( · ) of (4.3) and (4.5) satisfy the weak duality bounds (6.1) v ( y ) ≥ sup x> [ u ( x ) − xy ] , y > , equivalently u ( x ) ≤ inf y> [ v ( y ) + xy ] , x > . As a result, u ( x ) is finitely valued for all x > . Moreover, we have the limiting relations (6.2) lim sup x →∞ u ( x ) x ≤ , lim inf y →∞ v ( y ) y ≥ . TILITY FROM INTER-TEMPORAL WEALTH 27 Proof. Recall the inequality (2.17). By the same argument carried out in the measure space( Ω , G , µ ) we have, for any g ∈ C ( x ) and h ∈ D ( y ), using the polarity relations in (4.7) and(4.8), (cid:90) Ω U ( g ) d µ ≤ (cid:90) Ω U ( g ) d µ + xy − (cid:90) Ω gh d µ = (cid:90) Ω ( U ( g ) − gh ) d µ + xy ≤ (cid:90) Ω V ( h ) d µ + xy, x, y > , (6.3)the last inequality a consequence of (2.16). Maximising the left-hand-side of (6.3) over g ∈ C ( x )and minimising the right-hand-side over h ∈ D ( y ) gives u ( x ) ≤ v ( y ) + xy for all x, y > 0, and(6.1) follows.The assumption that v ( y ) < ∞ for all y > u ( x ) is finitely valuedfor some x > 0. Since U ( · ) is strictly increasing and strictly concave, and given the convexityof C , these properties are inherited by u ( · ), which is therefore finitely valued for all x > (cid:3) Above, we obtained concavity and monotonicity of u ( · ) by using convexity of C and theproperties of U ( · ). Similar arguments show that v ( · ) is strictly decreasing and strictly convex.We shall see these properties reproduced in proofs of existence and uniqueness of the optimisersfor u ( · ) , v ( · ).The next step is to give a compactness lemma for the dual domain. Lemma 6.3 (Compactness lemma for D ) . Let (˜ h n ) n ∈ N be a sequence in D . Then there existsa sequence ( h n ) n ∈ N with h n ∈ conv(˜ h n , ˜ h n +1 , . . . ) , which converges µ -a.e. to an element h ∈ D that is µ -a.e. finite.Proof. Delbaen and Schachermayer [6, Lemma A1.1] (adapted from a probability space tothe finite measure space ( Ω , G , µ )) implies the existence of a sequence ( h n ) n ∈ N , with h n ∈ conv(˜ h n , ˜ h n +1 , . . . ), which converges µ -a.e. to an element h that is µ -a.e. finite because D is bounded in L ( µ ) (the finiteness also from [6, Lemma A1.1]). By convexity of D , each h n , n ∈ N lies in D . Finally, by Fatou’s lemma, for every g ∈ C we have (cid:90) Ω gh d µ = (cid:90) Ω lim inf n →∞ gh n d µ ≤ lim inf n →∞ (cid:90) Ω gh n d µ ≤ , so that h ∈ D . (cid:3) Results in the style of Lemma 6.3 are standard in these duality proofs. We will see a similarresult for the primal domain C shortly.The next step in the chain of results we need is a uniform integrability result for the family( V − ( h )) h ∈D ( y ) . This will facilitate a proof of existence and uniqueness of the dual minimiser,and of the conjugacy for the value functions by establishing the first relation in (4.10). Lemma 6.4 (Uniform integrability of ( V − ( h )) h ∈D ( y ) ) . The family ( V − ( h )) h ∈D ( y ) is uniformlyintegrable, for any y > . The style of the proof is along identical lines to Kramkov and Schachermayer [21, Lemma3.2], but we give the proof for completeness. Proof of Lemma 6.4. Since V ( · ) is decreasing, we need only consider the case where V ( ∞ ) :=lim y →∞ V ( y ) = −∞ (otherwise there is nothing to prove). Let ϕ : ( − V (0) , − V ( ∞ )) (cid:55)→ (0 , ∞ ) denote the inverse of − V ( · ). Then ϕ ( · ) is strictly increasing. For any h ∈ D ( y ) (so (cid:82) Ω h d µ ≤ y )we have, for all y > (cid:90) Ω ϕ ( V − ( h )) d µ ≤ ϕ (0) + (cid:90) Ω ϕ ( − V ( h )) d µ = ϕ (0) + (cid:90) Ω h d µ ≤ ϕ (0) + y. Then, using l’Hˆopital’s rule and the change of variable ϕ ( x ) = y ⇐⇒ x = − V ( y ), andrecalling the function I ( · ) = − V (cid:48) ( · ) (the inverse of marginal utility U (cid:48) ( · )), we have(6.4) lim x →− V ( ∞ ) ϕ ( x ) x = lim x →∞ ϕ ( x ) x = lim y →∞ y − V ( y ) = lim y →∞ I ( y ) = + ∞ , on using the Inada conditions (2.5). The L ( µ )-boundedness of D ( y ) means we can apply thede la Vall´ee-Poussin theorem (Pham [28, Theorem A.1.2]) which, combined with (6.4), impliesthe uniform integrability of the family ( V − ( h )) h ∈D ( y ) . (cid:3) One can can now proceed to prove either existence of a unique optimiser in the dual problem,or conjugacy of the value functions. We proceed first with the former, followed by conjugacy. Lemma 6.5 (Dual existence) . The optimal solution (cid:98) h ( y ) ∈ D ( y ) to the dual problem (4.5) exists and is unique, so that v ( · ) is strictly convex.Proof. Fix y > 0. Let ( h n ) n ∈ N be a minimising sequence in D ( y ) for v ( y ) < ∞ . That is(6.5) lim n →∞ (cid:90) Ω V ( h n ) d µ = v ( y ) < ∞ . By the compactness lemma for D (and thus also for D ( y ) = y D ), Lemma 6.3, we can find asequence ( (cid:98) h n ) n ∈ N of convex combinations, so D ( y ) (cid:51) (cid:98) h n ∈ conv( h n , h n +1 , . . . ) , n ∈ N , whichconverges µ -a.e. to some element (cid:98) h ( y ) ∈ D ( y ). We claim that (cid:98) h ( y ) is the dual optimiser. Thatis, that we have(6.6) (cid:90) Ω V ( (cid:98) h ( y )) d µ = v ( y ) . From convexity of V ( · ) and (6.5) we deduce thatlim n →∞ (cid:90) Ω V ( (cid:98) h n ) d µ ≤ lim n →∞ (cid:90) Ω V ( h n ) d µ = v ( y ) , which, combined with the obvious inequality v ( y ) ≤ lim n →∞ (cid:82) Ω V ( (cid:98) h n ) d µ means that we alsohave, further to (6.5), lim n →∞ (cid:90) Ω V ( (cid:98) h n ) d µ = v ( y ) . In other words(6.7) lim n →∞ (cid:90) Ω V + ( (cid:98) h n ) d µ − lim n →∞ (cid:90) Ω V − ( (cid:98) h n ) d µ = v ( y ) < ∞ , and note therefore that both integrals in (6.7) are finite.From Fatou’s lemma, we have(6.8) lim n →∞ (cid:90) Ω V + ( (cid:98) h n ) d µ ≥ (cid:90) Ω V + ( (cid:98) h ( y )) d µ. From Lemma 6.4 we have uniform integrability of ( V − ( (cid:98) h n )) n ∈ N , so that(6.9) lim n →∞ (cid:90) Ω V − ( (cid:98) h n ) d µ = (cid:90) Ω V − ( (cid:98) h ( y )) d µ. TILITY FROM INTER-TEMPORAL WEALTH 29 Thus, using (6.8) and (6.9) in (6.7), we obtain v ( y ) ≥ (cid:90) Ω V ( (cid:98) h ( y )) d µ, which, combined with the obvious inequality v ( y ) ≤ (cid:82) Ω V ( (cid:98) h ( y )) d µ , yields (6.6). The unique-ness of the dual optimiser follows from the strict convexity of V ( · ), as does the strict convexityof v ( · ). For this last claim, fix y < y and λ ∈ (0 , λ (cid:98) h ( y ) + (1 − λ ) (cid:98) h ( y ) ∈D ( λy + (1 − λ ) y ) (yet must be sub-optimal for v ( λy + (1 − λ ) y ) as it is not guaranteed toequal (cid:98) h ( λy + (1 − λ ) y )) and therefore, using the strict convexity of V ( · ), v ( λy + (1 − λ ) y ) ≤ (cid:90) Ω V (cid:16) λ (cid:98) h ( y ) + (1 − λ ) (cid:98) h ( y ) (cid:17) d µ < λv ( y ) + (1 − λ ) v ( y ) . (cid:3) We now establish conjugacy of the value functions. The method is similar to the classicalmethod of proof in Kramkov and Schachermayer [21, Lemma 3.4], and works by bounding theelements in the primal domain to create a compact set for the weak ∗ topology σ ( L ∞ , L ) on L ∞ ( µ ), so as to apply the minimax theorem, involving a maximisation over a compact setand a minimisation over a subset of a vector space. This uses the fact that the dual domainis bounded in L ( µ ).For the convenience of the reader here is the minimax theorem as we shall apply it (seeStrasser [32, Theorem 45.8]). Theorem 6.6 (Minimax) . Let X be a σ ( E (cid:48) , E ) -compact convex subset of the topological dual E (cid:48) of a normed vector space E , and let Y be a convex subset of E . Assume that f : X × Y → R satisfies the following conditions: (1) x (cid:55)→ f ( x, y ) is continuous and concave on X for every y ∈ Y ; (2) y (cid:55)→ f ( x, y ) is convex on Y for every x ∈ X .Then: sup x ∈X inf y ∈Y f ( x, y ) = inf y ∈Y sup x ∈X f ( x, y ) . Here is the conjugacy result for the primal and dual value functions. Lemma 6.7 (Conjugacy) . The dual value function in (4.5) satisfies the conjugacy relation v ( y ) = sup x> [ u ( x ) − xy ] , for each y > , where u ( · ) is the primal value function in (4.3) .Proof. For n ∈ N denote by B n the set of elements in L ( µ ) lying in a ball of radius n : B n := (cid:8) g ∈ L ( µ ) : g ≤ n, µ − a . e . (cid:9) . The sets ( B n ) n ∈ N are σ ( L ∞ , L )-compact. Because each h ∈ D ( y ) is µ -integrable, D ( y ) isa closed, convex subset of the vector space L ( µ ), so we apply the minimax theorem asgiven in Theorem 6.6 to the compact set B n ( n fixed) and the set D ( y ), with the function f ( g, h ) := (cid:82) Ω ( U ( g ) − gh ) d µ , for g ∈ B n , h ∈ D ( y ), to give(6.10) sup g ∈B n inf h ∈D ( y ) (cid:90) Ω ( U ( g ) − gh ) d µ = inf h ∈D ( y ) sup g ∈B n (cid:90) Ω ( U ( g ) − gh ) d µ. Recall that a sequence ( g n ) n ∈ N in L ∞ ( µ ) converges to g ∈ L ∞ ( µ ) with respect to the weak ∗ topology σ ( L ∞ , L ) if and only if ( (cid:104) g n , h (cid:105) ) n ∈ N converges to (cid:104) g, h (cid:105) for each h ∈ L ( µ ). By the bipolarity relation C = D ◦ in (4.7), an element g ∈ L ( µ ) lies in C ( x ) if and only ifsup h ∈D ( y ) (cid:82) Ω gh d µ ≤ xy . Thus, the limit as n → ∞ on the left-hand-side of (6.10) is given as(6.11) lim n →∞ sup g ∈B n inf h ∈D ( y ) (cid:90) Ω ( U ( g ) − gh ) d µ = sup x> sup g ∈C ( x ) (cid:18)(cid:90) Ω U ( g ) d µ − xy (cid:19) = sup x> [ u ( x ) − xy ] . Now consider the right-hand-side of (6.10). Define V n ( y ) := sup TILITY FROM INTER-TEMPORAL WEALTH 31 Lemma 6.8 (Compactness lemma for C ) . Let (˜ g n ) n ∈ N be a sequence in C . Then there existsa sequence ( g n ) n ∈ N with g n ∈ conv(˜ g n , ˜ g n +1 , . . . ) , which converges µ -a.e. to an element g ∈ C that is µ -a.e. finite. To prove existence of a unique primal optimiser we also need a result analogous to Lemma6.4, on the uniform integrability of a sequence ( U + ( g n )) n ∈ N for g n ∈ C ( x ). The proof is in thestyle of Kramkov and Schachermayer [22, Lemma 1]. Lemma 6.9 (Uniform integrability of ( U + ( g n )) n ∈ N , g n ∈ C ( x )) . Let ( g n ) n ∈ N be a sequence in C ( x ) , for any fixed x > . The sequence ( U + ( g n )) n ∈ N is uniformly integrable.Proof. Fix x > 0. If U ( ∞ ) ≤ U ( ∞ ) > U + ( g n )) n ∈ N is not uniformly integrable, then, passing if need be to asubsequence still denoted by ( g n ) n ∈ N , we can find a constant α > A n ) n ∈ N of sets of ( Ω , G ) (so A n ∈ G , n ∈ N and A i ∩ A j = ∅ if i (cid:54) = j ) such that (cid:90) Ω U + ( g n ) A n d µ ≥ α, n ∈ N . (See for example Pham [28, Corollary A.1.1].) Define a sequence ( f n ) n ∈ N of elements in L ( µ )by f n := x + n (cid:88) k =1 g k A k , where x := inf { x > U ( x ) ≥ } .For any h ∈ D (so satisfying (cid:82) Ω h d µ ≤ 1) we have (cid:90) Ω f n h d µ = (cid:90) Ω (cid:32) x + n (cid:88) k =1 g k A k (cid:33) h d µ ≤ x + n (cid:88) k =1 (cid:90) Ω g k h A k d µ ≤ x + nx. Thus, f n ∈ C ( x + nx ) , n ∈ N .On the other hand, since U + ( · ) is non-negative and non-decreasing, (cid:90) Ω U ( f n ) d µ = (cid:90) Ω U + ( f n ) d µ = (cid:90) Ω U + (cid:32) x + n (cid:88) k =1 g k A k (cid:33) d µ ≥ (cid:90) Ω U + (cid:32) n (cid:88) k =1 g k A k (cid:33) d µ = n (cid:88) k =1 (cid:90) Ω U + (cid:16) g k A k (cid:17) d µ ≥ αn. Therefore,lim sup z →∞ u ( z ) z = lim sup n →∞ u ( x + nx ) x + nx ≥ lim sup n →∞ (cid:82) Ω U ( f n ) d µx + nx ≥ lim sup n →∞ (cid:18) αnx + nx (cid:19) = αx > , which contradicts the limiting weak duality bound in (6.2). This contradiction establishes theresult. (cid:3) One can can now proceed to prove existence of a unique optimiser in the primal problem.The method of proof is similar to the proof of dual existence, Lemma 6.5, with adjustments formaximisation as opposed to minimisation and concavity of U ( · ) replacing convexity of V ( · ) , so is included just for completeness. Lemma 6.10 (Primal existence) . The optimal solution (cid:98) g ( x ) ∈ C ( x ) to the primal problem (4.3) exists and is unique, so that u ( · ) is strictly concave.Proof. Fix x > 0. Let ( g n ) n ∈ N be a maximising sequence in C ( x ) for u ( x ) < ∞ (the finitenessproven in Lemma 6.2). That is(6.16) lim n →∞ (cid:90) Ω U ( g n ) d µ = u ( x ) < ∞ . By the compactness lemma for C (and thus also for C ( x ) = x C ), Lemma 6.8, we can finda sequence ( (cid:98) g n ) n ∈ N of convex combinations, so C ( x ) (cid:51) (cid:98) g n ∈ conv( g n , g n +1 , . . . ) , n ∈ N , whichconverges µ -a.e. to some element (cid:98) g ( x ) ∈ C ( x ). We claim that (cid:98) g ( x ) is the primal optimiser.That is, that we have(6.17) (cid:90) Ω U ( (cid:98) g ( x )) d µ = u ( x ) . By concavity of U ( · ) and (6.16) we havelim n →∞ (cid:90) Ω U ( (cid:98) g n ) d µ ≥ lim n →∞ (cid:90) Ω U ( g n ) d µ = u ( x ) , which, combined with the obvious inequality u ( x ) ≥ lim n →∞ (cid:82) Ω U ( (cid:98) g n ) d µ means that we alsohave, further to (6.16), lim n →∞ (cid:90) Ω U ( (cid:98) g n ) d µ = u ( x ) . In other words(6.18) lim n →∞ (cid:90) Ω U + ( (cid:98) g n ) d µ − lim n →∞ (cid:90) Ω U − ( (cid:98) g n ) d µ = u ( x ) < ∞ , and note therefore that both integrals in (6.18) are finite.From Fatou’s lemma, we have(6.19) lim n →∞ (cid:90) Ω U − ( (cid:98) g n ) d µ ≥ (cid:90) Ω U − ( (cid:98) g ( x )) d µ. From Lemma 6.9 we have uniform integrability of ( U + ( (cid:98) g n )) n ∈ N , so that(6.20) lim n →∞ (cid:90) Ω U + ( (cid:98) g n ) d µ = (cid:90) Ω U + ( (cid:98) g ( x )) d µ. Thus, using (6.19) and (6.20) in (6.18), we obtain u ( x ) ≤ (cid:90) Ω U ( (cid:98) g ( x )) d µ, which, combined with the obvious inequality u ( x ) ≥ (cid:82) Ω U ( (cid:98) g ( x )) d µ , yields (6.17). The unique-ness of the primal optimiser follows from the strict concavity of U ( · ), as does the strict con-cavity of u ( · ). For this last claim, fix x < x and λ ∈ (0 , λ (cid:98) g ( x ) + (1 − λ ) (cid:98) g ( x ) ∈C ( λx + (1 − λ ) x ) (yet must be sub-optimal for u ( λx + (1 − λ ) x ) as it is not guaranteed toequal (cid:98) g ( λx + (1 − λ ) x )) and therefore, using the strict concavity of U ( · ), u ( λx + (1 − λ ) x ) ≥ (cid:90) Ω U ( λ (cid:98) g ( x ) + (1 − λ ) (cid:98) g ( x )) d µ > λu ( x ) + (1 − λ ) u ( x ) . (cid:3) We now move on to further characterise the derivatives of the value functions, as well asthe primal and dual optimisers. The first result is on the derivative of the primal value valuefunction u ( · ) at zero (equivalently, the derivative of the dual value function v ( · ) at infinity).The proof of the following lemma is in the style of Kramkov and Schachermayer [21, Lemma3.5]. TILITY FROM INTER-TEMPORAL WEALTH 33 Lemma 6.11. The derivatives of the primal value function in (4.3) at zero and of the dualvalue function in (4.5) at infinity are given by (6.21) u (cid:48) (0) := lim x ↓ u (cid:48) ( x ) = + ∞ , − v (cid:48) ( ∞ ) := lim y →∞ ( − v (cid:48) ( y )) = 0 . Proof. By the conjugacy result in Lemma 6.7 between the value