Convergence of Optimal Expected Utility for a Sequence of Discrete-Time Markets
CConvergence of Optimal Expected Utility for aSequence of Discrete-Time Markets
David M. Kreps and Walter Schachermayer ∗ February 5, 2020
Abstract
We examine Kreps’ conjecture [17] that optimal expected utility in the classicBlack–Scholes–Merton (BSM) economy is the limit of optimal expected utility for asequence of discrete-time economies that “approach” the BSM economy in a naturalsense: The n th discrete-time economy is generated by a scaled n -step random walk,based on an unscaled random variable ζ with mean zero, variance one, and boundedsupport. We confirm Kreps’ conjecture if the consumer’s utility function U hasasymptotic elasticity strictly less than one, and we provide a counterexample tothe conjecture for a utility function U with asymptotic elasticity equal to 1, for ζ such that E [ ζ ] > . Fix a random variable ζ with mean zero, variance one, and bounded support. For n =1 , , . . . , construct a financial-market economy with two securities, a riskless bond , whichserves as numeraire (hence, has interest rate 0) and a risky security, called the stock , whichtrades against the bond in frictionless markets at time 0 , /n, /n, . . . , ( n − /n . Theprice process for the stock is generated as follows: For an i.i.d. sequence { ζ j ; j = 1 , , . . . } ,where each ζ k has the distribution of ζ , the law for the price of the stock at time k/n is S ( k/n ) := e ξ ( k/n ) where ξ ( k/n ) := k (cid:88) j =1 ζ j √ n . ( ξ (0) ≡ S (0) ≡ S (1) defined as above.Embed this model into the standard state space Ω = C [0 , ω denote a genericelement of Ω, with ω ( t ) the value of ω at time t . Endow Ω with the sup-norm topology; ∗ (Stanford Graduate School of Business and University of Vienna, Faculty of Mathematics, respec-tively.) Kreps’ research is supported by the Graduate School of Business, Stanford University. Schacher-mayer gratefully acknowledges support by the Austrian ScienceFund (FWF) under grant P28661 as wellas by the Vienna Science and Technology Fund (WWTF) through projects MA14-008 and MA16-021.Schachermayer also thanks the Department of Mathematics, Stanford University, for its hospitality whilethis paper was being written. F denote the Borel σ -field, and let { F t ; t ∈ [0 , } denote the standard filtration onΩ. For each n , let P n be the probability measure on Ω such that the joint distributionof ( ω (0) , ω (1 /n ) , . . . , ω (1)) matches the distribution of ( ξ (0) , ξ (1 /n ) , . . . , ξ (1)), and suchthat ω ( t ) for k/n < t < ( k + 1) /n is the linear interpolate of ω ( k/n ) and ω (( k + 1) /n ).And let S : Ω → R + be defined by S ( ω, t ) = e ω ( t ) .Donsker’s Theorem tells us that P n ⇒ P , where P is Wiener measure on C [0 , ω under P is a standard Brownian motion, starting at ω (0) = 0, and S ( ω ) under P is geometric Brownian motion, so that P , together with the riskless bond, prescribesthe simple continuous-time economy of Black and Scholes [5] and Merton [21] (hereafter,the BSM economy or model).We imagine an expected-utility-maximizing consumer who is endowed with initialwealth x , with which she purchases an initial portfolio of stock and bond. Thereafter,she trades in non-anticipatory and self-financing fashion in the stock and bond (that is,(a) the information she possesses at time k/n , on which basis she trades, is [only] thehistory of the stock price up to and including time k/n , and (b) any purchase of stock aftertime 0 is financed by the sale of bonds, and the proceeds of any sale of stock are investedin bonds), seeking to maximize the expectation of a utility function U : (0 , ∞ ) → R applied to the final dividend generated by the portfolio she holds at time 1.The question that forms the basis for this paper is: If we place this consumerin the n th discrete-time economy (where the stock and bond trade (only) at times0 , /n, /n, . . . , ( n − /n ), does the optimal expected utility she can attain approach,as n → ∞ , what she can optimally attain in the continuous-time BSM economy?Let u n ( x ) be the supremal expected utility she can attain in the n th discrete-timeeconomy if her initial wealth is x , and let u ( x ) be her supremal expected utility in the BSMeconomy. Kreps [17] obtains partial one-sided results, showing that lim inf n u n ( x ) ≥ u ( x ).And he proves lim n u n ( x ) = u ( x ) in the very special cases of U having either constantabsolute or relative risk aversion. But he only conjectures that the second “half”, orlim sup n u n ( x ) ≤ u ( x ) is true for gernal (concave and differentiable) U .Employing the notion of asymptotic elasticity of utility from Kramkov and Schacher-mayer [16] (and making extensive use of their analysis), we verify