Duality for optimal consumption under no unbounded profit with bounded risk
DDUALITY FOR OPTIMAL CONSUMPTION UNDER NO UNBOUNDEDPROFIT WITH BOUNDED RISK
MICHAEL MONOYIOS
Abstract.
We give a definitive treatment of duality for optimal consumption over the in-finite horizon, in a semimartingale incomplete market satisfying no unbounded profit withbounded risk (NUPBR). Rather than base the dual domain on (local) martingale deflators,we use a class of supermartingale deflators such that deflated wealth plus cumulative deflatedconsumption is a supermartingale for all admissible consumption plans. This yields a strongduality, because the enlarged dual domain of processes dominated by deflators is naturallyclosed, without invoking its closure. In this way we automatically reach the bipolar of theset of deflators. We complete this picture by proving that the set of processes dominated bylocal martingale deflators is dense in our dual domain, confirming that we have identifiedthe natural dual space. In addition to the optimal consumption and deflator, we charac-terise the optimal wealth process. At the optimum, deflated wealth is a supermartingaleand a potential, while deflated wealth plus cumulative deflated consumption is a uniformlyintegrable martingale. This is the natural generalisation of the corresponding feature in theterminal wealth problem, where deflated wealth at the optimum is a uniformly integrablemartingale. We use no constructions involving equivalent local martingale measures. Thisis natural, given that such measures typically do not exist over the infinite horizon and thatwe are working under NUPBR, which does not require their existence. The structure ofthe duality proof reveals an interesting feature compared with the terminal wealth problem.There, the dual domain is L -bounded, but here the primal domain has this property, andhence many steps in the duality proof show a marked reversal of roles for the primal anddual domains, compared with the proofs of Kramkov and Schachermayer [17, 18]. Contents
1. Introduction 22. The market 52.1. Admissible consumption plans 52.2. Deflators for consumption plans 63. The consumption problem and its dual 73.1. On stochastic clocks 93.2. The dual problem 104. The duality theorem 105. Abstract bipolarity and duality 135.1. Alternative dual domains 145.2. Abstract bipolarity 155.3. Abstract duality 166. Budget constraint and bipolarity relations 176.1. The budget constraint 176.2. Bipolar relations 186.3. Local martingale deflators versus consumption deflators 247. Proofs of the duality theorems 268. An example: Bessel process with stochastic volatility and correlation 37References 39
Date : June17, 2020. a r X i v : . [ q -f i n . P M ] J u l MICHAEL MONOYIOS Introduction
This paper gives a definitive treatment of duality for the optimal consumption and invest-ment problem for an agent maximising cumulative discounted utility from consumption overan infinite horizon. This problem has a long history, first being solved in a constant coefficientcomplete Brownian model by Merton [21] using dynamic programming methods. The samemodel was studied in great detail, considering also issues such as non-negativity constraintson consumption, and bankruptcy, by Karatzas et al [12] using similar methods. Dualitymethods for a finite horizon version of the problem to maximise utility from consumptionand terminal wealth, in a complete Itˆo process market, were developed by Karatzas, Lehoczkyand Shreve [13]. The infinite horizon problem for utility from consumption in a completeItˆo market was treated via duality methods by Huang and Pag`es [9]. In an incomplete Itˆomarket, duality methods for the finite horizon problem of maximising utility from terminalwealth were developed in a seminal paper by Karatzas et al [14]. These methods were ex-tended to finite horizon problems including consumption and portfolio constraints (includingmarket incompleteness) by Cvitani´c and Karatzas [4] and Shreve and Xu [30]. Duality meth-ods in an incomplete market with general semimartingale asset prices were then developedfor the terminal wealth problem in a masterly contribution by Kramkov and Schachermayer[17, 18].The consumption problem in a semimartingale incomplete market, over a possibly infinitehorizon, remained an open problem to treat via duality methods until fairly recently, when asignificant advance was made by Mostovyi [22]. Working under the no-arbitrage condition ofNo Free Lunch with Vanishing Risk (NFLVR), Mostovyi [22] was able to show that most of thetenets of duality theory for utility maximisation, as espoused by Kramkov and Schachermayer[17, 18], do hold true for the infinite horizon consumption problem. This was extended byChau et al [2] to cover the case where the no-arbitrage condition was weakened to the NoUnbounded Profit with Bounded Risk (NUPBR) condition, so that one need not insist on theexistence of equivalent local martingale measures (ELMMs). This is a general observation,first made in explicit terms by Karatzas and Kardaras [11], that all one needs for a well-posed utility maximisation problem is the existence of a suitable class of dual variables, ordeflators, which need not be densities of ELMMs, and which multiplicatively deflate primalvariables to create local martingales or supermartingales. This fact was implicit in Karatzaset al [14], which did not use ELMMs at all, and to some extent was an underlying themein the work of Kramkov and Schachermayer [17, 18] who, despite working under NFLVR(so ELMMs were definitively assumed to exist), expanded the dual domain to a class ofsupermartingale deflators and found counter-examples where the dual minimiser was not thedensity of an ELMM. We note that in both Mostovyi [22] and Chau et al [2] the formulationcould encompass other problems, by varying the measure (a stochastic clock) that was used toaggregate utility from consumption over time. (These papers also incorporated the stochasticclock into the wealth dynamics, which amounts to a change of variable from a traditionalconsumption rate, and we shall say more on this below.) By varying this clock the approachin [22, 2] can treat the finite horizon utility from consumption problem, the terminal wealthproblem, as well as the finite horizon problem of utility from both consumption and terminalwealth (but only with the same utility function for both objectives).Given the above history, it is as well to point out where there is still work to do and, asthis is the focus of this paper, let us now turn to this and describe the contribution.First, we obtain a stronger duality statement than in Mostovyi [22] and Chau et al [2], inthe following sense. In [22] and [2] the initial dual domain was based either on martingaledeflators (in [22], working under NFLVR) or on local martingale deflators (in [2], work-ing under NUPBR). The dual domain was then defined as the closure (in an appropriatetopology) of processes dominated by some element of the set of deflators in question. The
ERPETUAL CONSUMPTION DUALITY 3 authors of [22, 2] were forced into taking the aforementioned closure in order to obtain aclosed dual domain, which could then be shown to be the bipolar of the original domain ofdeflators, and thus also the polar of the primal domain. Contrast this with the result ofKramkov and Schachermayer [17, Lemma 4.1] in the terminal wealth problem. There, onebegins with a dual domain of supermartingales (such that deflated admissible wealth is asupermartingale for all strategies), then enlarges this domain to consider random variablesdominated by the terminal value of some deflator. No closure is taken, but it is neverthelessshown that the enlarged dual domain is naturally closed, so one reaches the bipolar of theset of deflators, and perfect bipolarity between the primal and dual domains is achieved.Herein lies our first contribution: we are able to extend the prescription of Kramkov andSchachermayer [17]. First, we base our dual domain on a set of supermartingales, this timesuch that deflated wealth plus cumulative deflated consumption is a supermartingale for alladmissible consumption plans. Then, again in the spirit of [17], we enlarge the dual domainto encompass processes dominated by the deflators. Crucially, no closure needs to be taken.We show that the enlarged dual domain is closed in the appropriate topology, so that wereach the bipolar of the original domain of supermartingales and obtain the duality betweenthe primal and dual optimisation problems without having to take a closure in defining theenlarged dual domain. Finally, we show that our enlarged dual domain coincides with theclosure of processes dominated by local martingale deflators, that is, the dual domain usedin Chau et al [2]. Thus, the set of processes dominated by local martingale deflators is densein our dual domain. This result (Proposition 5.1) is confirmation that we have chosen thedual domain in just the right way to achieve a strong duality statement. The underlyingbipolarity results are obtained by exploiting the Stricker and Yan [28] version of the Op-tional Decomposition Theorem (ODT), which uses deflators rather than ELMMs, so we donot use any constructions whatsoever involving equivalent measures. We shall say more onthis aspect very shortly.The second strengthening of the results in Mostovyi [22] and Chau et al [2] is fundamental.In addition to the optimal consumption, we characterise the associated optimal wealth process(and by extension the optimal strategy). Somewhat surprisingly, neither of [22] or [2] madeany statement whatsoever regarding the optimal wealth. This turns out to be a satisfyinganalysis which shows shows that, at the optimum, deflated wealth is a supermartingale andalso a potential, decaying to zero, while deflated wealth plus cumulative deflated consumptionat the optimum is a uniformly integrable martingale. This is natural, though to the bestof our knowledge has not been shown before in a general semimartingale infinite horizonconsumption problem. It is the natural generalisation of the Kramkov and Schachermayer[17, 18] terminal wealth result that, at the optimum, deflated wealth is transformed from asupermartingale to a uniformly integrable martingale.The next aspect of our work concerns the use of, or more accurately the avoidance of, anyconstructions involving ELMMs. We are working on an infinite horizon, and it is well knownthat in this case hardly any models will admit ELMMs, because the candidate change ofmeasure density is not a uniformly integrable martingale over the infinite timescale. Whilethis can be dealt with by eliminating the tail σ -algebra in some way when wishing to useequivalent measures restricted to a finite horizon σ -field, many papers routinely completethe stochastic basis, thus including ultimate events of measure zero. In these circumstances,even over a finite horizon, one can fail to have equivalent measures. This critique applies tothe proofs of [22, Lemma 4.2] and [2, Lemma 1]. While we conjecture that these proofs canbe made sound, we bypass any such pitfalls by exploiting the Stricker and Yan [28] version ofthe ODT and so avoiding ELMMs. As we are working under NUPBR, where ELMMs mightnot exist at all (a case is point is a stock driven by a three-dimensional Bessel process, whichwe use in an example of a utility maximisation problem in our framework in Section 8), it is MICHAEL MONOYIOS natural to construct proofs which avoid any use of ELMMs if possible, and this is what wedo.Finally, the proof of the main duality theorem in our approach reveals an interesting struc-ture of the consumption problem compared with the terminal wealth problem. In contrastto [22, 2], we do not incorporate a stochastic clock into the wealth dynamics, so our con-sumption rate is with respect to calendar time. The change of variable used in [22, 2] wasconvenient in those papers, as it allowed the authors to assume that a constant “consump-tion” stream was allowed. This amounts to, in essence, a decaying real consumption rate. (Itis manifestly the case that with a true consumption rate, one cannot guarantee being ableto consume at a constant rate for ever.) By choosing to work with the real consumptionrate, two aspects of the problem’s underlying structure emerge. First, it naturally leads tothe correct supermartingale constraint that one should apply at the outset: that deflatedwealth plus cumulative deflated consumption is a supermartingale. This leads to the correctchoice of dual domain. Second, it reveals a role reversal for the primal and dual domainscompared with the terminal wealth problem of Kramkov and Schachermayer [17, 18]. In[17, 18], because the constant wealth X ≡ L ( P ). But in the consumption problem it is the primal domain that is bounded in L (with respect to an appropriate measure), because the constant supermartingale Y ≡ c ≡ ERPETUAL CONSUMPTION DUALITY 5 The market
We have an infinite horizon financial market containing d stocks and a cash asset, on acomplete stochastic basis (Ω , F , F := ( F t ) t ≥ , P ), with the filtration F satisfying the usualconditions of right-continuity and augmentation with the P -null sets of F . We shall use thecash asset as num´eraire, so work with discounted quantities. The (discounted) stock pricevector is given by a positive d -dimensional c`adl`ag semimartingale S = ( S , . . . , S d ).An agent with initial capital x > c = ( c t ) t ≥ , assumed to satisfy the minimal condition (cid:82) t c s d s < ∞ , almost surely, ∀ t ≥
0. The associated wealth process is X , given by(2.1) X t = x + ( H · S ) t − (cid:90) t c s d s, t ≥ , x > . In (2.1), ( H · S ) denotes the stochastic integral and the trading strategy H is a predictable S -integrable vector process for for the number of units of each stock held. Write C t := (cid:90) t c s d s, t ≥ , for the non-decreasing cumulative consumption process. Then, with(2.2) X := x + ( H · S )denoting the wealth process of a self-financing portfolio corresponding to strategy H , we havethe decomposition(2.3) X = X − C. Admissible consumption plans.
We will assume solvency at all times, so X ≥ x >
0, we call the pair (
H, c ) (or (
X, c ))an x -admissible investment-consumption strategy. If, for a consumption process c we canfind a predictable S -integrable process H such that ( H, c ) is an x -admissible investment-consumption strategy, then we say that c is an x -admissible consumption process or, briefly,an admissible consumption plan. Denote the set of x -admissible consumption plans by A ( x ):(2.4) A ( x ) := (cid:26) c ≥ ∃ H such that X := x + ( H · S ) − (cid:90) · c s d s ≥ , a.s (cid:27) , x > . For x = 1 we write A ≡ A (1), and we note that A ( x ) = x A for x >
0. We observe that A isa convex set.For c ≡
0, the wealth process is that of a self-financing portfolio, with wealth process X asin (2.2). Define X ( x ) as the set of almost surely non-negative self-financing wealth processeswith initial value x > X ( x ) := (cid:8) X : X = x + ( H · S ) ≥ , a.s. (cid:9) , x > . As for the admissible consumption plans, we write
X ≡ X (1), with X ( x ) = x X for x > X is a convex set.Given the wealth decomposition in (2.3), an equivalent characterisation of the admissibleconsumption plans is that there exists a self-financing wealth process which dominates cu-mulative consumption (such a wealth process will necessarily be non-negative, so will lie in X ( x )). MICHAEL MONOYIOS
Deflators for consumption plans.