functions, the assertions in(6.21) are equivalent. We shall prove the second assertion.The function − v ( · ) is strictly concave and strictly increasing, so there is a finite non-negativelimit − v (cid:48) ( ∞ ) := lim y →∞ ( − v (cid:48) ( y )). Because − V ( · ) is increasing with lim y →∞ ( − V (cid:48) ( y )) = 0, forany (cid:15) > C (cid:15) > − V ( y ) ≤ C (cid:15) + (cid:15)y, ∀ y > 0. Using this, the L ( µ )-boundedness of D (so that (cid:82) Ω h d µ ≤ y, ∀ h ∈ D ( y )) and l’Hˆopital’s rule, we have, with (cid:82) Ω d µ =: δ > ≤ lim y →∞ − v (cid:48) ( y ) = lim y →∞ − v ( y ) y = lim y →∞ sup h ∈D ( y ) (cid:90) Ω − V ( h ) y d µ ≤ lim y →∞ sup h ∈D ( y ) (cid:90) Ω C (cid:15) + (cid:15)hy d µ ≤ lim y →∞ (cid:18) C (cid:15) δy + (cid:15) (cid:19) = (cid:15), and taking the limit as (cid:15) ↓ (cid:3) The final step in the series of lemmas that will furnish us with the proof of the abstractduality of Theorem 4.5 is to characterise the derivative of the primal value value function u ( · )at infinity (equivalently, the derivative of the dual value function v ( · ) at zero) along with aduality characterisation of the primal and dual optimisers. Lemma 6.12. (1) The derivatives of the primal value function in (4.3) at infinity and ofthe dual value function in (4.5) at zero are given by (6.22) u (cid:48) ( ∞ ) := lim x →∞ u (cid:48) ( x ) = 0 , − v (cid:48) (0) := lim y ↓ ( − v (cid:48) ( y )) = + ∞ . (2) For any fixed x > , with y = u (cid:48) ( x ) (equivalently x = − v (cid:48) ( y ) ), the primal and dualoptimisers (cid:98) g ( x ) , (cid:98) h ( y ) are related by (6.23) U (cid:48) ( (cid:98) g ( x )) = (cid:98) h ( y ) = (cid:98) h ( u (cid:48) ( x )) , µ -a.e. , and satisfy (6.24) (cid:90) Ω (cid:98) g ( x ) (cid:98) h ( y ) d µ = xy = xu (cid:48) ( x ) . (3) The derivatives of the value functions satisfy the relations (6.25) xu (cid:48) ( x ) = (cid:90) Ω U (cid:48) ( (cid:98) g ( x )) (cid:98) g ( x ) d µ, yv (cid:48) ( y ) = (cid:90) Ω V (cid:48) ( (cid:98) h ( y )) (cid:98) h ( y ) d µ, x, y > . Proof. Recall the inequality (2.16), which also applies to the value functions because they arealso conjugate by Lemma 6.7. We thus have, in addition to (2.16),(6.26) v ( y ) ≥ u ( x ) − xy, ∀ x, y > , with equality iff y = u (cid:48) ( x ) . With (cid:98) g ( x ) ∈ C ( x ) , x > (cid:98) h ( y ) ∈ D ( y ) , y > (cid:90) Ω (cid:98) g ( x ) (cid:98) h ( y ) d µ ≤ xy, x, y > . Using this as well as (2.16) and (6.26) we have(6.27) 0 ≤ (cid:90) Ω (cid:16) V ( (cid:98) h ( y )) − U ( (cid:98) g ( x )) + (cid:98) g ( x ) (cid:98) h ( y ) (cid:17) d µ ≤ v ( y ) − u ( x ) + xy, x, y > , The right-hand-side of (6.27) is zero if and only if y = u (cid:48) ( x ), due to (6.26), and the non-negative integrand must then be µ -a.e. zero, which by (2.16) can only happen if (6.23) holds,which establishes that primal-dual relation.Thus, for any fixed x > y = u (cid:48) ( x ), and hence equality in (6.27), we have0 = (cid:90) Ω (cid:16) V ( (cid:98) h ( y )) − U ( (cid:98) g ( x )) + (cid:98) g ( x ) (cid:98) h ( y ) (cid:17) d µ = v ( y ) − u ( x ) + (cid:90) Ω (cid:98) g ( x ) (cid:98) h ( y ) d µ = v ( y ) − u ( x ) + xy, y = u (cid:48) ( x ) , which implies that (6.24) must hold. Inserting the explicit form of (cid:98) h ( y ) = U (cid:48) ( (cid:98) g ( x )) into (6.24)yields the first relation in (6.25). Similarly, setting (cid:98) g ( x ) = I ( (cid:98) h ( y )) = − V (cid:48) ( (cid:98) h ( y )) into (6.24),with x = − v (cid:48) ( y ) (equivalent to y = u (cid:48) ( x )), yields the second relation in (6.25).It remains to establish the relations in (6.22), which are equivalent assertions. We shallprove the second one. This will use the fact that D is a subset of L ( µ ).From the second relation in (6.25) and the fact that(6.28) (cid:90) Ω gh d µ ≤ xy, ∀ g ∈ C ( x ) , h ∈ D ( y ) , x, y > , we see that, for any y > 0, we have − V (cid:48) ( (cid:98) h ( y )) ∈ C ( − v (cid:48) ( y )). Thus, for any h ∈ D , (6.28)implies that(6.29) − v (cid:48) ( y ) ≥ (cid:90) Ω − V (cid:48) ( (cid:98) h ( y )) h d µ, ∀ h ∈ D , which we shall make use of shortly.Since D ( y ) is a subset of L ( µ ), we have (cid:82) Ω (cid:98) h ( y ) d µ ≤ y , and hence(6.30) (cid:90) Ω (cid:98) h ( y ) y d µ ≤ , ∀ y > . Using Fatou’s lemma in (6.30) we have1 ≥ lim inf y ↓ (cid:90) Ω (cid:98) h ( y ) y d µ ≥ (cid:90) Ω lim inf y ↓ (cid:32) (cid:98) h ( y ) y (cid:33) d µ, which, given that (cid:98) h ( y ) /y is non-negative, gives that lim inf y ↓ ( (cid:98) h ( y ) /y ) < ∞ , µ -a.e. Therefore,writing (cid:98) h ( y ) =: y (cid:98) h y , which defines a unique element (cid:98) h y ∈ D , we have (cid:98) h := lim inf y ↓ (cid:98) h y = lim inf y ↓ (cid:98) h ( y ) y < ∞ , µ -a.e.Using this property and applying Fatou’s lemma to (6.29) we obtain, on using − V (cid:48) (0) = + ∞ ,+ ∞ ≥ lim inf y ↓ ( − v (cid:48) ( y )) ≥ lim inf y ↓ (cid:90) Ω − V (cid:48) ( y (cid:98) h y ) h d µ ≥ (cid:90) Ω lim inf y ↓ ( − V (cid:48) ( y (cid:98) h y )) h d µ = + ∞ , which gives us the second relation in (6.22). (cid:3) We have now established all results that give the duality in Theorem 4.5, so let us confirmthis. TILITY FROM INTER-TEMPORAL WEALTH 35 Proof of Theorem 4.5. Lemma 6.7 implies the relations (4.10) of item (i). The statements initem (ii) are implied by Lemma 6.10 and Lemma 6.5. Items (iii) and (iv) follow from Lemma6.11 and Lemma 6.12. (cid:3) We are almost ready to prove the concrete duality in Theorem 3.1, because Theorem 4.5readily implies nearly all of the assertions of Theorem 3.1. The outstanding assertion is thecharacterisation of the optimal wealth process in (3.2) and the associated uniformly integrablemartingale property of the process (cid:99) M := (cid:98) X ( x ) (cid:98) R ( y ) + (cid:82) · (cid:98) X s ( x ) (cid:98) Y s ( y ) d κ s . So we proceed toestablish these assertions in the proposition below, which turns out to be interesting in its ownright. We take as given the other assertions of Theorem 3.1, and in particular the optimalbudget constraint in (3.1). We shall confirm the proof of Theorem 3.1 in its entirety after theproof of the next result. Proposition 6.13 (Optimal wealth process) . Given the saturated budget constraint equalityin (3.1) , the optimal wealth process is characterised by (3.2) . The process (cid:99) M t := (cid:98) X t ( x ) (cid:98) R t ( y ) + (cid:90) t (cid:98) X s ( x ) (cid:98) Y s ( y ) d κ s , ≤ t < ∞ , is a uniformly integrable martingale, converging to an integrable random variable (cid:99) M ∞ , so themartingale extends to [0 , ∞ ] . The process (cid:98) X ( x ) (cid:98) R ( y ) is a potential, that is, a non-negativesupermartingale satisfying lim t →∞ E [ (cid:98) X t ( x ) (cid:98) R t ( y )] = 0 . Moreover, (cid:98) X ∞ ( x ) (cid:98) R ∞ ( y ) = 0 , almostsurely.Proof. It simplifies notation if we take x = y = 1, and is without loss of generality: although y = u (cid:48) ( x ) in (3.1), one can always multiply the utility function by an arbitrary constant so asto ensure that u (cid:48) (1) = 1. We thus have the optimal budget constraint(6.31) E (cid:20)(cid:90) ∞ (cid:98) X t (cid:98) Y t d κ t (cid:21) = 1 , for (cid:98) X ≡ (cid:98) X (1) ∈ X and (cid:98) Y ≡ (cid:98) Y (1) ∈ Y . Since (cid:98) X ∈ X , we know there exists an optimal wealthprocess (cid:98) X ≡ (cid:98) X (1) and an associated optimal trading strategy (cid:98) H , such that (cid:98) X = 1+( (cid:98) H · P ) ≥ (cid:99) M := (cid:98) X (cid:98) R + (cid:82) · (cid:98) X s (cid:98) Y s d κ s is a supermartingale over [0 , ∞ ). The supermartingalecondition, by the same arguments that led to the derivation of the budget constraint in Lemma2.5, leads to the inequality E (cid:104)(cid:82) ∞ (cid:98) X t (cid:98) Y t d κ t (cid:105) ≤ (cid:99) M must be a martingale over [0 , ∞ ). We shall show that this extends to [0 , ∞ ], alongwith the other claims in the lemma.Since (cid:99) M is a martingale, the (non-negative c`adl`ag) deflated wealth process (cid:98) X (cid:98) R is a mar-tingale minus a non-decreasing process, so is a non-negative c`adl`ag supermartingale, and thus(by Cohen and Elliott [5, Corollary 5.2.2], for example) converges to an integrable limitingrandom variable (cid:98) X ∞ (cid:98) R ∞ := lim t →∞ (cid:98) X t (cid:98) R t (and moreover (cid:98) X t (cid:98) R t ≥ E [ (cid:98) X ∞ (cid:98) R ∞ ] , t ≥ (cid:99) M clearly also converges to an integrable random variable, byvirtue of the budget constraint. Thus, (cid:99) M also converges to an integrable random variable (cid:99) M ∞ := (cid:98) X ∞ (cid:98) R ∞ + (cid:82) ∞ (cid:98) X t (cid:98) Y t d κ t . By Protter [30, Theorem I.13], the extended martingale over[0 , ∞ ], ( (cid:99) M t ) t ∈ [0 , ∞ ] is then uniformly integrable, as claimed.The martingale condition gives E (cid:20) (cid:98) X t (cid:98) R t + (cid:90) t (cid:98) X s (cid:98) Y s d κ s (cid:21) = 1 , ≤ t < ∞ . Taking the limit as t → ∞ , using monotone convergence in the second term within theexpectation and utilising (6.31) yields lim t →∞ E [ (cid:98) X t (cid:98) R t ] = 0 , so that (cid:98) X (cid:98) R is a potential, as claimed.Using the uniform integrability of (cid:99) M and taking the limit as t → ∞ in E [ (cid:99) M t ] = 1 , t ≥ t →∞ E [ (cid:99) M t ] = E (cid:104) lim t →∞ (cid:99) M t (cid:105) = E [ (cid:98) X ∞ (cid:98) R ∞ ] + 1 , on using (6.31). Hence, we get E [ (cid:98) X ∞ (cid:98) R ∞ ] = 0 and, since (cid:98) X ∞ (cid:98) R ∞ is non-negative, we deducethat (cid:98) X ∞ (cid:98) R ∞ = 0, almost surely as claimed.We can now assemble these ingredients to arrive at the optimal wealth process formula(3.2). Applying the martingale condition again, this time over [ t, u ] for some t ≥ 0, we have E (cid:20) (cid:98) X u (cid:98) R u + (cid:90) u (cid:98) X s (cid:98) Y s d κ s (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = (cid:98) X t (cid:98) R t + (cid:90) t (cid:98) X s (cid:98) Y s d κ s , ≤ t ≤ u < ∞ . Taking thew limit as u → ∞ and using the uniform integrability of (cid:99) M we obtain E (cid:20) lim u →∞ (cid:18) (cid:98) X u (cid:98) R u + (cid:90) u (cid:98) X s (cid:98) Y s d κ s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = (cid:98) X t (cid:98) R t + (cid:90) t (cid:98) X s (cid:98) Y s d κ s , t ≥ , which, on using (cid:98) X ∞ (cid:98) R ∞ = 0, re-arranges to (cid:98) X t (cid:98) R t = E (cid:20) (cid:90) ∞ t (cid:98) X s (cid:98) Y s d κ s (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , t ≥ , which establishes (3.2), and the proof is complete. (cid:3) Proof of Theorem 3.1. Given the definitions of the sets C ( x ) and D ( y ) in (4.2) and (4.4),respectively, and the identification of the abstract value functions in (4.3) and (4.5) withtheir concrete counterparts in (2.6) and (2.18), Theorem 4.5 implies all the assertions ofTheorem 3.1, with the exception of the optimal wealth process formula (3.2) and the uniformintegrability of (cid:99) M := (cid:98) X ( x ) (cid:98) R ( y ) + (cid:82) · (cid:98) X s ( x ) (cid:98) Y s ( y ) d κ s , which are established by Proposition6.13. (cid:3) Examples We end with two examples. The first uses an incomplete market model with strict local mar-tingale deflators, which is covered in our framework. The market features a three-dimensionalBessel process for the market price of risk (MPR) of a stock which also has a stochasticvolatility. We consider the problem 2.6 with the measure κ satisfying d κ t = exp( − αt ) d t fora constant discount rate α > 0, so that(7.1) κ t = 1 α (cid:0) − e − αt (cid:1) , t ≥ . Since κ is absolutely continuous with respect to Lebesgue measure, we use the formalism inRemark 3.3. We then specialise the example to the Black-Scholes model, to confirm that weobtain results consistent with the example presented by Bouchard and Pham [2, Section 4].The market is of course complete in this simple case.We shall use a constant relative risk aversion (CRRA) utility function of the power form:(7.2) U ( x ) = x p p , p < , p (cid:54) = 0 , x ∈ R + . TILITY FROM INTER-TEMPORAL WEALTH 37 The case p = 0 corresponds formally to logarithmic utility, U ( x ) = log( x ), and setting p = 0in the results for the power utility function does indeed recover the results for logarithmicutility, as can be verified by carrying out the analysis directly for that case. Example . Take an infinite horizon complete stochastic basis (Ω , F , F := ( F t ) t ≥ , P ), with F satisfying the usual hypotheses. Let ( W, W ⊥ ) be a two-dimensional Brownian motion. Wetake F to be the augmented filtration generated by ( W, W ⊥ ).Let B denote the process which solves the stochastic differential equationd B t = 1 B t d t + d W t =: λ t d t + d W t , B = 1 . The process B is the so-called three-dimensional Bessel process. The process λ := 1 /B will bethe market price of risk of a stock with price process P and stochastic volatility process σ > (cid:102) W := ρW + (cid:112) − ρ W ⊥ , and with ρ ∈ [ − , 1] some F -adapted stochastic correlation. We need not specify the dynamics of σ or ρ any further forthe purposes of the example. The stock price dynamics are given byd P t = σ t P t d B t = σ t P t ( λ t d t + d W t ) . Note that this model satisfies the so-called structure condition of Pham et al [29], because P admits the decomposition P = P + L + A with L ∈ M , loc a locally square-integrable localmartingale null at zero and A a predictable process of finite variation null at zero, and suchthat A = (cid:82) · (cid:98) λ s d (cid:104) L (cid:105) s for a predictable process (cid:98) λ .Take a constant relative risk aversion (CRRA) utility function as in (7.2), with the measure κ given by (7.1), so that γ t = e αt , t ≥ 0. The primal value function is u ( x ) := sup X ∈X ( x ) E (cid:20)(cid:90) ∞ e − αt U ( X t ) d t (cid:21) , x > . The wealth process satisfies(7.3) d X t = σ t π t ( λ t d t + d W t ) , X = x, where π = HS is the trading strategy expressed in terms of the wealth placed in the stock,with H the process for the number of shares.With E ( · ) denoting the stochastic exponential, the supermartingale deflators in this modelare given by local martingale deflators of the form(7.4) Z := E ( − λ · W − ψ · W ⊥ ) , for an arbitrary process ψ satisfying (cid:82) t ψ s d s < ∞ almost surely for all t ≥ 0, with each such ψ leading to a different deflator: this market is of course incomplete. Let Ψ denote the setof such integrands ψ . In the case that σ and ρ are deterministic, the market is completeand there is a unique local martingale deflator Z (0) := E ( − λ · W ). It is well-known (see forinstance Larsen [24, Example 2.2]) that Z (0) is a strict local martingale and, what is more,that Z (0) = λ and that λ is square integrable. The