that lim n u n ( x ) = u ( x )if the utility function U has asympototic elasticity less than 1. However, we show byexample that if the asymptotic elasticity of U is 1, it is possible that u ( x ) is finite whilelim n u n ( x ) = ∞ , both for all x > A substantial body of literature concerns utility maximization problems in financial mar-kets, going back to seminal work by R. Merton [20] and continuing, e.g., in ([6, 13, 16,8, 11, 24, 4]). As regards the continuity of utility maximization under weak convergenceof financial markets, positive results are obtained in ([10], [23], [25]); these results allassume that, in each discrete-time model, markets are complete.Our interest, motivated by the discussions in [17], is in cases where the discrete-time markets are incomplete. Since the seminal paper of Cox, Ross, and Rubinstein[7], financial economists have believed that, if ζ has two-element support (the so-calledbinomial case), and so markets are complete in each discrete-time economy, then these2iscrete-time economy for large n behave (in economic terms) like the continuous-timelimit, at least for the BSM continuous-time limit. But what if ζ has support of, say, sizethree, but there are only the two securities? Markets are incomplete for any finite n ; doesthis incompleteness mean very different economic outcomes? Or, if the probability laws P n that govern the discrete-time security-price processes converge weakly to P , is it thentrue that lim n u n ( x ) = u ( x )?It is already known that weak convergence is insufficient. Merton [20] observes thatif U has constant relative risk aversion with risk-aversion parameter less than 1/2, theoptimal strategy in the BSM economy is to short-sell bonds, leveraging to achieve a(fixed) fraction greater than 100% of current wealth in the risky asset. Suppose that,in our special discrete-time setting, where the security-price process is driven by scaledcopies of a single random variable ζ , ζ has support that is unbounded below. Trying toachieve such a leverage strategy in any of the finite-time economies would give a positiveprobability of bankruptcy, which is incompatible with these utility functions. The bestan investor can do for large enough n in these circumstances is to hold 100% of her wealthin the risky asset, which results in lim n u n ( x ) < u ( x ).On the other hand, weak convergence of P n to P alone does not preclude the possi-bility of asymptotic arbitrage ([12], [15]), in which case lim n u n ( x ) = ∞ , even when u ( x )is finite valued (and U is very well behaved); see [17], Chapter 7.By assuming in our setting that ζ has bounded support, we avoid the first problem.And, in our setting, asymptotic arbitrage is precluded; see [17], Proposition 7.1. Still,ill-behaved U can pose problems: Within our setting, we show that lim n u n ( x ) = u ( x ) forall x > U has asymptotic elasticity less than 1. But if U has asymptotic elasticity of1, even if markets are complete for each n , convergence can fail, and fail in spectacularfashion.The incomplete-market case on which we focus has recently been treated in a settingof greater generality by E. Bayraktar, Y. Dolinsky, and J. Guo [2]. Their paper assumesthat the financial markets ( S n ) ∞ n =1 are general semi-martingales and the limiting market S is a continuous semi-martingale. Also, the utility function U in [2] may measurably de-pend on the observed trajectory of the stock price. Hence their model includes our specialand paradigmatic case, where ( S n ) is induced by a single (scaled) random variable ζ and S is geometric Brownian motion. In this more general setting, they make assumptionssufficient to show that lim n u n ( x ) = u ( x ). The key assumptions in [2] are Assumption 2.3(ii), that a certain family of random variables is uniformly integrable, and Assumption2.5, which effectively assumes away the possibility of asymptotic arbitrage. Lemma 2.2of [2] provides some fairly strong conditions under which Assumption 2.3 (ii) is satisfied,conditions that are not related to the concept of asymptotic elasticity. In comparison, we deduce the uniform integrability of certain corresponding families of dual random vari-ables (see (8.8) and (8.14) below) from the assumption that U has asymptotic elasticityless than 1. And, in our more limited setting, the impossibility of asymptotic arbitrageis a conclusion. This paper [2] was put on ArXiv in November 2018; the authors kindly brought their paper to ourattention after a first version of the current paper appeared on ArXiv in July 2019. The references totheir results refer to the ArXiv version of [2] from September 2019. The utility function, its conjugate function, andasymptotic elasticity
We always assume the following:
Assumption (3.1).