The dual domain for our infinite horizon utilitymaximisation problem from inter-temporal consumption will be an extension of the one usedby Kramkov and Schachermayer [17, 18] for the terminal wealth problem. We shall refer tothe processes in the dual domain as deflators (or, sometimes, as consumption deflators , if weneed to distinguish them from the corresponding deflators in the absence of consumption).Define the set of positive c`adl`ag processes such that deflated wealth plus cumulative de-flated consumption is a supermartingale for every admissible consumption plan:(2.5) Y ( y ) := (cid:8) Y > , c`adl`ag , Y = y : XY + (cid:82) c s Y s d s is a supermartingale, ∀ c ∈ A (cid:9) . Using A rather than A ( x ) in (2.5) is without loss of generality, given A ( x ) = x A , x >
0. Asusual, we write
Y ≡ Y (1) and we have Y ( y ) = y Y for y >
0. In (2.5), the wealth process X isthe one on the left-hand-side of (2.1) or (2.3) with x = 1, so incorporating consumption. Wenote that, since ( X, c ) ≡ (1 ,
0) is an admissible consumption-investment pair, each Y ∈ Y ( y )is a supermartingale. The set Y is easily seen to be convex.Because c ≡ Y ( y ) includes the wealth deflators that were used byKramkov and Schachermayer [17, 18]. We shall write Y to denote such deflators, and theset of wealth deflators will be denoted by Y ( y ): Y ( y ) := (cid:8) Y > , c`adl`ag , Y = y : X Y is a supermartingale, for all X ∈ X (cid:9) . As before, we write Y ≡ Y (1) and we have Y ( y ) = y Y for y >
0. Since X ≡ X , each Y ∈ Y ( y ) is a supermartingale. The wealth deflators are also known as supermartingale deflators . Clearly, the set Y is convex.The set Z of local martingale deflators (LMDs) is composed of positive c`adl`ag local mar-tingales Z with unit initial value such that deflated self-financing wealth X Z , for all X ∈ X ,is a local martingale:(2.6) Z := (cid:8) Z > , c`adl`ag , Z = 1 : X Z is a local martingale, for all X ∈ X (cid:9) . Since the local martingale X Z ≥ X ∈ X , it is also a supermartingale and, since X ≡ X , each Z ∈ Z is also a supermartingale. The set Z contains the densityprocesses of equivalent local martingale measures (ELMMs) in situations where those wouldexist. We shall not, however, be using any constructions involving ELMMs, even restrictedto a finite horizon. We shall say more on this in Section 2.2.1.We observe that we have the inclusions Y ⊇ Y ⊇ Z . The standing no-arbitrage assumption we shall make is that the set of supermartingaledeflators is non-empty:(2.7) Y (cid:54) = ∅ . It is well-known that (2.7) is equivalent to the no unbounded profit with bounded risk(NUPBR) condition (also known as no arbitrage of the first kind, or NA ), weaker thanthe no free lunch with vanishing risk (NFLVR) condition, the latter requiring the existenceof ELMMs, which is often problematic over the infinite horizon, as we discuss in Section 2.2.1.There are a number of equivalent characterisations of NUPBR, including that the set Z ofLMDs is non-empty: see Karatzas and Kardaras [11], Kardaras [16], Takaoka and Schweizer[29] and Chau et al [2], as well as the recent overview by Kabanov, Kardaras and Song [10].2.2.1. Completion of the stochastic basis and equivalent measures.
As indicated earlier, weshall avoid completely any constructions which invoke equivalent local martingale measures(ELMMs), even restricted to a finite horizon. This is partly for aesthetic reasons: since wework under NUPBR and assume only the existence of various classes of deflators, which is
ERPETUAL CONSUMPTION DUALITY 7 the minimal requirement for well posed utility maximisation problems, it seems natural toseek proofs which use only deflators. This is what we do.There is some 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, ultimate events (as time t ↑ ∞ ) of 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, seeKaratzas and Shreve [15, Section 1.7].) One then has to proceed with caution when invokingarguments which utilise equivalent measures, by finding a consistent way to eliminate thetail σ -algebra from the picture when restricting to a finite horizon T < ∞ .One route forward is to not complete the space, as in Huang and Pag`es [9], in an infinitehorizon consumption model in a complete Brownian market. This is sound, though care isneeded to ensure that no results are used which require the usual hypotheses 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 (cid:83) ≤ t ≤ T (cid:48) F t , for some 0 ≤ T ≤ T (cid:48) < ∞ . This can bedone in a consistent way, and relies on an application of Carath´eodory’s extension theorem(Rogers and Williams [26, Theorem II.5.1]). One can then obtain equivalent measures in aninfinite horizon model when restricting such measures to any finite horizon. This procedureis carried out in a Brownian filtration in Karatzas and Shreve [15, Section 1.7], with acautionary example [15, Example 1.7.6], showing that augmenting the σ -field generated byBrownian motion over any finite horizon with null sets of the corresponding tail σ -algebrawould render invalid 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 infinite horizon financial model.We avoid having to invoke the above fixes, since we avoid all constructions involvingELMMs. In particular, in Section 6 we establish bipolarity results between the primal anddual domains using only the Stricker and Yan [28] version of the optional decompositiontheorem, relying on deflators rather than equivalent measures.We raise this issue because many papers appear to use a complete stochastic basis on aninfinite horizon, and at the same time then use equivalent measures over a finite horizon,without any statement about the elimination of the tail σ -field. This applies to some proofsin papers tackling the infinite horizon consumption problem (see Mostovyi [22, Lemma 4.2]and Chau et al [2, Lemma 1]). Presumably, the authors’ implicit assumption was that the tail σ -algebra was eliminated in a consistent way when invoking such arguments. But it shouldbe said that no such qualifying statements were made. We conjecture that the arguments in[22, 2] can be rendered sound by amendments as described above. Fortunately this is not anissue we need to address, as we bypass all these problems by new arguments which avoid theuse of ELMMs entirely.3. The consumption problem and its dual
Let U : R + → R be a utility function, strictly concave, strictly increasing, continuouslydifferentiable on R + and satisfying the Inada conditions(3.1) lim x ↓ U (cid:48) ( x ) = + ∞ , lim x →∞ U (cid:48) ( x ) = 0 . MICHAEL MONOYIOS
To guarantee a well-posed consumption problem, one could also impose here the reasonableasymptotic elasticity condition of Kramkov and Schachermayer [17]:(3.2) AE( U ) := lim sup x →∞ xU (cid:48) ( x ) U ( x ) < . The condition in (3.2) was shown in [17] to be a minimal condition, in an arbitrary marketmodel, to guarantee that the terminal wealth utility maximisation problem satisfied all thetenets of a general duality theory. It was later shown, again by Kramkov and Schachermayer[18], that if one instead assumes a market model such that the weak condition of a finitedual value function holds, then this alternative set-up gives a consistent duality theory.Furthermore, finiteness of the dual problem, along with a minimal condition on the primalvalue function (to be finitely valued for at least one value of initial capital) so as to excludea trivial problem, implies the reasonable asymptotic elasticity condition. For this reason, weshall follow the spirit of Kramkov and Schachermayer [18] and just impose weak finitenessconditions on the primal and dual value functions so as to exclude trivial problems, and thenlater make the (standard) remark in the style of [18, Note 2] on how these are consistentwith (3.2) (see Remark 5.7).Let κ : (0 , ∞ ) → R + be a positive finite measure which will determine how utility ofconsumption is discounted through time, assumed to be almost surely absolutely continuouswith respect to Lebesgue measure and satisfying(3.3) d κ t d t ≤ , almost surely , t ≥ . For later use, define the positive process γ = ( γ t ) t ≥ as the reciprocal of ( d κ t / d t ) t ≥ :(3.4) γ t := (cid:18) d κ t d t (cid:19) − , t ≥ . Define the primal value function from optimal consumption by(3.5) u ( x ) := sup c ∈A ( x ) E (cid:20)(cid:90) ∞ U ( c t ) d κ t (cid:21) , x > . To exclude a trivial problem, we shall assume throughout that u ( x ) > −∞ for all x > E (cid:2)(cid:82) ∞ min[0 , U ( c t )] d κ t (cid:3) > −∞ .The supremum in (3.5) is written as one over consumption processes. This should notobscure the fact that an optimal consumption process must also determine an associatedoptimal wealth process (equivalently an optimal trading strategy). This is clear from thedefinition in (2.4), where the consumption process is defined with reference to the associatedinvestment strategy. Indeed, in traditional formulations of the problem, this is acknowledgedin the notation by writing the value function as a supremum over a pair of controls involvingeither ( X, c ) or (
H, c ). Our goal is to find an optimal consumption process (cid:98) c , but to alsocharacterise the associated optimal wealth process (cid:98) X . Note that no such characterisationof the optimal wealth process was given in either of Mostovyi [22] or Chau et al [2]. Thisturns out to be an interesting feature of the analysis, with a nice result (Proposition 7.14)incorporated into the main duality theorem: at the optimum, the deflated wealth processis a supermartingale and a potential, while the deflated wealth plus cumulative deflatedconsumption is a uniformly integrable martingale. These results are the natural extensionsof the result for the terminal wealth problem in Kramkov and Schachermayer [17, 18], inwhich optimal deflated wealth is a uniformly integrable martingale. Example . The example we are pri-marily interested in is the case where d κ t = e − αt d t , for some positive impatience parameter ERPETUAL CONSUMPTION DUALITY 9 α > γ t = e αt , t ≥
0, which isthe factor which inflates the natural deflators in the dual problem, as we shall see.The problem in (3.5) is then E (cid:2)(cid:82) ∞ e − αt U ( c t ) d t (cid:3) → max! We shall illustrate the solution ofsuch a problem with a stock driven by a three-dimensional Bessel process, and with stochasticvolatility and correlation, in Example 8.1.3.1. On stochastic clocks.
We discuss briefly some variations of the problem (3.5) whichcan be incorporated into our framework (but which are not the main focus of our analysis).In Mostovyi [22] and Chau et al [2] the measure κ is taken to be a stochastic clock , thatis, a non-decreasing, c`adl`ag adapted process satisfying κ = 0 , κ ∞ ≤ K < ∞ , a.s. , P [ κ ∞ > > , for some finite positive constant K . As shown by Mostovyi [22, Examples 2.5–2.9], by appro-priate choice of the stochastic clock a number of different problems can be included withinthe framework of (3.5), such as the terminal wealth problem, the finite horizon consumptionproblem, the finite horizon consumption and terminal wealth problem (though only withthe same utility function for both consumption and terminal wealth), as well as the infinitehorizon problem in Example 3.1. The same observation applies to our problem, provided wechoose the measure κ to be a stochastic clock of the appropriate type. Our primary focus,however, is to give a definitive treatment of the traditional infinite horizon discounted utilityof consumption problem.Note also that in [22, 2], the stochastic clock was incorporated into the wealth dynamics:for some process ¯ c , (2.1) was replaced by X t = x + ( H · S ) t − (cid:90) t ¯ c s d κ s , t ≥ , x > . Thus, the process ¯ c (let us call it a pseudo-consumption rate, to distinguish it from our vari-able) of those papers involves a change of variable from our consumption rate. The approachin [22, 2] allows for a constant positive pseudo-consumption rate, which can sometimes bemathematically convenient. With a true consumption rate and an infinite horizon, a constantconsumption plan is not possible. Each approach can be converted to the other, as we nowillustrate.For concreteness, suppose the measure κ is as in (3.3). The pseudo-consumption rate ¯ c isthen related to the real consumption rate by ¯ c t = γ t c t , t ≥
0. The problems considered in[22, 2] are of the form(3.6) E (cid:20)(cid:90) ∞ U ( t, γ t c t ) d κ t (cid:21) → max !for some time dependent utility function U ( · , · ). (This utility was also stochastic in [22, 2],but this makes no difference to the argument here.) To make the problem in (3.6) equivalentto our problem in (3.5) requires U ( t, γ t c t ) = U ( c t ) almost surely for all t ≥
0, and this iseasy to satisfy. For example, if γ t = e αt , t ≥ U ( · ) = log( · ) is logarithmic utility, wechoose U ( t, ¯ c ) = log(¯ c ) − αt . If U ( c ) = c p /p, p < , p (cid:54) = 0 is power utility, then we choose U ( t, ¯ c ) = e − αpt ¯ c p /p . Hence, we can always restore a problem of the form in (3.5) (equivalentto the problems in [22, 2] up to an additive or multiplicative constant, typically).We choose in this work to adopt the classical definition of consumption. Part of ourreason for doing so is to make very transparent the underlying supermartingale constraint ondeflated wealth plus cumulative deflated consumption that one must apply, if one is to showhow the program of Kramkov and Schachermayer [17, 18], suitably modified and extended,creates a natural procedure for characterising the classical consumption duality. As will beseen, this reveals an interesting role reversal of the primal and dual domains in many steps ofthe proofs, compared with the terminal wealth problem, because it turns out that the primal domain in the consumption problem is L -bounded (with respect to a suitable measure), butit is the dual domain that has this property in the terminal wealth case. 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 ˜ c is the un-discounted consumption process, then theproblem in (3.5) is E (cid:2)(cid:82) ∞ U (˜ c t /A t ) d κ t (cid:3) → max! We can define another utility function (cid:101) U : R → R such that (cid:101) U ( A t , ˜ c t ) = U (˜ c t /A t ) , t ≥
0, and the problem in (3.5) can then betransported to one in terms of the raw (un-discounted) consumption rate. For example, if γ t = e αt , t ≥ U ( · ) = log( · ) is logarithmic utility, we choose (cid:101) U ( A, ˜ c ) = log(˜ c ) − log( A ).If U ( c ) = c p /p, p < , p (cid:54) = 0 is power utility, then we choose (cid:101) U ( A, ˜ c ) = A − p ˜ c p /p . Remark . In the problem (3.5) we can allow U ( · ) to be stochastic,so to also depend on ω ∈ Ω in an optional way, as done by Mostovyi [22]. The analysis isunaffected, as the reader can easily verify, so one can read the proofs with a stochastic utilityin mind and with dependence on ω ∈ Ω suppressed throughout.3.2.