strict local martingale property is inheritedby Z in (7.4), for any choice of integrand ψ .The supermartingales R ∈ R are given by R = exp (cid:0) − (cid:82) · β s d s (cid:1) Z and the inter-temporalwealth deflators Y ∈ Y by Y = βR , that is,(7.5) Y t = β t exp (cid:18) − (cid:90) t β s d s (cid:19) Z t , t ≥ , with β ∈ B , so (cid:82) · β s d s < ∞ almost surely. The process M := XR + (cid:82) · X s Y s d s is given as(7.6) M t := X t R t + (cid:90) t X s Y s d s = x + (cid:90) t R s ( σ s π s − λ s X s ) d W s − (cid:90) t X s R s ψ s d W ⊥ s , t ≥ , which is a non-negative local martingale and thus a supermartingale.The convex conjugate of the utility function is V ( y ) := − y q /q, y > 0, where q < , q (cid:54) = 0 isthe conjugate variable to p , satisfying 1 − q = (1 − p ) − . The dual value function is given by v ( y ) := inf Y ∈Y E (cid:20)(cid:90) ∞ e − αt V ( yY t e αt ) d t (cid:21) , y > . The dual minimisation involves both an optimisation over the local martingale deflators Z ∈ Z as well as over the auxiliary dual control β ∈ B , since the wealth-path deflators Y ∈ Y aregiven by (7.5).Denote the unique dual minimiser by (cid:98) Y ∈ Y , given by (cid:98) Y = (cid:98) β exp (cid:18) − (cid:90) · (cid:98) β s d s (cid:19) (cid:98) Z = (cid:98) β (cid:98) R, where (cid:98) β ∈ B is the optimal auxiliary dual control, (cid:98) R ∈ R denotes the optimal incarnation ofthe supermartingale R and (cid:98) Z denotes the optimal local martingale deflator, given by (cid:98) Z := E ( − λ · W − (cid:98) ψ · W ⊥ ) , for some optimal integrand (cid:98) ψ in (7.4). For use below, define the non-negative martingale H by(7.7) H t := E (cid:20) (cid:90) ∞ e − α (1 − q ) s (cid:98) Y qs d s (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , t ≥ . Using (3.4), the optimal wealth process is given by(7.8) ( (cid:98) X t ( x )) − (1 − p ) = u (cid:48) ( x )e αt (cid:98) Y t , t ≥ . By (3.5) the optimisers satisfy the saturated budget constraint(7.9) E (cid:20)(cid:90) ∞ (cid:98) X t ( x ) (cid:98) Y t d t (cid:21) = x. The relations (7.8) and (7.9) yield(7.10) (cid:98) X t ( x ) = xH e − α (1 − q ) t (cid:98) Y − (1 − q ) t , t ≥ . Using the result (7.10) in the right-hand-side of (3.6), the optimal wealth process then alsosatisfies (cid:98) X t ( x ) (cid:98) R t = xH E (cid:20) (cid:90) ∞ t e − α (1 − q ) s (cid:98) Y qs d s (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , t ≥ . More pertinently, the optimal martingale (cid:99) M , corresponding to the process in (7.6) at theoptimum, is computed as(7.11) (cid:99) M t := (cid:98) X t ( x ) (cid:98) R t + (cid:90) t (cid:98) X s ( x ) (cid:98) Y s d s = xH H t , t ≥ , so is indeed a martingale.By martingale representation, (cid:99) M will have a stochastic integral representation which, with-out loss of generality, can be written in the form(7.12) (cid:99) M t = x + (cid:90) t (cid:98) R s (cid:98) X s ( x )( ϕ s − qλ s ) d W s + (cid:90) t (cid:98) R s (cid:98) X s ( x ) ξ s d W ⊥ s , t ≥ , for some integrands ϕ, ξ . Comparing with the representation in (7.6) at the optimum yieldsthe optimal trading strategy in terms of the optimal portfolio proportion (cid:98) θ := (cid:98) π/ (cid:98) X ( x ), and TILITY FROM INTER-TEMPORAL WEALTH 39 the optimal integrand (cid:98) ψ , as(7.13) (cid:98) θ t := (cid:98) π t (cid:98) X t ( x ) = λ t σ t (1 − p ) + ϕ t σ t , (cid:98) ψ t = − ξ t , t ≥ . In particular, the process ϕ records the correction to the Merton-type strategy λ/ ( σ (1 − p ))due to the stochastic volatility and correlation.This is as far as one can go without computing explicitly the dual minimiser (cid:98) Y , which istypically impossible in closed form for power utility, except for some special cases such as aBlack-Scholes model (as we shall show further below).For the special case of logarithmic utility, one can set p = 0 and q = 0 in the results forpower utility, which gives that H = 1 /α is constant, and so (cid:99) M = x is also constant, yielding (cid:98) θ t = λ t σ t , (cid:98) ψ t = 0 , t ≥ , giving the classic myopic trading strategy for logarithmic utility (and the correction to theMerton strategy satisfies ϕ = qλ = 0 for q = 0, as it should).In particular, since (cid:98) ψ ≡ 0, the dual optimiser is given as(7.14) (cid:98) Y = (cid:98) β exp (cid:18) − (cid:90) · (cid:98) β s d s (cid:19) Z (0) , for some optimal auxiliary dual control (cid:98) β ∈ B , with Z (0) = E ( − λ · W ) the minimal localmartingale deflator. Moreover, setting q = 0 in (7.10) and using H = 1 /α gives the optimalwealth process in the form(7.15) (cid:98) X t ( x ) = αx e − αt (cid:98) Y t , t ≥ . But, using the optimal strategy (cid:98) π = ( λ/σ ) (cid:98) X ( x ) in the wealth SDE (7.3), we also computethat(7.16) (cid:98) X t ( x ) = xZ (0) t , t ≥ . Equating the two expressions for (cid:98) X ( x ) in (7.15) and (7.16), and then using (7.14), yields thatthe optimal