The utility function U is strictly increasing, strictly concave, andcontinuously differentiable, and satisfies the Inada conditions that lim x → U (cid:48) ( x ) = ∞ and lim x →∞ U (cid:48) ( x ) = 0 . Moreover, without loss of generality and for notational conveniencelater, we assume unless otherwise specified that U is normalized so that lim x →∞ U ( x ) > ,without precluding the possibility that lim x →∞ U ( x ) = ∞ . (Of course, lim x → U ( x ) can beeither finite or −∞ .) We let V denote the conjugate function to U : For y > V ( y ) = sup x> (cid:2) U ( x ) − xy (cid:3) , for y > . The following results are standard (see, e.g., [16]) and follow from Assumption (3 . • Let I : (0 , ∞ ) → (0 , ∞ ) be the inverse of U (cid:48) ; that is, I ( y ) = ( U (cid:48) ) − ( y ) . Then forevery y ∈ (0 , ∞ ), V ( y ) = U ( I ( y )) − yI ( y ). • The function y → V ( y ) is strictly convex, continuously differentiable, and strictlydecreasing. • V ( ∞ ) = U (0) and V (0) = U ( ∞ ), where the values of U and V at 0 and ∞ areinterpreted as the limits as x and y approach 0 and ∞ , respectively. • V (cid:48) ( y ) = − I ( y ), so lim y → V (cid:48) ( y ) = −∞ and lim y →∞ V (cid:48) ( y ) = 0 • U ( x ) = inf y> (cid:2) V ( y ) + xy ] , for x > . The notion of asymptotic elasticity of U , defined in [16], plays an important role in ouranalysis. For the utility function U , its asymptotic elasticity, written AE( U ), is definedby AE( U ) := lim sup x →∞ xU (cid:48) ( x ) U ( x )If, for instance, U ( x ) = x α /α for α ∈ (0 , U ) = α .The concavity of U implies that AE( U ) ≤ U is bounded above andif U ( ∞ ) >
0, then AE( U ) = 0. But if U ( ∞ ) = ∞ , AE( U ) can equal 1; an example iswhere U ( x ) = x/ ln( x ) for sufficiently large x . Many of our results depend on the assumption that AE( U ) <
1, which derives froma comparison of the average and marginal utilities provided by U as the argument of U approaches ∞ : AE( U ) < γ < , U (cid:48) ( x ) < γ U ( x ) x , for all large enough x. The conditions (3 .