The dual problem.
Let V : R + → R denote the convex conjugate of U ( · ), 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 x> [ V ( y ) + xy ] , x > , as well as V ( · ) = − I ( · ) = − ( U (cid:48) ) − ( · ), where I ( · ) denotes the inverse of marginal utility. Inparticular, we have the inequality(3.7) V ( y ) ≥ U ( x ) − xy, ∀ x, y > , with equality iff U (cid:48) ( x ) = y. For each consumption deflator Y ∈ Y ( y ) defined in (2.5), define a process Y γ by(3.8) Y γt := γ t Y t , t ≥ , where γ was defined in (3.4). For later use, denote the set of such processes by (cid:101) Y ( y ):(3.9) (cid:101) Y ( y ) := { Y γ : Y γ is given by (3.8), with Y ∈ Y ( y ) } , y > , so the set (cid:101) Y ( y ) is in one-to-one correspondence with the set Y ( y ) of consumption deflators.As usual, we write (cid:101) Y ≡ (cid:101) Y (1), and we have (cid:101) Y ( y ) = y (cid:101) Y for y > v : R + → R defined by(3.10) v ( y ) := inf Y ∈Y ( y ) E (cid:20)(cid:90) ∞ V ( γ t Y t ) d κ t (cid:21) , y > . We shall assume throughout that v ( y ) < ∞ for all y > The duality theorem
Here is the main result, the perpetual consumption duality. It is somewhat stronger andmathematically more robust than previous results. We describe how the theorem differsfrom, and in which senses it strengthens, existing results, after presenting the theorem.
Theorem 4.1 (Perpetual consumption duality under NUPBR) . Define the primal consump-tion problem by (3.5) and the corresponding dual problem by (3.10) . Assume (2.7) , (3.1) andthat u ( x ) > −∞ , ∀ x > , v ( y ) < ∞ , ∀ y > . Then:
ERPETUAL CONSUMPTION DUALITY 11 (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) c ( x ) ∈ A ( x ) and (cid:98) Y ( y ) ∈ Y ( y ) exist and are unique,so that u ( x ) = E (cid:20)(cid:90) ∞ U ( (cid:98) c 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 > . (iii) With y = u (cid:48) ( x ) (equivalently, x = − v (cid:48) ( y ) ), the primal and dual optimisers are relatedby (4.1) U (cid:48) ( (cid:98) c t ( x )) = γ t (cid:98) Y t ( y ) , equivalently , (cid:98) c t ( x ) = − V (cid:48) ( γ t (cid:98) Y t ( y )) , t ≥ , and satisfy (4.2) E (cid:20)(cid:90) ∞ (cid:98) c t ( x ) (cid:98) Y t ( y ) d t (cid:21) = xy. Moreover, the associated optimal wealth process (cid:98) X ( x ) is given by (4.3) (cid:98) X t ( x ) (cid:98) Y t ( y ) = E (cid:20) (cid:90) ∞ t (cid:98) c 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:98) X ( x ) (cid:98) Y ( y ) + (cid:82) · (cid:98) c s ( x ) (cid:98) Y s ( y ) d s is a uniformly integrable martingale. (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) c t ( x )) (cid:98) c 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) . The proof of Theorem 4.1 will be given in Section 7, and will rely on bipolarity results andan abstract version of the duality stated in Section 5, with the bipolarity results proven inSection 6. A duality result of this form was established by Mostovyi [22] under NFLVR. Thiswas strengthened to a result under NUPBR by Chau et al [2]. Compared to these papers,Theorem 4.1 makes a stronger statement in other ways.First, we characterise the optimal wealth process, a statement that was missing from[22, 2]. This turns out to be a nice result to prove (see Proposition 7.14), showing that theoptimal process (cid:98) X (cid:98) Y is a supermartingale and a potential, while (cid:98) X (cid:98) Y + (cid:82) · (cid:98) c s (cid:98) Y s d s is a uniformlyintegrable martingale. This is the natural extension of the result that the optimally deflatedoptimal wealth process is a uniformly integrable martingale in the terminal wealth problem(see Kramkov and Schachermayer [17, 18]), and confirms that the supermartingale conditionwe placed on the process XY + (cid:82) · c s Y s d s for admissibility is the right criterion to start from.Further, as we shall see in the course of proving Theorem 4.1, the dual domain Y ( y ) willneed to be enlarged, in a spirit akin to Kramkov and Schachermayer [17, 18], to considerprocesses which are dominated by some element of the original dual domain. This enlarge-ment, as is known from the terminal wealth scenario of [17, 18], is needed in order to reachthe bipolar of the original dual domain, so that the (enlarged) dual domain is closed in anappropriate topology. This in turn guarantees that a unique dual optimiser will exist. Thisis one of the key contributions made in [17, 18]. One does not assume a priori that eitherthe primal or dual domains are closed.Here, for the consumption problem, we shall see that we do not need to enlarge the primaldomain, only the dual domain. Mostovyi [22] and Chau et al [2] found a similar phenomenon,but with the important caveat that they took the enlarged dual domain to be the closure (inthe appropriate topology) of the set of processes dominated by local martingale deflators (in[2]) or martingale deflators (in [22]) . Here, we do not explicitly make the dual domain closed (in the manner of [22, 2]) byconstruction, so we obtain a stronger result. We merely enlarge the dual domain in a man-ner analogous to Kramkov and Schachermayer [17, 18], by considering processes dominatedby consumption deflators, and then show that the enlarged domain is closed using super-martingale convergence results which exploit so-called Fatou convergence of processes. Wealso prove that our enlarged domain coincides with the closure of processes dominated bylocal martingale deflators (see Proposition 5.1), so coincides with that used in [2]. In otherwords, the domain used in [2] is dense in our domain. The proof of Proposition 5.1 will alsoreveal why the supermartingale convergence results, used to show that our enlarged domainis closed, cannot provide the same result for the (pre-closure) domains used in [22, 2]. Ba-sically, the limiting supermartingale is just that, a supermartingale, and it cannot be shownto be a (local) martingale deflator.This all reveals that, in a real sense, we have found just the right dual domain for a strongduality statement.Lastly, regarding some steps underlying the proof of Theorem 4.1, and in particular thearguments in Section 6 used to establish bipolarity relations connecting the primal and dualdomains, our proofs make no use at any point of constructions involving equivalent measures,such as ELMMs, but use only deflators. Since we are working under NUPBR this is natural,and in some senses even desirable. Moreover, as we have alluded to in Section 2.2.1, there arepotential complications in using equivalent measures when working on an infinite horizon.Such constructions involving ELMMs are used in Mostovyi [22, Lemma 4.2] and Chau et al[2, Lemma 1]. In both of these works, the stochastic basis is complete, so while we conjecturethat the proofs in those papers can be rendered correct with suitable adjustments, there aregood reasons for taking the course we follow here.In our scenario, therefore, we provide an unambiguously robust route through the proofswhich avoids any use of ELMMs. This, in addition to the features described above, of showingthat the naturally enlarged dual domain is closed, without taking its closure to guaranteethis, makes Theorem 4.1 a quite distinct infinite horizon consumption duality result fromthose in [22, 2].
Remark . As discussed inSection 3.1, one can incorporate a stochastic clock into the wealth dynamics, as done byMostovyi [22] and Chau et al [2]. Our entire program works with this change, and we pointout here how Theorem 4.1 would be altered. We modify the wealth dynamics (2.1) to X t = x + ( H · S ) t − (cid:90) t c s d κ s , t ≥ , x > , where κ is a stochastic clock of the form described in Section 3.1. The process c was denotedby ¯ c in Section 3.1, but for a clean notation we shall not make this adjustment here. Theprimal value function is still given by (3.5). The consumption deflators are also unchanged,but the key supermartingale constraint in (2.5) is altered to reflect the change of consumptionvariable, to: XY + (cid:82) c s Y s d κ s is a supermartingale, ∀ c ∈ A , where admissible consumption plans are still those for which the wealth process X is non-negative. In other words, one simply alters the measure used in the cumulative deflatedconsumption term in the fundamental supermartingale constraint. As a result, the form ofthe dual problem in (3.10) is altered to v ( y ) := inf Y ∈Y ( y ) E (cid:20)(cid:90) ∞ V ( Y t ) d κ t (cid:21) , y > , so one loses the extraneous process γ in the argument of V ( · ) in the definition of the dualvalue function, and hence in the expression for this function in item (ii) of the theorem. ERPETUAL CONSUMPTION DUALITY 13
Similar adjustments occur in the remaining results of Theorem 4.1. Thus, item (iii) of thetheorem is altered to:With y = u (cid:48) ( x ) (equivalently, x = − v (cid:48) ( y )), the primal and dual optimisers are related by U (cid:48) ( (cid:98) c t ( x )) = (cid:98) Y t ( y ) , equivalently , (cid:98) c t ( x ) = − V (cid:48) ( (cid:98) Y t ( y )) , t ≥ , and satisfy E (cid:20)(cid:90) ∞ (cid:98) c t ( x ) (cid:98) Y t ( y ) d κ t (cid:21) = xy, with the associated optimal wealth process (cid:98) X ( x ) given by (cid:98) X t ( x ) (cid:98) Y t ( y ) = E (cid:20) (cid:90) ∞ t (cid:98) c 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:98) X ( x ) (cid:98) Y ( y ) + (cid:82) · (cid:98) c s ( x ) (cid:98) Y s ( y ) d κ s is a uniformly integrable martingale.Item (iv) of the theorem is altered to: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) c t ( x )) (cid:98) c 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) . Abstract bipolarity and duality
In this section we state a bipolarity result in abstract form, leading to an abstract dualitytheorem, from which Theorem 4.1 will follow. Proofs of these results will follow in subsequentsections.Set Ω := [0 , ∞ ) × Ω. Let G denote the optional σ -algebra on Ω , that is, the sub- σ -algebraof B ([0 , ∞ )) ⊗ F generated by evanescent sets and stochastic intervals of the form (cid:74) T, ∞ (cid:74) forarbitrary stopping times T . Define the measure µ := κ × P on ( Ω , G ). On the resulting finitemeasure 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 (3.5) and (3.10) are nowconsidered as subsets of L ( µ ). The abstract primal domain C ( x ) is identical to the set ofadmissible consumption plans, now considered as a subset of L ( µ ):(5.1) C ( x ) := { g ∈ L ( µ ) : g = c, µ -a.e., for some c ∈ A ( x ) } , x > . As always we write
C ≡ C (1), with C ( x ) = x C for x >
0, and the set C is convex. (Since C = A we do not really need to introduce the new notation, and do so only for some notationalsymmetry in the abstract formulation.) In the abstract notation, the primal value function(3.5) is written as(5.2) u ( x ) := sup g ∈C ( x ) (cid:90) Ω U ( g ) d µ, x > . For the dual problem, the abstract dual domain is an enlargement of the original domainto accommodate processes dominated by the original dual variables. To this end, define theset(5.3) D ( y ) := { h ∈ L ( µ ) : h ≤ γY, µ -a.e., for some Y ∈ Y ( y ) } , y > . As usual, we write
D ≡ D (1), we have D ( y ) = y D for y >
0, and the set D is convex. Withthis notation, and since V ( · ) is decreasing, the dual problem (3.10) takes the form(5.4) v ( y ) := inf h ∈D ( y ) (cid:90) Ω V ( h ) d µ, y > . The enlargement of the dual domain from Y (equivalently, (cid:101) Y in (3.9)) to D is needed forthe same reason as in Kramkov and Schachermayer [17, 18] in the context of the terminalwealth problem (where one enlarged the dual domain from supermartingale deflators toelements of L ( P ) that were dominated by terminal values of supermartingale deflators).The enlargement will ensure that D is closed with respect to convergence in measure µ (proven in Lemma 6.7). This in turn ensures that we reach a perfect bipolarity betweenthe primal and dual domains (as given in Proposition 5.5), which is a key ingredient inestablishing full duality between the primal and dual problems. Contrast this enlargementwith the approach taken in Chau et al [2] and Mostovyi [22] as described immediately below.5.1. Alternative dual domains.
In Chau et al [2] (respectively, Mosotvyi [22]) the dualdomain was not based on the deflators Y ∈ Y but instead on the local martingale deflators Z ∈ Z (respectively, equivalent martingale deflators). Thus, translated into our formulation(so using a true rather than a pseudo-consumption rate), Chau et al [2] use, in place of D ( y ),a domain defined as the closure, with respect to the topology of convergence in measure µ ,of a set D ( y ), where D ( y ) is defined analogously to D ( y ) but with local martingale deflatorsreplacing the consumption deflators. Thus, with A ≡ cl( A ) denoting the closure of any set A ⊆ L ( µ ), we have(5.5) D ( y ) ≡ cl( D ) := cl (cid:8) h ∈ L ( µ ) : h ≤ yγZ, for some Z ∈ Z (cid:9) , y > . As usual we write D ≡ D (1), and D ( y ) = yD for y >
0, with the same convention for D . Inthis formulation, therefore, the dual value function is represented as in (5.4) but with D ( y )in place of D ( y ).The salient point here is the fact that the closure of D ( y ) has been taken in (5.5). Thereason for this will become transparent in the proofs of Section 6, but we outline the issuehere, and state a nice result (Proposition 5.1) which connects the domains D , D and D .In the approach of [2] (and also of [22], with martingale deflators in place of local martingaledeflators), if one does not take the aforementioned closure, it becomes impossible (as faras we can see) to prove that the dual domain is closed. It thus becomes impossible toobtain a perfect bipolarity between the primal and dual domains, on which the dualityproofs ultimately rest. The technical reason for this is that the closed property of D isestablished (see Lemma 6.7) using a supermartingale convergence result based on Fatouconvergence of processes. The limiting supermartingale in this procedure is known only tobe a supermartingale in Y , so is not guaranteed to be a local martingale deflator. This is thedriving force behind our choice of dual domain based on a supermartingale criterion. Theapproach in [2, 22] is simply not amenable to this procedure, which is why those papers hadto invoke the closure in (5.5).In this way, we strengthen the duality theorems in [22, 2], by not forcing the dual domainto be closed by construction. This point is well made by Rogers [25], who observes thathaving to take the closure of the dual domain in its definition “makes the statement of themain result somewhat weaker”. We do denigrate in any way, however, the advances made in[22, 2].What is more, we have the proposition below, which reaffirms in some sense that ourchoice of dual domain is the correct one: we have chosen it in just the right way to reach thebipolar of the original dual domain and hence the polar of the primal domain. Proposition 5.1.