auxiliary dual control is also constant, and given by(7.17) (cid:98) β t = α, t ≥ . These results for logarithmic utility can of course be obtained by going directly through theanalysis from scratch in the manner above. Indeed, one can directly compute the dual valuefunction, as follows. Using the defintion (3.3) along with V ( y ) = − (1 + log( y )) for logarithmicutility, one expresses the dual value function as v ( y ) = 1 α ( V ( y ) − 1) + inf β ∈B ,ψ ∈ Ψ E (cid:20)(cid:90) ∞ e − αt (cid:18)(cid:90) t ( β s + 12 ( λ s + ψ s )) d s − log( β t ) (cid:19) d t (cid:21) . The optimisations over ψ and β can be carried out separately. Clearly, the term involving ψ is minimised by (cid:98) ψ ≡ 0, while an integration by parts in the remaining integrals yields v ( y ) = 1 α ( V ( y ) − 1) + inf β ∈B E (cid:20)(cid:90) ∞ e − αt (cid:18) α λ t + β t α − log( β t ) (cid:19) d t (cid:21) . The minimisation over β can then be carried out pointwise, yielding (7.17) and giving thedual optimiser for logarithmic utility: (cid:98) Y t = α exp ( − αt ) Z (0) t , t ≥ Example . If we specialise Example 7.1 to the casewhere λ and σ are constant, we are in a Black-Scholes market and the computations forpower utility can be carried out explicitly. We show this in order to verify that our formalismreproduces the results of the example in Bouchard and Pham [2, Section 4]. The market isnow complete, and there is a unique local martingale deflator given by Z = E ( − λW ). Thewealth-path deflators take the form Y = β exp (cid:18) − (cid:90) · β s d s (cid:19) Z, for some β ∈ B .With this structure, the same method as for Example 7.1 yields the same representation(7.10) for the optimal wealth process, where in this case the dual minimiser is given by(7.18) (cid:98) Y = (cid:98) β exp (cid:18) − (cid:90) · (cid:98) β s d s (cid:19) E ( − λW ) , for some optimal auxiliary dual control (cid:98) β ∈ B , and the martingale H in (7.7) has the samerepresentation with the dual minimiser in (7.18) in place.The process M of (7.6) is this time given by the same expression but without the integralinvolving ψ , so we have M t := X t R t + (cid:90) t X s Y s d s = x + (cid:90) t R s ( σπ s − λX s ) d W s , t ≥ . The optimal martingale (cid:99) M once again has the representation in (7.11), and has a stochasticintegral representation of the form in (7.12) but without the integral with respect to W ⊥ ,and we once again find an expression of the form in (7.13) for the optimal trading strategy.Our goal is to now compute the dual minimiser, by computing (cid:98) β , and to thus show that thecorrection ϕ to the Merton strategy is zero in this case.To compute (cid:98) β we examine the dual value function, which is expressed in the form v ( y ) = inf β ∈B V ( y ) E (cid:20)(cid:90) ∞ exp (cid:18) − α (1 − q ) t − q (cid:90) t β s d s (cid:19) β qt Z qt d t (cid:21) . Given the constant parameters of the model, one now makes the (not unreasonable) ansatzthat (cid:98) β is deterministic, and in fact constant. With this conjecture, one passes the expectationinside the integral, uses(7.19) E [ Z qu | F t ] = E ( − qλW ) t exp (cid:18) − q (1 − q ) λ u (cid:19) , ≤ t ≤ u, and computes the resultant expression to arrive at v ( y ) = inf β V ( y ) (cid:32) β q qβ + (1 − q )( α + qλ ) (cid:33) . Straightforward differentiation gives the (constant) optimiser as (cid:98) β = α + 12 qλ , and (7.18) then gives the dual minimiser. With this in place, one expresses the martingale H in the form H t = (cid:18) α + 12 qλ (cid:19) q E (cid:20) (cid:90) ∞ exp (cid:18)(cid:18) α + 12 qλ (cid:19) u (cid:19) Z qu (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , t ≥ . TILITY FROM INTER-TEMPORAL WEALTH 41 Once again, we take the expectation inside the integral and use (7.19), and we arrive at H t = (cid:18) α + 12 qλ (cid:19) − (1 − q ) E ( − qλW ) t , t ≥ . This in turn yields that the optimal martingale (cid:99) M is given by (cid:99) M t = x H t H = x E ( − qλW ) t , t ≥ , and the optimal wealth process is given by the representation (7.10) as (cid:98) X t ( x ) = x E ( − qλW ) t Z t , t ≥ . Thus, the processes (cid:99) M , (cid:98) X ( x ) are related according to (cid:98) X t ( x ) Z t = (cid:99) M t , t ≥ . We can now compute the optimal trading strategy. Using the dynamics of the wealth processfor any strategy π , given by (7.3) with constant parameters, we have that(7.20) X t Z t = x + (cid:90) t ( σπ s − λX s ) d W s , t ≥ . On the other hand, at the optimum, since (cid:98) X ( x ) Z = x E ( − qλW ), we have(7.21) (cid:98) X t ( x ) Z t = x − qλ (cid:90) t (cid:98) X s Z s d W s , t ≥ . Equating (7.20) at the optimum with (7.21) gives the optimal trading strategy as (cid:98) θ t ≡ (cid:98) π t (cid:98) X t ( x ) = λσ (1 − p ) , t ≥ , so the optimal strategy is the Merton strategy, as expected. References [1] C. 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