1) on U include lim x →∞ U ( x ) >
0; this is solely so that AE( U ) ≥ x . Becauselim x →∞ U ( x ) − U ( x ) x − x = lim x →∞ U ( x ) x , for all x > , AE( U ) < γ < x > , U (cid:48) ( x ) < γ U ( x ) − U ( x ) x − x , for all large enough x, where “large enough” depends on the value of x . As noted in [26], the concept of asymptotic elasticity connects to the limiting behaviorof relative risk aversion by an application of de l’Hˆopital’s rule as follows: If the limitof the coefficient of relative risk aversion, lim x →∞ − xU (cid:48)(cid:48) ( x ) /U (cid:48) ( x ), exists and is strictlypositive, then lim x →∞ xU (cid:48) ( x ) /U ( x ) exists and is less than 1; that is, U has asymptoticelasticity less than 1. Since it is believed to be “common” for economic agents to havenon-increasing relative risk aversion, this belief implies that agents with this commonproperty have asymptotic elasticity less than one. As is well known, the continuous-time BSM economy admits a unique equivalent martin-gale measure denoted by P ∗ ; that is, a probability measure on Ω that is probabilisticallyequivalent to P and such that { S ( ω, t ); t ∈ [0 , } is a martingale (over the natural fil-tration { F t } ). This measure P ∗ has Radon-Nikodym derivative with respect to P givenby (cid:18) d P ∗ d P (cid:19) ( ω ) = exp (cid:18) − ω (1)2 − (cid:19) . And, as is well known, this economy has “complete markets.” That is, the consumer canconstruct (as a stochastic integral) any measurable positive contingent claim X that shecan afford, where what she can afford is given by the single budget constraint E P ∗ [ X ] ≤ x , where E P ∗ [ · ] denotes expectation with respect to P ∗ . Hence, with wealth x , theconsumer’s problem is toMaximize E P [ U ( X )] , subject to E P ∗ [ X ] ≤ x. Let u ( x ) := sup (cid:8) E P [ U ( X )] : E P ∗ [ X ] ≤ x (cid:9) . That is, u ( x ) is the supremum of expected-utility level that the consumer can achieve inthe BSM economy, starting with wealth x .It is convenient for later purposes to define the density function Z : C [0 , → (0 , ∞ )by Z ( ω ) := exp (cid:18) − ω (1)2 − (cid:19) . Z is the unique continuous (in ω ) version of the random variable d P ∗ /d P . Ofcourse, E P∗ [ X ] = E P [ X · Z ] for any random variable X such that (at least) one of theexpectations makes sense. And, in this notation u ( x ) = sup (cid:8) E P [ U ( X )] : E P [ X · Z ] ≤ x } . We have the following from Cox and Huang [6], Karatzas, Lehoczy, and Shreve [14],and [16]. (See, in particular, [16], Theorem 2.0.)4.1. u ( x ) ≥ U ( x ) (since the consumer can always buy and hold x bonds).4.2. x → u ( x ) is continuously differentiable, strictly increasing, and concave.4.3. From 4.1 and 4.2, if u ( x ) < ∞ for any x >
0, then u ( x ) < ∞ for all x > u ( x ) < ∞ , and if the consumer’s problem has a solution (that is, if the supremumis attained) , then there exists y ( x ) > X ( ω ) = I (cid:0) y ( x ) · Z ( ω ) (cid:1) , where y ( x ) = u (cid:48) ( x ) and (as noted earlier) I = ( U (cid:48) ) − . If the consumer’s problem has a solution at wealth level x >
0, then it has asolution for all wealth levels x (cid:48) > x (cid:48) < x .However, it is possible that, at least for some x , u ( x ) < ∞ and yet the supremumthat defines u ( x ) is not attained by any contingent claim X . (An example is givenin [16], Section 5; we produce examples below.) If (for the given utility function U ) this is true for some finite x , let x be the infimum of all x for which there is nosolution (but u ( x ) < ∞ ); there is a solution at x , and so the range of x for whichthere is a solution is the interval (0 , x ].The function x → u ( x ) is continuously differentiable and the “Lagrange multi-plier function” x → y ( x ) = u (cid:48) ( x ) is continuous and strictly decreasing on (0 , x ) . v be the conjugate function to u . That is, v ( y ) = sup { u ( x ) − xy : x > } , for y > . Then v ( y ) = E P (cid:2) V (cid:0) yZ (cid:1)(cid:3) . The function y → v ( y ) is convex and nonincreasing, and it is strictly decreasingand continuously differentiable where it is finite. Of course, it may be that u ( x ) ≡ ∞ , in which case v ( y ) ≡ ∞ . But suppose u ( x ) < ∞ for some, and therefore for all, x >