With respect to the topology of convergence in measure µ , the set D := (cid:8) h ∈ L ( µ ) : h ≤ γZ, for some Z ∈ Z (cid:9) , is dense in the set D ≡ D (1) of (5.3) . That is, we have D = D ≡ cl( D ) . ERPETUAL CONSUMPTION DUALITY 15
The proof of Proposition 5.1 will be given in Section 6, alongside the proof of the bipolarityresult in Proposition 5.5 that is the subject of the next subsection.5.2.
Abstract bipolarity.
The abstract duality theorem relies on the abstract bipolarityresult in Proposition 5.5 below which connects the sets C and D . The result is of course inthe spirit of Kramkov and Schachermayer [17, Proposition 3.1].We shall sometimes employ the notation (cid:104) g, h (cid:105) := (cid:90) Ω gh d µ, g, h ∈ L ( µ ) . Let us recall some definitions, particularly the concepts of set solidity and the polar of aset.
Definition 5.2 (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 topologyof convergence in measure µ . Definition 5.3 (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 [1, Theorem 1.3], originally proven in a probability space, and adapted here to themeasure space ( Ω , G , µ ). Theorem 5.4 (Bipolar theorem, Brannath and Schachermayer [1], 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 5.5 (Abstract bipolarity) . Under the condition (2.7) , 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 ◦ , (5.6) h ∈ D ⇐⇒ (cid:104) g, h (cid:105) ≤ , ∀ g ∈ C , that is, D = C ◦ ;(5.7)(iii) C and D are bounded in L ( µ ) , and C is also bounded in L ( µ ) . The proof of Proposition 5.5 will be given in Section 6, where we shall establish the infinitehorizon budget constraint, giving a necessary condition for admissible consumption plans,and a reverse implication, leading to a sufficient condition for admissibility, culminating inthe full bipolarity relations once we enlarge the dual domain. The derivations in Section 6are quite distinct from previous approaches, and are the bedrock of the mathematical results.As indicated earlier, we shall establish the bipolarity results without any recourse whatsoeverto constructions involving ELMMs, by exploiting ramifications of the Stricker and Yan [28]version of the optional decomposition theorem.
Abstract duality.
Armed with the abstract bipolarity in Proposition 5.5, we have thefollowing abstract version of the convex duality relations between the primal problem (5.2)and its dual (5.4). The theorem shows that all the natural tenets of utility maximisationtheory, as established by Kramkov and Schachermayer [17] in the terminal wealth problemunder NFLVR, extend to infinite horizon inter-temporal problems under NUPBR, with weakunderlying assumptions on the primal and dual domains.
Theorem 5.6 (Abstract duality theorem) . Define the primal value function u ( · ) by (5.2) and the dual value function by (5.4) . Assume that the utility function satisfies the Inadaconditions (3.1) and that (5.8) u ( x ) > −∞ , ∀ x > , v ( y ) < ∞ , ∀ y > . Then, with Proposition 5.5 in place, we have: (i) u ( · ) and v ( · ) are conjugate: (5.9) 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,so that 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 > . (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 5.6 will be given in Section 7, and uses as its starting point thebipolarity result in Proposition 5.5.The duality proof itself follows some of the classical steps (with adaptations) of Kramkovand Schachermayer [17, 18], but there is an interesting role reversal for the primal anddual sets. In the terminal wealth problem, the dual domain is bounded in L ( P ), becausethe constant wealth process : Ω (cid:55)→ c ≡ L ( µ ). Instead, it turns out that L ( µ )-boundednessis satisfied by the primal domain. The upshot is that, in a number of places, the methodof proof used in [17, 18] for a property of the primal domain is applied in our case to acorresponding property in the dual domain (and vice versa). Examples include the proofs ofuniform integrability of the families ( U + ( g )) g ∈C ( x ) and ( V − ( h )) h ∈D ( y ) , a reversed applicationof the minimax theorem (replacing a maximisation with a minimisation and so forth) inproving conjugacy of the value functions, and some characterisations of the derivatives ofthe value functions at zero and infinity. We shall point out these features when provingthe results. This is one of the reasons for our choosing to give a complete, self-containedtreatment with full proofs.We conclude this section with a small remark (that is by now standard, but does needstating) on reasonable asymptotic elasticity as an alternative to assuming finiteness of thedual value function. ERPETUAL CONSUMPTION DUALITY 17
Remark . In Theorem 5.6 we have assumed only theminimal conditions in (5.8) to guarantee non-trivial primal and dual problems. It is well-known that, in place of the second condition in (5.8) of a finitely-valued dual problem, wecould have imposed the reasonable asymptotic elasticity condition of Kramkov and Schacher-mayer [17] as given in (3.2), along with the assumption that u ( x ) < ∞ for some x >
0. Then,as in Kramkov and Schachermayer [18, Note 2], these conditions would have implied that v ( y ) < ∞ for all y > Budget constraint and bipolarity relations
The budget constraint.
The first step in the proof of the duality theorem is to es-tablish bipolarity relations between the primal and dual domains. We shall do this in stages,first deriving the infinite horizon budget constraint. This yields the form of the dual problemas a byproduct. The derivation also lends itself to a discussion of the rationale for choosingthe dual domain to be the set Y ( y ) of consumption deflators, and what would have been theramifications of instead choosing the wealth deflators or the local martingale deflators as thedual variables. Lemma 6.1 (Infinite horizon budget constraint) . Let c ∈ A ( x ) be any admissible consump-tion plan and let Y ∈ Y ( y ) be any consumption deflator. We then have the infinite horizonbudget constraint : (6.1) E (cid:20)(cid:90) ∞ c t Y t d t (cid:21) ≤ xy, ∀ c ∈ A ( x ) , Y ∈ Y ( y ) . Proof.
Recall the wealth process X incorporating consumption in (2.1), and the decom-position in (2.3) into a self-financing wealth process X minus cumulative consumption C = (cid:82) · c s d s . We thus have, on using the Itˆo product rule on the process CY and re-arranging,(6.2) XY + (cid:90) · c s Y s d s = X Y − (cid:90) · C s − d Y s , for any c ∈ A ( x ) and any Y ∈ Y ( y ). Since XY + (cid:82) · c s Y s d s is a supermartingale and XY ≥ E (cid:20)(cid:90) t c s Y s d s (cid:21) ≤ xy, t ≥ . Letting t ↑ ∞ and using monotone convergence we obtain (6.1). (cid:3) Remark . The derivation of Lemma 6.1 allows usto give some of the rationale for choosing the dual domain as we did.Suppose instead that we chose the dual domain to be the set Y ( y ) of supermartingaledeflators. In that case, we would reach the analogue of (6.2) in the form XY + (cid:90) · c s Y s d s = X Y − (cid:90) · C s − d Y s , for any Y ∈ Y ( y ). The right-hand-side is a difference of supermartingales, so not necessarilya supermartingale, and we would fail to achieve the infinite horizon budget constraint.Suppose, on the other hand, that we chose the dual domain to be constructed from the set Z of local martingale deflators. This is the route taken by Chau et al [2] and by Mostovyi[22] (except that the deflators were martingales in [22], in tandem with the NFLVR scenarioin that paper.) We would then reach the analogue of (6.2) in the form XZ + (cid:90) · c s Z s d s = X Z − (cid:90) · C s − d Z s , for any Z ∈ Z . Now, X Z is a non-negative local martingale and thus a supermartingale, sousing this and that XZ ≥
0, we would obtain(6.3) E (cid:20)(cid:90) t c s Z s d s (cid:21) ≤ x − E (cid:20)(cid:90) t C s − d Z s (cid:21) , t ≥ . The process M := (cid:82) · C s − d Z s is a local martingale. With ( T n ) n ∈ N a localising sequence for M , so that E (cid:104)(cid:82) T n C s − d Z s (cid:105) = 0 , n ∈ N , (6.3) would convert to E (cid:104)(cid:82) T n c s Z s d s (cid:105) ≤ x, n ∈ N . Letting n ↑ ∞ and using monotone convergence we would obtain a budget constraint E (cid:2)(cid:82) ∞ c t Z t d t (cid:3) ≤ x . So far so good. The difficulty in taking this route would arise later,when enlarging the dual domain to try to reach its bipolar. One seeks to enlarge the dualdomain to processes which are dominated by some process in the original dual domain, andthen to show that the enlarged domain is closed with respect to convergence in measure µ .The closedness proof relies on exploiting Fatou convergence of supermartingales. The limitin this procedure is known to be a supermartingale, but there is no guarantee that it is alocal martingale deflator. So duality would ultimately fail, unless the enlarged dual domainwas made closed by explicit construction. This is why Mostovyi [22] (respectively, Chau etal [2]) used a construction of the form in (5.5), invoking the closure. Ultimately, as stated inProposition 5.1, all avenues reach the same goal, but the difference is that in our approachwe did not have to invoke a closure.From Lemma 6.1 we obtain the form of the dual problem to (3.5) by bounding the achiev-able utility in the familiar way. For any c ∈ A ( x ) and Y ∈ Y ( y ) we have E (cid:20)(cid:90) ∞ U ( c t ) d κ t (cid:21) ≤ E (cid:20)(cid:90) ∞ U ( c t ) d κ t (cid:21) + xy − E (cid:20)(cid:90) ∞ c t Y t d t (cid:21) = E (cid:20)(cid:90) ∞ ( U ( c t ) − c t γ t Y t ) d κ t (cid:21) + xy ≤ E (cid:20)(cid:90) ∞ V ( γ t Y t ) d κ t (cid:21) + xy, x, y > , the last inequality a consequence of (3.7). This motivates the definition of the dual problemassociated with the primal problem (3.5), with dual value function v ( · ) defined by (3.10).6.2. Bipolar relations.
In economic terms, the budget constraint (6.1) says that initialcapital can finance future consumption, and constitutes a necessary condition for admissibleconsumption processes. Indeed, another way of defining admissible consumption plans is toinsist that, at any time t ≥
0, current wealth (suitably deflated) must finance future deflatedconsumption. We would thus require X t Y t ≥ E (cid:20) (cid:90) ∞ t c s Y s d s (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , t ≥ , for all deflators Y ∈ Y ( y ). Re-arranging the above inequality, we have E (cid:20) (cid:90) ∞ c s Y s d s (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) ≤ X t Y t + (cid:90) t c s Y s d s, t ≥ . Taking expectations, one recovers the infinite horizon budget constraint provided that thesupermartingale condition in (2.5) holds. This is another justification for the choice of dualdomain as we have presented it.Setting x = y = 1 in (6.1), the budget constraint gives us that, for c ∈ A and Y ∈ Y , wehave E (cid:2)(cid:82) ∞ c t Y t d t (cid:3) ≤
1. We thus have the implications(6.4) c ∈ A = ⇒ E (cid:20)(cid:90) ∞ c t Y t d t (cid:21) ≤ , ∀ Y ∈ Y , ERPETUAL CONSUMPTION DUALITY 19 and(6.5) Y ∈ Y = ⇒ E (cid:20)(cid:90) ∞ c t Y t d t (cid:21) ≤ , ∀ c ∈ A . We wish to establish the reverse implications in some form, if need be by enlarging thedomains. First, we establish the reverse implication to (6.4) in Lemma 6.4 below. Thisrequires some version of the Optional Decomposition Theorem (ODT), whose original formis due to El Karoui and Quenez [6] in a Brownian setting. This was generalised to the locallybounded semimartingale case by Kramkov [19] , extended to the non-locally bounded caseby F¨ollmer and Kabanov [7], and to models with constraints by F¨ollmer and Kramkov [8].The relevant version of the ODT for us is the one due to Stricker and Yan [28], whichuses deflators (and in particular LMDs) rather then ELMMs. In the proof of Lemma 6.4we shall apply a part of the Stricker and Yan ODT which applies to the super-hedging ofAmerican claims, so is designed to construct a process which can super-replicate a payoff atan arbitrary time. The salient observation is that this result can also be used to dominatea consumption stream, which is how we shall employ it. For clarity and convenience of thereader, we state here the ODT results we need, and afterwards specify precisely which resultsfrom [28] 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 (2.6). Theorem 6.3 (Stricker and Yan [28] 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 admitsa decomposition of the form W = W + ( φ · S ) − A, where φ is a predictable S -integrable process such that Z ( φ · S ) 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 (6.6) W t = ess sup Z ∈Z ,T ∈T ( t ) Z t E [ Z T b T |F t ] , t ≥ . Part (i) of Theorem 6.3 is taken from [28, Theorem 2.1]. Part (ii) is a combination of [28,Lemma 2.4 and Remark 2].The following lemma establishes the reverse implication to (6.4).
Lemma 6.4.
Suppose c is a non-negative adapted c`adl`ag process that satisfies, for all Y ∈ Y , (6.7) E (cid:20)(cid:90) ∞ c t Y t d t (cid:21) ≤ . Then c ∈ A .Proof. Since c is assumed to satisfy (6.7) for all deflators Y ∈ Y , and since Z ⊆ Y , (6.7)is satisfied for any Z ∈ Z . For such a local martingale deflator, and for any stopping time T ∈ T , the integration by parts formula gives(6.8) C T Z T = (cid:90) T C s − d Z s + (cid:90) T c s Z s d s, T ∈ T , where C := (cid:82) · c s d s is the non-decreasing candidate cumulative consumption process. Theprocess M := (cid:82) · C s − d Z s is a local martingale. Let ( T n ) n ∈ N be a localising sequence for M , an almost surely increasing sequence of stopping times with lim n →∞ T n = ∞ a.s. such that thestopped process M T n t := M t ∧ T n , t ≥ n ∈ N .Therefore, E (cid:104)(cid:82) T ∧ T n C s − d Z s (cid:105) = 0 for each n ∈ N . Using this along with the finiteness of T ∈ T and the uniform integrability of M T n , we have E (cid:20)(cid:90) T C s − d Z s (cid:21) = E (cid:20) lim n →∞ (cid:90) T ∧ T n C s − d Z s (cid:21) = lim n →∞ E (cid:20)(cid:90) T ∧ T n C s − d Z s (cid:21) = 0 . Using this in (6.8) we obtain E [ Z T C T ] = (cid:90) T Z s c s d s ≤ , the last inequality a consequence of the assumption (6.7) and Z ⊆ Y . Since Z ∈ Z and T ∈ T were arbitrary, we have sup Z ∈Z ,T ∈T E [ Z T C T ] ≤ < ∞ . Thus, from part (ii) of Theorem 6.3, there exists a c`adl`ag process W that dominates C ,so W t ≥ C t , a . s ., ∀ t ≥
0, and ZW is a super-martingale for each Z ∈ Z . From (6.6), thesmallest such W given by W t = ess sup Z ∈Z ,T ∈T ( t ) Z t E [ Z T C T |F t ] , t ≥ , so that W ≤
1. Further, by part (i) of Theorem (6.3), there exists a predictable S -integrableprocess H and an adapted increasing process A , with A = 0, such that W has decomposition W = W + ( H · S ) − A , with Z ( H · S ) a local martingale for each Z ∈ Z , and E [ Z T A T ] < ∞ for all Z ∈ Z and T ∈ T .Since W dominates C , we can define a process X by X t := 1 + ( H · S ) t , t ≥ , which also dominates C , 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 C , so that c ∈ A . (cid:3) We can now assemble consequences of the budget constraint and of Lemma 6.4 which,combined with the bipolar theorem, gives the following polarity properties of the set A . Lemma 6.5 (Polarity properties of A ) . The set
A ≡ A (1) of admissible consumption planswith initial capital x = 1 is a closed, convex and solid subset of L ( µ ) . It is equal to thepolar of the set (cid:101) Y ≡ (cid:101) Y (1) of (3.9) with respect to measure µ : (6.9) A = (cid:101) Y ◦ , so that (6.10) A ◦ = (cid:101) Y ◦◦ , and A is equal to its bipolar: (6.11) A ◦◦ = A . Proof.
Lemma 6.4, combined with the implication in (6.4), gives the equivalence c ∈ A ⇐⇒ E (cid:20)(cid:90) ∞ c t Y t d t (cid:21) ≤ , ∀ Y ∈ Y . ERPETUAL CONSUMPTION DUALITY 21
In terms of the measure κ of (3.3) and the set (cid:101) Y in (3.9) of processes γY, Y ∈ Y , we have c ∈ A ⇐⇒ E (cid:20)(cid:90) ∞ c t Y γt d κ t (cid:21) ≤ , ∀ Y γ ∈ (cid:101) Y . Equivalently, in terms of the measure µ , we have(6.12) c ∈ A ⇐⇒ (cid:90) Ω cY γ d µ ≤ , ∀ Y γ ∈ (cid:101) Y . The characterisation (6.12) is the dual representation of A : A = (cid:110) c ∈ L ( µ ) : (cid:104) c, Y γ (cid:105) ≤ , for each Y γ ∈ (cid:101) Y (cid:111) . This says that A is the polar of (cid:101) Y , establishing (6.9) and thus (6.10).Part (i) of the bipolar theorem, Theorem 5.4, along with (6.9), imply that A 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 5.4 gives A ◦◦ ⊇ A with A ◦◦ the smallest closed, convex, solid set containing A ,But since A is itself closed, convex and solid, we have (6.11). (cid:3) Remark . There are other ways to obtain the closed, convex and solid properties of A .First, the equivalence (6.12) along with Fatou’s lemma yields that the set A is closed withrespect to the topology of convergence in measure µ . To see this, let ( c n ) n ∈ N be a sequencein A which converges µ -a.e. to an element c ∈ L ( µ ). For arbitrary Y γ ∈ (cid:101) Y we obtain, viaFatou’s lemma and the fact that c n ∈ A for each n ∈ N , (cid:90) Ω cY γ d µ ≤ lim inf n →∞ (cid:90) Ω c n Y γ d µ ≤ , so by (6.12), c ∈ A , and thus A is closed. Further, it is straightforward to establish theconvexity of A from its definition. Finally, solidity of A is also clear: if one can dominatea consumption plan c ∈ A with a self-financing wealth process, then one can also dominateany smaller consumption plan with the same portfolio.The next step is to attempt to reach some form of reverse polarity result to (6.9). It ishere that the enlargement of the dual domain from (cid:101) Y to the set D of (5.3) comes into play.To see why this enlargement is needed, we first observe from (6.5) that we have(6.13) Y γ ∈ (cid:101) Y = ⇒ (cid:104) c, Y γ (cid:105) ≤ , ∀ c ∈ A , which implies that(6.14) (cid:101) Y ⊆ A ◦ . We do not have the reverse inclusion, because we do not have the reverse implication to(6.13), so cannot write a full bipolarity relation between sets A and (cid:101) Y . The enlargementfrom (cid:101) Y to the set D resolves the issue, yielding the consumption bipolarity of Lemma 6.8below. This procedure, in the spirit of Kramkov and Schachermayer [17], requires us toestablish that the enlarged domain is closed in an appropriate topology. Here is the relevantresult. Lemma 6.7.
The enlarged dual domain
D ≡ D (1) of (5.3) is closed with respect to thetopology of convergence in measure µ . The proof of Lemma 6.7 will be given further below. First, we use the result of the lemmato establish the consumption bipolarity result below.
Lemma 6.8 (Consumption bipolarity) . Given Lemma 6.7, the set D is a closed, convex andsolid subset of L ( µ ) , and the the sets A and D satisfy the bipolarity relations (6.15) A = D ◦ , D = A ◦ . Proof.
For any h ∈ D there will exist an element Y γ ∈ (cid:101) Y such that h ≤ Y γ , µ -almosteverywhere. Hence, the implication (6.13) holds true with D in place of (cid:101) Y : h ∈ D = ⇒ (cid:104) c, h (cid:105) ≤ , ∀ c ∈ A , which yields the analogue of (6.14):(6.16) D ⊆ A ◦ . Combining (6.10) and (6.16) we have(6.17)
D ⊆ (cid:101) Y ◦◦ . Part (ii) of the bipolar theorem, Theorem 5.4, says that (cid:101) Y ◦◦ ⊇ (cid:101) Y and that (cid:101) Y ◦◦ is thesmallest closed, convex, solid set which contains (cid:101) Y . But D is also closed, convex and solid(closed due to Lemma 6.7, convexity following easily from the convexity of (cid:101) Y , and solidity isobvious), and by definition D ⊇ (cid:101) Y , so we also have(6.18) D ⊇ (cid:101) Y ◦◦ . Thus, (6.17) and (6.18) give(6.19) D = (cid:101) Y ◦◦ . In other words, in enlarging from (cid:101) Y to D we have succeeded in reaching the bipolar of theformer.Combining (6.19) and (6.10) we see that D is the polar of A ,(6.20) D = A ◦ , so we have the second equality in (6.15). From (6.20) we get D ◦ = A ◦◦ which, combinedwith (6.11), yields the first equality in (6.15), and the proof is complete. (cid:3) It remains to prove Lemma 6.7, which we used above. We recall the concept of Fatouconvergence of stochastic processes from F¨ollmer and Kramkov [8], that will be needed.
Definition 6.9 (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 [8, 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.With this preparation, we can now prove Lemma 6.7. ERPETUAL CONSUMPTION DUALITY 23
Proof of Lemma 6.7.
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 ≤ γ (cid:98) Y n , µ -a.e for some supermartingale (cid:98) Y n ∈ Y .From F¨ollmer and Kramkov [8, Lemma 5.2] there exists a sequence ( Y n ) n ∈ N of supermartin-gales with each Y n ∈ conv( (cid:98) Y n , (cid:98) Y n +1 , . . . ), and a supermartingale Y , such that ( Y n ) n ∈ N isFatou convergent on a dense subset τ of R + to Y .Define a supermartingale sequence ( (cid:98) V n ) n ∈ N by (cid:98) V n := X (cid:98) Y n , with X ∈ X . Once againfrom [8, Lemma 5.2] there exists a sequence ( V n ) n ∈ N of supermartingales with each V n ∈ conv( (cid:98) V n , (cid:98) V n +1 , . . . ) = X conv( (cid:98) Y n , (cid:98) Y n +1 , . . . ), and a supermartingale V , such that ( V n ) n ∈ N is Fatou convergent on τ to V . Since V n ∈ X conv( (cid:98) Y n , (cid:98) Y n +1 , . . . ) for each n ∈ N , wehave V n = X Y n , for Y n ∈ conv( (cid:98) Y n , (cid:98) Y n +1 , . . . ). Because the sequence ( Y n ) n ∈ N is Fatouconvergent on τ to the supermartingale Y , the sequence ( V n ) n ∈ N = ( X Y n ) n ∈ N is Fatouconvergent on τ to the supermartingale V = X Y . Since X Y is a supermartingale and X ∈ X , and because c ≡ Y ⊆ Y )we have Y ∈ Y .Finally, because h n ≤ γ (cid:98) Y n , µ -a.e.for each n ∈ N , we also have h ≤ γY, µ -a.e. (as we verifybelow) and thus h ∈ D , so D is closed.To verify that h n ≤ γ (cid:98) Y n , n ∈ N , µ -a.e. implies h ≤ γY, µ -a.e., we observe that, since Y n = (cid:80) N ( n ) k = n λ k (cid:98) Y k , we have(6.21) γY n = N ( n ) (cid:88) k = n λ k γ (cid:98) Y k ≥ N ( n ) (cid:88) k = n λ k h k , µ -a.e.Because the sequence ( Y n ) n ∈ N Fatou converges on the dense subset τ to the supermartingale Y ∈ Y , the left-hand-side of (6.21) Fatou converges on τ to γY . The right-hand-side of(6.21) converges in measure µ to h , so we conclude that h ≤ γY on a dense subset of R + ,and hence that h ≤ γY, µ -a.e., and the proof is complete. (cid:3) Remark . In the proof above, we could instead have defined the supermartingale sequence( (cid:98) V n ) n ∈ N by (cid:98) V n := X (cid:98) Y n + (cid:82) · c s (cid:98) Y ns d s for c ∈ A and X the associated wealth process, andthe proof would go through in the same manner. But it is simpler to consider the case c ≡ X is self-financing. The conclusionis unaltered because the limiting supermartingale Y ∈ Y ⊂ Y is indeed a consumptiondeflator.With the consumption bipolarity of Lemma 6.8, we have in fact established Proposition5.5, so let us confirm this. Proof of Proposition 5.5.
With the identification C = A (from the definition (5.1)), and theproperties of A established in Lemma 6.5, we have all the claimed properties of C in items(i) and (ii). The corresponding assertions for D follow from Lemma 6.8.Let us show that C is bounded in L ( µ ), and thus also in L ( µ ). Set Y ≡ C (cid:51) g = c ∈ A and D (cid:51) h ≤ γY = γ , we obtain (on using γ t ( d κ t / d t ) = 1 , t ≥ (cid:82) Ω g d µ ≤
1, so C is bounded in L ( µ ) and hence in L ( µ ).For the L -boundedness of D , we shall find a positive element g ∈ C and show that D isbounded in L ( g d µ ), and hence bounded in L ( µ ). Choose A (cid:51) c t ≡ c t := e − δt , t >
0, forsome δ >
1. It is easy to verify that with x = 1 and H ≡ X ≥ , µ -a.e.,so c ∈ A . We observe that C (cid:51) g ≡ c is strictly positive except on a set of µ -measure zero.We then have, for any h ∈ D , so h ≤ γY for some Y ∈ Y (satisfying E [ Y t ] ≤ , t ≥ (cid:90) Ω gh d µ ≤ E (cid:20)(cid:90) ∞ e − δt Y t d t (cid:21) = (cid:90) ∞ e − δt E [ Y t ] d t ≤ δ . Thus, D is bounded in L ( g d µ ) and hence bounded in L ( µ ). (cid:3) The L ( µ )-boundedness of the primal domain C is to be contrasted with the terminal wealthproblem of Kramkov and Schachermayer [17, 18], in which the dual domain is bounded in L ( P ). This is the source of a switching of roles of the primal and dual domains in theconsumption problem compared with the terminal wealth problem, and will manifest itselfon numerous occasions in the course of proving the duality theorem in the next section.6.3. Local martingale deflators versus consumption deflators.