0. While u ( x ) is necessarily concave, differentiable,and strictly increasing, it is not in general true that lim x →∞ u (cid:48) ( x ) = 0. That is, themarginal (maximal expected) utility of wealth need not approach zero as the wealth levelgoes to ∞ . Roughly speaking, this can happen when a consumer can purchase ever largeramounts of consumption on events of ever smaller probability, but where the ratio of theamount purchased to the probability of the event approaches infinity at a rapid enoughrate. This idea was exploited by Kramkov and Schachermayer [16] for any utility function U that satisfies AE( U ) = 1, by choosing, based on U , specific measures that are differentfrom but play an analogous role to P ∗ and P . Here, P ∗ and P are fixed – they comefrom BSM – so we show this sort of possibility through the selection of specific utilityfunctions U . 6dmitting this is possible (we show that it is), consider the implications for v and, inparticular, for v around the value y , where y = lim x →∞ u (cid:48) ( x ). To the left of y ( y < y ),we have v ( y ) = ∞ . To the right, v ( y ) is finite. But what is the limiting behavior of v as y approaches y from the right? In theory, we could have lim y (cid:38) y v ( y ) = ∞ . Orlim y (cid:38) y v ( y ) < ∞ , and lim y (cid:38) y v (cid:48) ( y ) = ∞ . Or lim y (cid:38) y v ( y ) < ∞ , and lim y (cid:38) y v (cid:48) ( y ) < ∞ .All of these are possible. In fact, giving the full catalog, we have Proposition 1: Proposition 1.
Assume that U satisfies the conditions (3 . . It is possible that u ( x ) = ∞ (for all x ≥ ). But if u ( x ) < ∞ for some x , hence for all x , it must be that x → u ( x ) isstrictly increasing. Moreover, we have the following possibilities.a. For some utility functions U , lim x →∞ u (cid:48) ( x ) = 0 , in which case v ( y ) is finite for all y > . (Since lim y → v ( y ) = lim x →∞ u ( x ) and lim y →∞ v ( y ) = lim x → u ( x ) , thefunction v can have limit ∞ or a finite limit as y approaches ; and v can havelimit −∞ or a finite limit as y → ∞ .)b. For other utility functions U , lim x →∞ u (cid:48) ( x ) > . If we denote lim x →∞ u (cid:48) ( x ) by y ,then v ( y ) = ∞ for y < y , while v ( y ) < ∞ for y > y . As for the behavior of v as y (cid:38) y , we have the following possibilities: i. lim y (cid:38) y v ( y ) = ∞ ii. lim y (cid:38) y v ( y ) < ∞ and lim y (cid:38) y v (cid:48) ( y ) = ∞ ; iii. lim y (cid:38) y v ( y ) < ∞ and lim y (cid:38) y v (cid:48) ( y ) < ∞ .Moreover, all are possible for any value of y > .Finally, AE ( u ) ≤ AE ( U ) ; hence, AE ( U ) < implies lim x →∞ u (cid:48) ( x ) = 0 . That is, asymp-totic elasticity less than 1 removes the cases given by part b. The possibility outlined in part a is simple to show: Take utility functions withconstant relative risk aversion, for which solutions are well known and fit case a . And thefinal assertion needs no proof; it derives from [16], Theorem 2.2. To give examples of thethree possibilities outlined in part b requires some calculations. Since this is a diversionfrom our main message, we leave this to Section 10. In fact, while part b seems to be themost intriguing aspect of the proposition, we note that for the proof of Theorem 1 belowwe only rely on the final assertion of Proposition 1. Proposition 2.