We can now return tothe discussion of Section 5.1, in which we made comparisons with the approaches to bipolarityin Chau et al [2] and Mostovyi [22]. This will lead us to the proof of Proposition 5.1. Theproof will demonstrate that the approach in [22, 2] can get a fair way towards establishingbipolarity between A and the set (cid:101) Z , defined analogously to (cid:101) Y in (3.9), by (cid:101) Z := { Z γ : Z γ := γZ, Z ∈ Z} . One can get a little further by enlarging to D , but it is then necessary to invoke the closure D to reach full bipolarity. This establishes the result of the proposition, and shows how thedual domain we chose is not too big, and not too small, to establish bipolarity. A byproductof the proof is that it shows how results analogous to Mostovyi [22, Lemma 4.2] and Chau etal [2, Lemma 1], which give an equivalence between an admissible consumption plan and anappropriate budget constraint involving either local martingale deflators (in [2]) or martingaledeflators (in [22]) can be established without recourse to constructions involving equivalentmeasures, by judicious use of the Stricker and Yan [28] ODT, rather like the proof of Lemma6.4. As we pointed out in Section 2.2.1, this is both an aesthetic and mathematically desirablefeature. Proof of Proposition 5.1.
Consider a consumption plan with initial capital x = 1. Using thesame arguments as in Remark 6.2 we establish the analogue of (6.3) for x = 1:(6.22) E (cid:20)(cid:90) t c s Z s d s (cid:21) ≤ − E (cid:20)(cid:90) t C s − d Z s (cid:21) , t ≥ . The process M := (cid:82) · C s − d Z s is a local martingale. With ( T n ) n ∈ N a localising sequence for M , so that E (cid:104)(cid:82) T n C s − d Z s (cid:105) = 0 , n ∈ N , (6.22) converts to E (cid:104)(cid:82) T n c s Z s d s (cid:105) ≤ , n ∈ N .Letting n ↑ ∞ and using monotone convergence we obtain a budget constraint in the form E (cid:2)(cid:82) ∞ c t Z t d t (cid:3) ≤
1. We thus have the implications analogous to (6.4) and (6.5):(6.23) c ∈ A = ⇒ E (cid:20)(cid:90) ∞ c t Z t d t (cid:21) ≤ , ∀ Z ∈ Z , and(6.24) Z ∈ Z = ⇒ E (cid:20)(cid:90) ∞ c t Z t d t (cid:21) ≤ , ∀ c ∈ A . We can then establish the reverse implication to (6.23) in exactly the same manner as inthe proof of Lemma 6.4. In other words, if c is a non-negative process satisfying the budgetconstraint, then it is an admissible consumption plan. That is, we have(6.25) E (cid:20)(cid:90) ∞ c t Z t d t (cid:21) ≤ , ∀ Z ∈ Z = ⇒ c ∈ A . Thus, following the same arguments as in the proof of Lemma 6.5, we have, from (6.23) and(6.25), A = (cid:110) c ∈ L ( µ ) : (cid:104) c, γZ (cid:105) ≤ , for each Z γ = γZ ∈ (cid:101) Z (cid:111) , ERPETUAL CONSUMPTION DUALITY 25 so that A is the polar of (cid:101) Z : A = (cid:101) Z ◦ , implying(6.26) A ◦ = (cid:101) Z ◦◦ , and that A is equal to its bipolar: A ◦◦ = A , by the same arguments as in the proof of Lemma 6.5.Now, (6.24) gives us that (cid:101) Z ⊆ A ◦ , by the same arguments that led to (6.14). We do not have the reverse inclusion, becausewe do not have the reverse implication to (6.24), so cannot write a full bipolarity relationbetween sets A and (cid:101) Z . To this end, one can try enlarging the dual domain from (cid:101) Z to D , inthe same manner that we enlarged from (cid:101) Y to the set D when using consumption deflators asdual variables. This yields, in the same manner as we established (6.16),(6.27) D ⊆ A ◦ . Combining (6.26) and (6.27) we have(6.28) D ⊆ (cid:101) Z ◦◦ . Here is the crucial point: to establish the reverse inclusion to (6.28) would require thatthe set D is closed with respect to the topology of convergence in measure µ . But thearguments we used for the proof of Lemma 6.7 to establish this property for the domain D ,break down when applied to the set D , because the limiting supermartingale in the Fatouconvergence argument is known only to be a supermartingale, and cannot be shown to be alocal martingale deflator. So we are forced to enlarge D itself to its closure D .With this enlargement to D , we first show that (6.27), and hence (6.28), extend from D to D . Suppose ( h n ) n ∈ N is a sequence in D ⊆ D that converges µ -a.e. to some element h ∈ L ( µ ). Then h ∈ D , since D is closed in µ -measure (and a µ -a.e. convergent sequencemust also converge in measure µ ). Using Fatou’s lemma and that h n ∈ D we have, for any c ∈ A (cid:104) c, h (cid:105) = (cid:104) c, lim n →∞ h n (cid:105) ≤ lim n →∞ (cid:104) c, h n (cid:105) ≤ . Thus, we get the implication h ∈ D = ⇒ (cid:104) c, h (cid:105) ≤ , ∀ c ∈ A , so we extend (6.27) and, inparticular, (6.28) from D to D : D ⊆ (cid:101) Z ◦◦ , which is the analogue of (6.17). Finally, using the bipolar theorem in the same manner asthe last part of the proof of Lemma 6.8, we establish bipolarity between D and A :(6.29) A = D ◦ , D = A ◦ . Comparing (6.29) with (6.15) shows that we have D = D , so D is dense in D , and the proofis complete. (cid:3) Proofs of the duality theorems
In this section we prove the abstract duality of Theorem 5.6, from which the concreteduality of Theorem 4.1 is then deduced. Throughout this section, we have in place the resultof Proposition 5.5, as this bipolarity is the starting point of the duality proof. The proof ofTheorem 5.6 proceeds via a series of lemmas. Some of them have a similar flavour to thesteps in the celebrated Kramkov and Schachermayer [17, 18] abstract duality proof, but inmany places the roles of the primal and dual domains are reversed compared to [17, 18]. Thisis because in [17, 18] the dual domain is L ( P )-bounded, but here it is the primal domainthat is L ( µ )-bounded.Let us state the basic properties that are taken as given throughout this section. Fact 7.1.
Throughout this section, assume that the utility function satisfies the Inada con-ditions (3.1), that the sets C and D satisfy all the properties in Proposition 5.5, and that theabstract primal and dual value functions in (5.2) and (5.4) satisfy the minimal conditions in(5.8). All subsequent lemmata and propositions in this section implicitly take Fact 7.1 as given.
The first step is to establish weak duality.
Lemma 7.2 (Weak duality) . The primal and dual value functions u ( · ) and v ( · ) of (5.2) and (5.4) satisfy the weak duality bounds (7.1) 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 (7.2) lim sup x →∞ u ( x ) x ≤ , lim inf y →∞ v ( y ) y ≥ . Proof.
For any g ∈ C ( x ) and h ∈ D ( y ), using the polarity relations in (5.6) and (5.7) we maybound the achievable utility according to (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 > , (7.3)the last inequality a consequence of (3.7). Maximising the left-hand-side of (7.3) over g ∈ C ( x )and minimising the right-hand-side over h ∈ D ( y ) gives u ( x ) ≤ v ( y ) + xy for all x, y > 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 con-vex. We shall see these properties reproduced in proofs of existence and uniqueness of theoptimisers for u ( · ) , v ( · ).The next step is to give a compactness lemma for the primal domain. ERPETUAL CONSUMPTION DUALITY 27
Lemma 7.3 (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.Proof. Delbaen and Schachermayer [5, Lemma A1.1] (adapted from a probability space tothe finite measure space ( Ω , G , µ )) implies the existence of a sequence ( g n ) n ∈ N , with g n ∈ conv(˜ g n , ˜ g n +1 , . . . ), which converges µ -a.e. to an element g that is µ -a.e. finite because C isbounded in L ( µ ) (the finiteness also following from [5, Lemma A1.1]). By convexity of C ,each g n , n ∈ N lies in C . Finally, by Fatou’s lemma, for every h ∈ D we have (cid:90) Ω gh d µ = (cid:90) Ω lim inf n →∞ g n h d µ ≤ lim inf n →∞ (cid:90) Ω g n h d µ ≤ , so that g ∈ C . (cid:3) Results in the style of Lemma 7.3 are standard in these duality proofs. We will see asimilar result for the dual domain D shortly. Typically, the program is to first prove such aresult in the dual domain and to follow this with a uniform integrability result for the family( V − ( h )) h ∈D ( y ) . This facilitates a proof of existence and uniqueness of the dual minimiser,and of the conjugacy for the value functions by establishing the first relation in (5.9).Here, as we have alluded to earlier, the natural course of events is switched on its head: oneworks instead first in the primal domain, with the next step to prove a uniform integrabilityresult for the family ( U + ( g )) g ∈C ( x ) . This leads to existence and uniqueness of the primalmaximiser, and to conjugacy in the form of the second (bi-conjugate) relation in (5.9). Thestyle of proof in the dual domain for the classical program transfers to the primal domainhere. This switching of the roles of the primal and dual domains will be an almost continualfeature of the analysis of this section, and we shall point out further instances of it in duecourse. All this stems from the L ( µ )-boundedness of the primal (as opposed to the dual)domain in the consumption problem, as pointed out in the first paragraph of this section.Here is the next step in this chain of results. Lemma 7.4 (Uniform integrability of ( U + ( g )) g ∈C ( x ) ) . The family ( U + ( g )) g ∈C ( x ) is uniformlyintegrable, for any x > . The style of the proof is along identical lines to Kramkov and Schachermayer [17, Lemma3.2], but there it was applied to the concave function − V ( · ) and in the dual domain to provethe uniform integrability of ( V − ( h )) h ∈D ( y ) . We are witnessing the switching of the roles of C and D . Proof of Lemma 7.4.
Since U ( · ) is increasing, we need only consider the case where U ( ∞ ) :=lim x →∞ U ( x ) = + ∞ (otherwise there is nothing to prove). Let ϕ : ( U (0) , U ( ∞ )) (cid:55)→ (0 , ∞ )denote the inverse of U ( · ). Then ϕ ( · ) is strictly increasing. For any g ∈ C ( x ) (so (cid:82) Ω g d µ ≤ x )we have, for all x > (cid:90) Ω ϕ ( U + ( g )) d µ ≤ ϕ (0) + (cid:90) Ω ϕ ( U ( g )) d µ = ϕ (0) + (cid:90) Ω g d µ ≤ ϕ (0) + x. Then, using l’Hˆopital’s rule and the change of variable ϕ ( x ) = y ⇐⇒ x = U ( y ), we have(7.4) lim x → U ( ∞ ) ϕ ( x ) x = lim x →∞ ϕ ( x ) x = lim y →∞ yU ( y ) = lim y →∞ U (cid:48) ( y ) = + ∞ , on using the Inada conditions (3.1). The L ( µ )-boundedness of C ( x ) means we can applythe de la Vall´ee-Poussin theorem (Pham [23, Theorem A.1.2]) which, combined with (7.4),implies the uniform integrability of the family ( U + ( g )) g ∈C ( x ) . (cid:3) Remark . There is another way to establish Lemma 7.4 which matches more closely thestyle of proof in Kramkov and Schachermayer [18, Lemma 1], and which we shall see appliedto the dual domain in Lemma 7.10 to establish uniform integrability of ( V − ( h )) h ∈D ( y ) . Wemention this method here, because at first glance the method of [18, Lemma 1] will not workto establish Lemma 7.4, due to the fact that D is not bounded in L ( µ ). However, as weshow here, a slight adjustment to the proof can rectify matters. Here is the argument.Let ( g n ) n ∈ N be a sequence in C ( x ), for any fixed x >
0. We want to show that the sequence( U + ( g n )) n ∈ N is uniformly integrable.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 [23, Corollary A.1.1].) Define, for some g ∈ C , a sequence ( f n ) n ∈ N of elements in L ( µ ) by(7.5) f n := x g + n (cid:88) k =1 g k A k , where x := inf { x > U ( x ) ≥ } . (It is here where we are amending the arguments inKramkov and Schachermayer [18, Lemma 1]: there, one defines the sequence ( f n ) n ∈ N by f n := x + (cid:80) nk =1 g k A k , but an examination of the rest of the argument we now give showsthat this will require (cid:82) Ω h d µ ≤ , ∀ h ∈ D , which we do not have, because the constantconsumption stream c ≡ (cid:82) Ω gh d µ ≤ , ∀ g ∈C , h ∈ D , which allows the alternative definition of the sequence ( f n ) n ∈ N in (7.5) to makethings work.)For any h ∈ D we have (cid:90) Ω f n h d µ = (cid:90) Ω (cid:32) x g + 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 g + 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 (7.2). This contradiction establishesthe result. ERPETUAL CONSUMPTION DUALITY 29
One can can now proceed to prove either existence of a unique optimiser in the primalproblem, or conjugacy of the value functions. We proceed first the former, followed byconjugacy.