For the sequence of discrete-time economies as described in Section 1,and a utility function U that satisfies conditions (3 . , lim inf n →∞ u n ( x ) ≥ u ( x ) for all x > . Note: If u ( x ) = ∞ , this proposition still applies, implying that lim n →∞ u n ( x ) = ∞ .7 roof. Kreps ([17], Proposition 5.2) states that if, in the BSM economy, a bounded andcontinuous contingent claim X satisfies E P [ U ( X )] = z and E P ∗ [ X ] = x (so that u ( x ) ≥ z ),then for every (cid:15) >
0, there exists N such that, for all n > N , the consumer in the n thdiscrete-time economy can synthesize a claim X n for an initial investment of x such that E P n [ U ( X n )] ≥ z − (cid:15) . Suppose we know that u ( x ) < ∞ and, for the given x , a solution to the consumer’sproblem exists (that is, the sup that defines u ( x ) is a max). We then know, since thesolution is of the form X = I ( yZ ) for some multiplier y > X : Ω → (0 , ∞ ) is a continuous function of ω . By truncating the solution X ,we get approximately u ( x ), with what is a bounded and continuous claim. Hence, weconclude that lim inf n u n ( x ) ≥ u ( x ) . The cases where u ( x ) < ∞ but no solution exists and where u ( x ) = ∞ are a bit moredelicate, because we don’t know, a priori, that we approach the upper bound (finite inthe first case, ∞ in the second) with bounded and continuous contingent claims. But wecan show this is so. Suppose for some level z , there is a measurable contingent claim X such that E P [ U ( X )] = z and E P ∗ [ X ] = x . In this context, of course X ≥ (cid:15) >
0. We first replace X with a bounded claim X (cid:48) , bounded away from ∞ above and away from 0 below, in two steps. First, for α < X α := αX + (1 − α ) x . Of course, E P ∗ [ X α ] = x . And by a double application of monotoneconvergence (split E P [ U ( X α )] into E P [ U ( X α ) { X α ≥ x } ] + E P [ U ( X α ) { X α 4. Of course, X α is bounded below by (1 − α ) x . As for the upper bound, cap X α o atsome large β . That is, let X α o ,β be X α o ∧ β . For large enough β o , this is bounded aboveand will satisfy E P [ U ( X α o ,β o ] > z − (cid:15)/ 2, while capping X α o can only relax the budgetconstraint.So, it is wlog to assume that our original X (that gives expected utility close to z and satisfies the budget constraint for x ) is bounded above and bounded away from zero.Now apply a combination of Luzin’s Theorem and Tietze’s Extension Theorem: We canapproximate X with a continuous function X (cid:48) that differs from X on a set of arbitrarilysmall measure and that satsifies the same upper and lower bounds as X ; this allows thechoice of X (cid:48) to satisfy E P [ U ( X (cid:48) )] > z − (cid:15)/ 4. It may be that E P ∗ [ X (cid:48) ] > x , but thelast (cid:15)/ X (cid:48) with X (cid:48) − ( E P ∗ [ X (cid:48) ] − x ), giving a bounded and continuouscontingent claim that costs x (or less) and provides expected utility z − (cid:15) , at which point[17], Proposition 5.2 can be applied to prove (in general) Proposition 2. In order to tackle the reverse inequality of the one in Proposition 2 we need some prepa-ration. The proof of this proposition relies on Theorem 1 in [18], which says that any bounded and continuouscontingent claim x can be synthesized with “ x -controlled risk” in the n th discrete-time economy for largeenough n , where “approximately synthesized” means: For given (cid:15) > n (dependingon (cid:15) ), the synthesized claim, x n satisfies P n (cid:0) | x n − x | > (cid:15) (cid:1) < (cid:15) ; and “ x -controlled risk” means that thesynthesized claim x n satisfies x n ( ω ) ∈ (cid:0) inf ω (cid:48) x ( ω (cid:48) ) , sup ω (cid:48) x ( ω (cid:48) ) (cid:1) with P n -probability 1. n th discrete-time economy, the consumer faces three types of constraints:1. She has a level of initial wealth x , and her initial portfolio cannot exceed x in value.2. Between times 0 and 1, any