Lemma 7.6 (Primal existence) . The optimal solution (cid:98) g ( x ) ∈ C ( x ) to the primal problem (5.2) 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 7.2). That is(7.6) 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 7.3, 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(7.7) (cid:90) Ω U ( (cid:98) g ( x )) d µ = u ( x ) . By concavity of U ( · ) and (7.6) 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 (7.6), lim n →∞ (cid:90) Ω U ( (cid:98) g n ) d µ = u ( x ) . In other words(7.8) 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 (7.8) are finite.From Fatou’s lemma, we have(7.9) lim n →∞ (cid:90) Ω U − ( (cid:98) g n ) d µ ≥ (cid:90) Ω U − ( (cid:98) g ( x )) d µ. From Lemma 7.4 we have uniform integrability of ( U + ( (cid:98) g n )) n ∈ N , so that(7.10) lim n →∞ (cid:90) Ω U + ( (cid:98) g n ) d µ = (cid:90) Ω U + ( (cid:98) g ( x )) d µ. Thus, using (7.9) and (7.10) in (7.8), 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 (7.7). 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 guaranteedto equal (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 establish conjugacy of the value functions. Compared with the classical method ofproof in Kramkov and Schachermayer [17, Lemma 3.4], our method is similar, but instead ofbounding the elements in the primal domain to create a compact set for the weak ∗ topology σ ( L ∞ , L ) on L ∞ ( µ ), we bound the elements in the dual domain. Accordingly, we applya mirror image of the usual statement of the minimax theorem (see Strasser [27, Theorem45.8]) and consider a minimisation over a compact set and a maximisation over a subset of avector space, as opposed to the usual maximisation over a compact set and a minimisationover a subset of a vector space. This reversal is once again appropriate because the primaldomain is a subset of L ( µ ), whereas in the terminal wealth problem the dual domain is asubset of L ( P ). The consequence is that we prove the second (bi-conjugate) relation in (5.9),as opposed to the first. Here is the minimax theorem as we shall use it, easily proven byreversing the sign of the function in the usual minimax theorem and converting maximisationto minimisation, and vice versa. Theorem 7.7 (Minimax) . Let X be a convex subset of a normed vector space E and let Y bea σ ( E (cid:48) , E ) -compact convex, subset of the topological dual E (cid:48) of E . Assume that f : X ×Y → R satisfies the following conditions: (1) x (cid:55)→ f ( x, y ) is concave on X for every y ∈ Y ; (2) y (cid:55)→ f ( x, y ) is continuous and convex on Y for every x ∈ X .Then: inf y ∈Y sup x ∈X f ( x, y ) = sup x ∈X inf y ∈Y f ( x, y ) . Here is the conjugacy result for the primal and dual value functions.
Lemma 7.8 (Conjugacy) . The primal value function in (5.2) satisfies the bi-conjugacy re-lation u ( x ) = inf y> [ v ( y ) + xy ] , for each x > , where v ( · ) is the dual value function in (5.4) .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) h ∈ L ( µ ) : h ≤ n, µ − a . e . (cid:9) . The sets ( B n ) n ∈ N are σ ( L ∞ , L )-compact. Because each g ∈ C ( x ) is µ -integrable, C ( x ) isa closed, convex subset of the vector space L ( µ ), so we apply the minimax theorem asgiven in Theorem 7.7 to the compact set B n ( n fixed) and the set C ( x ), with the function f ( g, h ) := (cid:82) Ω ( V ( h ) + gh ) d µ , for g ∈ C ( x ) , h ∈ B n , to give(7.11) inf h ∈B n sup g ∈C ( x ) (cid:90) Ω ( V ( h ) + gh ) d µ = sup g ∈C ( x ) inf h ∈B n (cid:90) Ω ( V ( h ) + gh ) d µ. By the bipolarity relation D = C ◦ in (5.7), an element h ∈ L ( µ ) lies in D ( y ) if and only ifsup g ∈C ( x ) (cid:82) Ω gh d µ ≤ xy . Thus, the limit as n → ∞ on the left-hand-side of (7.11) is givenas(7.12) lim n →∞ inf h ∈B n sup g ∈C ( x ) (cid:90) Ω ( V ( h ) + gh ) d µ = inf y> inf h ∈D ( y ) (cid:18)(cid:90) Ω V ( h ) d µ + xy (cid:19) = inf y> [ v ( y ) + xy ] . Now consider the right-hand-side of (7.11). Define U n ( x ) := inf
The right-hand-side of (7.11) is then given assup g ∈C ( x ) inf h ∈B n (cid:90) Ω ( V ( h ) + gh ) d µ = sup g ∈C ( x ) (cid:90) Ω U n ( g ) d µ =: u n ( x ) , so that taking the limit as n → ∞ and equating this with the limit obtained in (7.12), wehave(7.13) lim n →∞ u n ( x ) = inf y> [ v ( y ) + xy ] ≥ u ( x ) , with the inequality due to the weak duality bound in (7.1). Consequently, we will be done ifwe can now show that we also have lim n →∞ u n ( x ) ≤ u ( x ) . Evidently, ( u n ( x )) n ∈ N is a decreasing sequence satisfying the limiting inequality in (7.13).Let (˜ g n ) n ∈ N be a maximising sequence in C ( x ) for lim n →∞ u n ( x ), so such thatlim n →∞ (cid:90) Ω U n (˜ g n ) d µ = lim n →∞ u n ( x ) . The compactness lemma for C , Lemma 7.3, implies the existence of a sequence ( g n ) n ∈ N in C ( x ), with g n ∈ conv(˜ g n , ˜ g n +1 , . . . ), which converges µ -a.e. to an element g ∈ C ( x ). Now, U n ( x ) = U ( x ) for x ≥ I ( n ), where I ( · ) = − V (cid:48) ( · ) is the inverse of U (cid:48) ( · ) (and U n ( · ) → U ( · )as n → ∞ ). So we deduce from Lemma 7.4 that the sequence ( U + n ( g n )) n ∈ N is uniformlyintegrable, and hence that(7.14) lim n →∞ (cid:90) Ω U + n ( g n ) d µ = (cid:90) Ω U + ( g ) d µ. On the other hand, from Fatou’s lemma, we have(7.15) lim n →∞ (cid:90) Ω U − n ( g n ) d µ ≥ (cid:90) Ω U − ( g ) d µ, so (7.14) and (7.15) give(7.16) lim n →∞ (cid:90) Ω U n ( g n ) d µ ≤ (cid:90) Ω U ( g ) d µ. Finally, using concavity of U n ( · ) and (7.16), we obtainlim n →∞ u n ( x ) = lim n →∞ (cid:90) Ω U n (˜ g n ) d µ ≤ lim n →∞ (cid:90) Ω U n ( g n ) d µ ≤ (cid:90) Ω U ( g ) d µ ≤ u ( x ) , and the proof is complete. (cid:3) We now move on to the dual side of the analysis. We begin with a similar compactnesslemma to Lemma 7.3, but now for the dual domain. The proof is identical to the proof ofLemma 7.3 so is omitted.
Lemma 7.9 (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. Next, we have an analogous result to Lemma 7.4, but for the dual variables, concerningthe uniform integrability of a sequence ( V − ( h n )) n ∈ N for h n ∈ D ( y ) (which will subsequentlylead to a lemma on existence and uniqueness of the dual optimiser). The proof is in thestyle of Kramkov and Schachermayer [18, Lemma 1], but there the technique was applied toa corresponding primal result akin to Lemma 7.4. Lemma 7.10 (Uniform integrability of ( V − ( h n )) n ∈ N , h n ∈ D ( y )) . Let ( h n ) n ∈ N be a sequencein D ( y ) , for any fixed y > . The sequence ( V − ( h n )) n ∈ N is uniformly integrable.Proof. Fix y >
0. If V ( ∞ ) ≥ V ( ∞ ) < V − ( h n )) n ∈ N is not uniformly integrable, then, passing if need be to asubsequence still denoted by ( h 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 V − ( h n ) A n d µ ≥ α, n ∈ N . (See for example Pham [23, Corollary A.1.1].) Define a sequence ( f n ) n ∈ N of elements in L ( µ ) by f n := y + n (cid:88) k =1 h k A k , where y := inf { y > V ( y ) ≤ } . For any g ∈ C (so satisfying (cid:82) Ω g d µ ≤
1) we have (cid:90) Ω gf n d µ = (cid:90) Ω g (cid:32) y + n (cid:88) k =1 h k A k (cid:33) d µ ≤ y + n (cid:88) k =1 (cid:90) Ω gh k A k d µ ≤ y + ny. Thus, f n ∈ D ( y + ny ) , n ∈ N .On the other hand, since V − ( · ) is non-negative and non-decreasing, (cid:90) Ω V ( f n ) d µ = − (cid:90) Ω V − ( f n ) d µ = − (cid:90) Ω V − (cid:32) y + n (cid:88) k =1 h k A k (cid:33) d µ ≤ − (cid:90) Ω V − (cid:32) n (cid:88) k =1 h k A k (cid:33) d µ = − n (cid:88) k =1 (cid:90) Ω V − (cid:16) h k A k (cid:17) d µ ≤ − αn. Therefore,lim inf z →∞ v ( z ) z = lim inf n →∞ v ( y + ny ) y + ny ≤ lim inf n →∞ (cid:82) Ω V ( f n ) d µy + ny ≤ lim inf n →∞ (cid:18) − αny + ny (cid:19) = − αy < , which contradicts the limiting weak duality bound in (7.2). This contradiction establishesthe result. (cid:3) One can can now proceed to prove existence of a unique optimiser in the dual problem.The proof is very much on the same lines as the proof of primal existence (Lemma 7.6), withadjustments for minimisation as opposed to maximisation and convexity of V ( · ) replacingconcavity of U ( · ) , so is included just for completeness. Lemma 7.11 (Dual existence) . The optimal solution (cid:98) h ( y ) ∈ D ( y ) to the dual problem (5.4) 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(7.17) lim n →∞ (cid:90) Ω V ( h n ) d µ = v ( y ) < ∞ . ERPETUAL CONSUMPTION DUALITY 33
By the compactness lemma for D (and thus also for D ( y ) = y D ), Lemma 7.9, 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.That is, that we have(7.18) (cid:90) Ω V ( (cid:98) h ( y )) d µ = v ( y ) . From convexity of V ( · ) and (7.17) 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 (7.17), lim n →∞ (cid:90) Ω V ( (cid:98) h n ) d µ = v ( y ) . In other words(7.19) 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 (7.19) are finite.From Fatou’s lemma, we have(7.20) lim n →∞ (cid:90) Ω V + ( (cid:98) h n ) d µ ≥ (cid:90) Ω V + ( (cid:98) h ( y )) d µ. From Lemma 7.10 we have uniform integrability of V − ( (cid:98) h n )) n ∈ N , so that(7.21) lim n →∞ (cid:90) Ω V − ( (cid:98) h n ) d µ = (cid:90) Ω V − ( (cid:98) h ( y )) d µ. Thus, using (7.20) and (7.21) in (7.19), 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 (7.18). Theuniqueness of the dual optimiser follows from the strict convexity of V ( · ), as does the strictconvexity of 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 notguaranteed to equal (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 move on to further characterise the derivatives of the value functions, as wellas the primal and dual optimisers. The first result is on the derivative of the primal valuevalue function u ( · ) at infinity (equivalently, the derivative of the dual value function v ( · ) atzero). Once again, because of the switching of the roles of the primal and dual sets in ourproofs compared with those of the terminal wealth problem, the proof of the following lemmamatches closely the proof in Kramkov and Schachermayer [17, Lemma 3.5] of the derivativeof v ( · ) at infinity (giving the derivative of u ( · ) at zero). Lemma 7.12.
The derivatives of the primal value function in (5.2) at infinity and of thedual value function in (5.4) at zero are given by (7.22) u (cid:48) ( ∞ ) := lim x →∞ u (cid:48) ( x ) = 0 , − v (cid:48) (0) := lim y ↓ ( − v (cid:48) ( y )) = + ∞ . Proof.
By the conjugacy result in Lemma 7.8 between the value functions, the assertions in(7.22) are equivalent. We shall prove the first assertion.The primal value function u ( · ) is strictly concave and strictly increasing, so there is a finitenon-negative limit u (cid:48) ( ∞ ) := lim x →∞ u (cid:48) ( x ). Because U ( · ) is increasing with lim x →∞ U (cid:48) ( x ) =0, for any (cid:15) > C (cid:15) such that U ( x ) ≤ C (cid:15) + (cid:15)x, ∀ x >
0. Using this, the L ( µ )-boundedness of C (so that (cid:82) Ω g d µ ≤ x, ∀ g ∈ C ( x )) and l’Hˆopital’s rule, we have, with (cid:82) Ω d µ =: δ >
0, 0 ≤ lim x →∞ u (cid:48) ( x ) = lim x →∞ u ( x ) x = lim x →∞ sup g ∈C ( x ) (cid:90) Ω U ( g ) x d µ ≤ lim x →∞ sup g ∈C ( x ) (cid:90) Ω C (cid:15) + (cid:15)gx d µ ≤ lim x →∞ (cid:18) C (cid:15) δx + (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 Theorem 5.6is to characterise the derivative of the primal value value function u ( · ) at zero (equivalently,the derivative of the dual value function v ( · ) at infinity) along with a duality characterisationof the primal and dual optimisers. Lemma 7.13. (1)
The derivatives of the primal value function in (5.2) at zero and ofthe dual value function in (5.4) at infinity are given by (7.23) u (cid:48) (0) := lim x ↓ u (cid:48) ( x ) = + ∞ , − v (cid:48) ( ∞ ) := lim y →∞ ( − v (cid:48) ( y )) = 0 . (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 (7.24) U (cid:48) ( (cid:98) g ( x )) = (cid:98) h ( y ) = (cid:98) h ( u (cid:48) ( x )) , µ -a.e. , and satisfy (7.25) (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 (7.26) 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 (3.7), which also applies to the value functions because they arealso conjugate by Lemma 7.8. We thus have, in addition to (3.7),(7.27) 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 (3.7) and (7.27) we have(7.28) 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 > , ERPETUAL CONSUMPTION DUALITY 35
The right-hand-side of (7.28) is zero if and only if y = u (cid:48) ( x ), due to (7.27), and the non-negative integrand must then be µ -a.e. zero, which by (3.7) can only happen if (7.24) holds,which establishes that primal-dual relation.Thus, for any fixed x > y = u (cid:48) ( x ), and hence equality in (7.28), 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 (7.25) must hold. Inserting the explicit form of (cid:98) h ( y ) = U (cid:48) ( (cid:98) g ( x )) into(7.25) yields the first relation in (7.26). Similarly, setting (cid:98) g ( x ) = I ( (cid:98) h ( y )) = − V (cid:48) ( (cid:98) h ( y )) into(7.25), with x = − v (cid:48) ( y ) (equivalent to y = u (cid:48) ( x )), yields the second relation in (7.26).It remains to establish the relations in (7.23), which are equivalent assertions. We shallprove the first one. This will use the fact that C is a subset of L ( µ ). In the terminal wealthcase, one typically proves the second assertion using the property that the dual domain lieswithin L ( P ). This is the switching of the roles of the primal and dual domains in theconsumption problem, that we have witnessed throughout this section.From the first relation in (7.26) and the fact that(7.29) (cid:90) Ω gh d µ ≤ xy, ∀ g ∈ C ( x ) , h ∈ D ( y ) , x, y > , we see that, for any x >
0, we have U (cid:48) ( (cid:98) g ( x )) ∈ D ( u (cid:48) ( x )). Thus, for any g ∈ C , (7.29) impliesthat(7.30) u (cid:48) ( x ) ≥ (cid:90) Ω U (cid:48) ( (cid:98) g ( x )) g d µ, ∀ g ∈ C , which we shall make use of shortly.Since C ( x ) is a subset of L ( µ ), we have (cid:82) Ω (cid:98) g ( x ) d µ ≤ x , and hence(7.31) (cid:90) Ω (cid:98) g ( x ) x d µ ≤ , ∀ x > . Using Fatou’s lemma in (7.31) we have1 ≥ lim inf x ↓ (cid:90) Ω (cid:98) g ( x ) x d µ ≥ (cid:90) Ω lim inf x ↓ (cid:18) (cid:98) g ( x ) x (cid:19) d µ, which, given that (cid:98) g ( x ) /x is non-negative, gives that lim inf x ↓ ( (cid:98) g ( x ) /x ) < ∞ , µ -a.e. Therefore,writing (cid:98) g ( x ) =: x (cid:98) g x , which defines a unique element (cid:98) g x ∈ C , we have (cid:98) g := lim inf x ↓ (cid:98) g x = lim inf x ↓ (cid:98) g ( x ) x < ∞ , µ -a.e.Using this property and applying Fatou’s lemma to (7.30) we obtain, on using U (cid:48) (0) = + ∞ ,+ ∞ ≥ lim inf x ↓ u (cid:48) ( x ) ≥ lim inf x ↓ (cid:90) Ω U (cid:48) ( x (cid:98) g x ) g d µ ≥ (cid:90) Ω lim inf x ↓ U (cid:48) ( x (cid:98) g x ) g d µ = + ∞ , which gives us the first relation in (7.23). (cid:3) We have now established all results that give the duality in Theorem 5.6, so let us confirmthis.