trades she makes must be self-financing.3. She has available only the trades that the price process permits. In a word, herfinal consumption bundle must be a synthesizable contingent claim.The importance of 3 is that, for ζ having support with more than two elements, theconsumer does not face “complete markets”.However, we know that any final consumption bundle X that she constructs in then’th economy subject to these three constraints must satisfy E Q ∗ n [ X ] = x, where E Q ∗ n denotes expectation with respect to any probability measure Q ∗ n that is anequivalent martingale measure (emm) for P n . We fix one particular emm for each P n , namely the emm, which we hereafter denoteby P ∗ n , provided by the Esscher transform: (cid:18) d P ∗ n d P n (cid:19) ( ω ) = exp (cid:2) − a n ω (1) − b n (cid:3) for constants a n and b n , chosen such that P ∗ n is a martingale probability measure. Specif-ically, a n is fixed by the “martingale equation” that E P n (cid:20) d P ∗ n d P n e ω (( k +1) /n ) (cid:12)(cid:12)(cid:12)(cid:12) F k/n (cid:21) = e ω ( k/n ) , and b n is then fixed as a normalizing constant, given the value of a n . Moreover, it canbe shown that a n = 1 / E [ ζ ]24 √ n + o (1 / √ n ) where E [ ζ ] is the third moment of ζ , andthat lim n b n = 1 / 8. (The notation E [ · ] is used to denote expectations over ζ .) Of course, P n ⇒ P (weakly on C [0 , 1] endowed with the sup-norm topology) and, for this specificequivalent martingale measure, P ∗ n ⇒ P ∗ . So, suppose we pose the following problem for the consumer:Maximize E P n [ U ( X )] , subject to E P ∗ n [ X ] = x, (6.1)where E P n [ · ] denotes expectation with respect to P n and E P ∗ n [ · ] denotes expectation withrespect to the specific emm P ∗ n . In words, we allow the consumer any consumptionclaim X she wishes to purchase, subject only to the constraint that she can afford X atthe “prices” given by d P ∗ n /d P n .Let Z n be the function on C [0 , 1] given by Z n ( ω ) = exp (cid:2) − a n ω (1) − b n (cid:3) . Hence, Z n isa specific version of the random variable d P ∗ n /d P n and the constraint E P ∗ n [ X ] = x can berewritten as E P n [ Z n X ] = x . That is, we can think simply of a consumer facing complete Because we assume that ζ has mean zero and variance one, we know that P n admits emms. For a detailed derivation, see[17], Lemma 5.1. Z n the “pricing kernel” for contingent claims. Using this interpretation, wedenote the supremal utility the consumer can obtain in the problem (4.1) as u Z n n ( x ) := sup (cid:8) E P n [ U ( X )] , subject to E P n [ Z n X ] ≤ x (cid:9) . (6.2)The point of this is that the problem (6.1) relaxes the constraints that actually facethe consumer in the n th discrete-time economy; in (6.1) she faces“complete markets”; inher real problem, she faces further “synthesizability” constraints. Hence, we know that u Z n n ( x ) ≥ u n ( x ) for all x > n = 1 , , . . . (6.3)If we can show that lim n u Z n n ( x ) = u ( x ), we will know that lim sup n u n ( x ) ≤ u ( x ). This,together with Proposition 2, will establish that lim n u n ( x ) = u ( x ). So this is what we setout to do. In fact, we add one more plot element. As we have stated above, Z is the func-tion Z ( ω ) = exp( − ω (1) / − / , which is a version (the unique continuous version) of d P ∗ /d P . Define u Zn ( x ) = sup (cid:8) E P n [ U ( X )] : E P n [ ZX ] ≤ x (cid:9) . (7.1)In words, u Zn ( x ) is the supremal expected utility that the consumer can attain if she facescomplete markets and “prices” Z in the n th discrete-time economy. That is, moving fromthe consumer’s problem in the BSM model to the problem described by (7.1) changes theconsumer’s probability assessment from P to P n but not the “prices” she faces. In movingfrom (7.1) to (6.2), we keep the probability assessment as P n but change the prices from Z to Z n . This “taking it one step at a time” is useful