Proof of Theorem 5.6.
Lemma 7.8 implies the relations (5.9) of item (i). The statements initem (ii) are implied by Lemma 7.6 and Lemma 7.11. Items (iii) and (iv) follow from Lemma7.12 and Lemma 7.13. (cid:3)
We are almost ready to prove the concrete duality in Theorem 4.1, because Theorem5.6 readily implies nearly all of the assertions of Theorem 4.1. The outstanding assertionis the characterisation of the optimal wealth process in (4.3) and the associated uniformlyintegrable martingale property of the deflated wealth plus cumulative deflated consumptionprocess (cid:98) X ( x ) (cid:98) Y ( y ) + (cid:82) · (cid:98) c s ( x ) (cid:98) Y s ( y ) d s . So we proceed to establish these assertions in theproposition below, which turns out to be interesting in its own right. We take as given theother assertions of Theorem 4.1, and in particular the optimal budget constraint in (4.2).We shall confirm the proof of Theorem 4.1 in its entirety after the proof of the next result. Proposition 7.14 (Optimal wealth process) . Given the saturated budget constraint equalityin (4.2) , the optimal wealth process is characterised by (4.3) . The process (cid:99) M t := (cid:98) X t ( x ) (cid:98) Y t ( y ) + (cid:90) t (cid:98) c s ( x ) (cid:98) Y s ( y ) d s, ≤ t < ∞ , is a uniformly integrable martingale, converging to an integrable random variable (cid:99) M ∞ , sothe martingale extends to [0 , ∞ ] . The process (cid:98) X ( x ) (cid:98) Y ( y ) a potential, that is, a non-negativesupermartingale satisfying lim t →∞ E [ (cid:98) X t ( x ) (cid:98) Y t ( y )] = 0 . Moreover, (cid:98) X ∞ ( x ) (cid:98) Y ∞ ( 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 (4.2), one can always multiply the utility function by an arbitrary constant soas to ensure that u (cid:48) (1) = 1. We thus have the optimal budget constraint(7.32) E (cid:20)(cid:90) ∞ (cid:98) c t (cid:98) Y t d t (cid:21) = 1 , for (cid:98) c ≡ (cid:98) c (1) ∈ A and (cid:98) Y ≡ (cid:98) Y (1) ∈ Y . Since (cid:98) c ∈ A , 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 · S ) − (cid:90) · (cid:98) c s d s ≥ , and such that (cid:99) M := (cid:98) X (cid:98) Y + (cid:82) · (cid:98) c s (cid:98) Y s d s is a supermartingale over [0 , ∞ ). The supermartin-gale condition, by the same arguments that led to the derivation of the budget constraint inLemma 6.1, leads to the inequality E (cid:104)(cid:82) ∞ (cid:98) c t (cid:98) Y t d t (cid:105) ≤ (cid:99) M must be a martingale over [0 , ∞ ). We shall show that this extends to [0 , ∞ ],along with 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) Y is a mar-tingale minus a non-decreasing process, so is a non-negative c`adl`ag supermartingale, andthus (by Cohen and Elliott [3, Corollary 5.2.2], for example) converges to an integrable lim-iting random variable (cid:98) X ∞ (cid:98) Y ∞ := lim t →∞ (cid:98) X t (cid:98) Y t (and moreover (cid:98) X t (cid:98) Y t ≥ E [ (cid:98) X ∞ (cid:98) Y ∞ ] , 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) Y ∞ + (cid:82) ∞ (cid:98) c t (cid:98) Y t d t . By Protter [24, Theorem I.13], the extended martingale over[0 , ∞ ], ( (cid:99) M t ) t ∈ [0 , ∞ ] is then uniformly integrable, as claimed. ERPETUAL CONSUMPTION DUALITY 37
The martingale condition gives E (cid:20) (cid:98) X t (cid:98) Y t + (cid:90) t (cid:98) c s (cid:98) Y s d s (cid:21) = 1 , ≤ t < ∞ . Taking the limit as t → ∞ , using monotone convergence in the second term within theexpectation and utilising (7.32) yieldslim t →∞ E [ (cid:98) X t (cid:98) Y t ] = 0 , so that (cid:98) X (cid:98) Y 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) Y ∞ ] + 1 , on using (7.32). Hence, we get E [ (cid:98) X ∞ (cid:98) Y ∞ ] = 0 and, since (cid:98) X ∞ (cid:98) Y ∞ is non-negative, we deducethat (cid:98) X ∞ (cid:98) Y ∞ = 0, almost surely as claimed.We can now assemble these ingredients to arrive at the optimal wealth process formula(4.3). 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) Y u + (cid:90) u (cid:98) c s (cid:98) Y s d s (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = (cid:98) X t (cid:98) Y t + (cid:90) t (cid:98) c s (cid:98) Y s d 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) Y u + (cid:90) u (cid:98) c s (cid:98) Y s d s (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) = (cid:98) X t (cid:98) Y t + (cid:90) t (cid:98) c s (cid:98) Y s d s, t ≥ , which, on using (cid:98) X ∞ (cid:98) Y ∞ = 0, re-arranges to (cid:98) X t (cid:98) Y t = E (cid:20) (cid:90) ∞ t (cid:98) c s (cid:98) Y s d s (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , t ≥ , which establishes (4.3), and the proof is complete. (cid:3) Proof of Theorem 4.1.
Given the definitions of the sets C ( x ) and D ( y ) in (5.1) and (5.3),respectively, and the identification of the abstract value functions in (5.2) and (5.4) withtheir concrete counterparts in (3.5) and (3.10), Theorem 5.6 implies all the assertions ofTheorem 4.1, with the exception of the optimal wealth process formula (4.3) and the uniformintegrability of (cid:98) X ( x ) (cid:98) Y ( y ) + (cid:82) · (cid:98) c s ( x ) (cid:98) Y s ( y ) d s , which are established by Proposition 7.14. (cid:3) An example: Bessel process with stochastic volatility and correlation
We end with an example of an infinite horizon consumption problem in an incompletemarket model with strict local martingale deflators, which is covered in our framework.
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.We take 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 well-known three-dimensional Bessel process. The process λ := 1 /B will be the market price of risk of a stock with price process S and stochastic volatility process Y >
0, driven by the correlated Brownian motion (cid:102) W := ρW + (cid:112) − ρ W ⊥ , and with ρ ∈ [ − ,
1] some F -adapted stochastic correlation. We need not specify the dynamics of Y or ρ any further for the purposes of the example. The stock price dynamics are given byd S t = Y t S t d B t = Y t S t ( λ t d t + d W t ) . Take a constant relative risk aversion (CRRA) utility function: U ( x ) := x p /p, p < , p (cid:54) =0 , x >
0. The results for logarithmic utility U ( · ) = log( · ) can be recovered by setting p = 0in the final formulae, and this can be verified by carrying out the analysis directly for thatcase. Take the measure κ to be given by d κ t = e − αt d t , for a positive discount rate α , sothat γ t = e αt , t ≥
0. The primal value function is u ( x ) := sup c ∈A ( x ) E (cid:20)(cid:90) ∞ e − αt U ( c t ) d t (cid:21) , x > . The wealth process incorporating consumption satisfiesd X t = Y t π t ( λ t d t + d W t ) − c t d 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 deflators in this model are given bylocal martingale deflators of the form(8.1) 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. In the case that Y and ρ are deterministic, the market is complete and there is a unique deflator Z (0) := E ( − λ · W ). Itis well-known (see for instance Larsen [20, Example 2.2]) that Z (0) is a strict local martingaleand, what is more, that Z (0) = λ and that λ is square integrable. The strict local martingaleproperty is inherited by Z in (8.1), for any choice of integrand ψ .The deflated wealth plus cumulative deflated consumption process M is then given by(8.2) M t := X t Z t + (cid:90) t c s Z s d s = x + (cid:90) t Z s ( Y s π s − λ s X s ) d W s − (cid:90) t X s Z 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) = 0is the conjugate variable to p , satisfying 1 − q = (1 − p ) − . The dual value function is givenby v ( y ) := inf Z ∈Z E (cid:20)(cid:90) ∞ e − αt V ( yZ t e αt ) d t (cid:21) , y > . Denote the unique dual minimiser by (cid:98) Z , given by (cid:98) Z := E ( − λ · W − (cid:98) ψ · W ⊥ ) , for some optimal integrand (cid:98) ψ in (8.1). For use below, define the non-negative martingale H by H t := E (cid:20) (cid:90) ∞ e − α (1 − q ) s (cid:98) Z qs d s (cid:12)(cid:12)(cid:12)(cid:12) F t (cid:21) , t ≥ . Using Theorem 4.1, and in particular (4.1), the optimal consumption process is given by(8.3) ( (cid:98) c t ( x )) − (1 − p ) = u (cid:48) ( x )e αt (cid:98) Z t , t ≥ . By (4.2) the optimisers satisfy the saturated budget constraint(8.4) E (cid:20)(cid:90) ∞ (cid:98) c t ( x ) (cid:98) Z t d t (cid:21) = x. ERPETUAL CONSUMPTION DUALITY 39
The relations (8.3) and (8.4) yield (cid:98) c t ( x ) = xH e − α (1 − q ) t (cid:98) Z − (1 − q ) t , t ≥ . Using (4.3), the optimal wealth process is then given by (cid:98) X t ( x ) (cid:98) Z t = xH E (cid:20) (cid:90) ∞ t e − α (1 − q ) s (cid:98) Z 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 (8.2) at theoptimum, is computed as (cid:99) M t := (cid:98) X t ( x ) (cid:98) Z t + (cid:90) t (cid:98) c s ( x ) (cid:98) Z 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,without loss of generality, can be written in the form (cid:99) M t = x + (cid:90) t (cid:98) Z s (cid:98) X s ( x )( ϕ s − qλ s ) d W s + (cid:90) t (cid:98) Z s (cid:98) X s ( x ) β s d W ⊥ s , t ≥ , for some integrands ϕ, β . Comparing with the representation in (8.2) at the optimum yieldsthe optimal trading strategy in terms of the optimal portfolio proportion (cid:98) θ := (cid:98) π/ (cid:98) X ( x ) andthe optimal integrand (cid:98) ψ in the form (cid:98) θ t := (cid:98) π t (cid:98) X t ( x ) = λ t Y t (1 − p ) + ϕ t Y t , (cid:98) ψ t = − β t , t ≥ . In particular, the process ϕ records the correction to the Merton-type strategy λ/ ( Y (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) Z , whichis typically impossible in closed form for power utility. For the special case of logarithmicutility, one can set p = 0 and q = 0 in the results for power utility, to show that the process H = 1 /α is constant, and (cid:99) M = x is also constant, yielding (cid:98) θ t = λ t Y t , (cid:98) ψ t = 0 , t ≥ , giving the classic myopic trading strategy for logarithmic utility and, in particular, that thedual optimiser is the minimal deflator: (cid:98) Z = Z (0) = E ( − λ · W ). The optimal consumptionand wealth processes are given explicitly as (cid:98) c t ( x ) = α e − αt xZ (0) t , (cid:98) X t ( x ) = e − αt xZ (0) t , t ≥ , so that we have the classical relation (cid:98) c ( x ) = α (cid:98) X ( x ), as is always the case for infinite horizonlogarithmic utility from consumption. The results for logarithmic utility can of course beobtained by going directly through the analysis from scratch in the manner above. References [1]
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