in the analysis to follow. In this section, we prove the following result: Theorem 1. Suppose that the utility function U satisfies conditions (3 . and that AE ( U ) < . Then, for all x > , the value function x → u ( x ) is finite-valued and lim n →∞ u n ( x ) = u ( x ) . (8.1)The proof of Theorem 1 will take several steps and consumes this entire section. Webegin with a lemma. Lemma 1. For any constant γ , lim n →∞ E P n (cid:2) exp( γω (1) (cid:3) = E P (cid:2) exp( γω (1)) (cid:3) . Proof of Lemma 1. Under P n , ω (1) = (cid:80) nk =1 ζ k / √ n for { ζ k } an i.i.d. sequence of randomvariables with the law of ζ . Hence, E P n (cid:2) exp( γω (1)) (cid:3) = E P n (cid:20) exp (cid:18) γ n (cid:88) k =1 ζ k √ n (cid:19)(cid:21) = (cid:18) E (cid:20) exp (cid:18) γ ζ √ n (cid:19)(cid:21)(cid:19) n . 10 Taylor series approximation to exp( γζ/ √ n ) is 1 + γζ/ ( √ n ) + γ ζ / (2 n ) + o (1 /n ), wherethe o (1 /n ) term is uniform in the value of ζ because ζ has bounded support. Therefore, E P n (cid:2) exp( γω (1)) (cid:3) = (cid:18) E (cid:20) γζ √ n + γ ζ n + o (1 /n ) (cid:21)(cid:19) n = (cid:0) γ / (2 n ) + o (1 /n ) (cid:1) n . The term on the rhs converges to e γ / , which is (of course) E P (cid:2) exp( γω (1)) (cid:3) . Now we turn to the proof of Theorem 1: Step 1. Because AE ( U ) < , u ( x ) < ∞ for all x > . This step rates a remark: For “general” price processes as investigated, for instance, in[16], having asymptotic elasticity less than 1 does not guarantee that the optimal expectedutility is finite. The result here strongly depends on the price processes being given bythe BSM model.The second key to Step 1 is the following bound:There exist L > α > V ( y ) ≤ Ly − α , for all y ∈ (0 , ∞ ) . (8.2)Corollary 6.1 in [16] establishes this bound as a consequence of AE( U ) < 1, but only for0 < y ≤ y , for some y > If we have that V ( ∞ ) = U (0) < 0, we get the bound for all y > 0. And, for purposes of this theorem, it is without loss of generality to shift U by aconstant. So if U (0) ≥ 0, simply replace U with U ( x ) − U (0) − c , for a suitable constant c > 0. Then we have, and henceforth assume, 8.2 for all y ∈ (0 , ∞ ) . Hence, we may estimate v ( y ) = E P (cid:20) V (cid:18) y dP ∗ dP (cid:19)(cid:21) ≤ L E P (cid:20) y − α exp (cid:18) − ω (1)2 − (cid:19) − α (cid:21) = L y − α e α/ E (cid:20) exp (cid:18) α ω (1) (cid:19)(cid:21) < ∞ , (8.3)as the latter expectation is just an exponential moment of a Gaussian variable. Hence,by ([16], Theorem 2.0), the dual value function y → v ( y ) as well as the primal valuefunction x → u ( x ) have finite values; and we deduce as well from part c of Proposition1, that AE( u ) ≤ AE( U ) < Step 2. Using the notation (7.1) , define u Z ∞ ( x ) for x > by u Z ∞ ( x ) := lim sup n →∞ u Zn ( x ) . (8.4) Then u Z ∞ ( x ) < ∞ for all x > . Define for each n the conjugate function v Zn , which is v Zn ( y ) = E P n [ V ( yZ )] . (8.5)11y the same argument that gave (8.3), we have v Zn ( y ) = E P n (cid:2) V (cid:0) yZ (cid:1)(cid:3) ≤ L y − α e α/ E P n (cid:20) exp (cid:18) α ω (1) (cid:19)(cid:21) . (8.6)Lemma 1 tells us that the expectation on the rhs of (8.6) converges, which implies that v Zn ( y ) is uniformly bounded in n for fixed y , which, by standard arguments concerningconjugate functions, proves that u Z ∞ ( x ) is finite for each x . Step 3. Indeed, lim n u Zn ( x ) exists and equals u ( x ) for all x > . We show that lim n v Zn ( y ) = v ( y ) for all y > 0, which proves step 3, again using standardarguments concerning conjugate functions.Compare v Zn ( y ) and v ( y ): v Zn ( y ) = E P n [ V ( yZ )] and v ( y ) = E P [ V ( yZ )] . If V were a bounded function (of course, V is continuous), the conclusion would followimmediately from P n ⇒ P . But V is typically not bounded, and so we must show thatthe contributions to the expectations from the “tails” can be uniformly controlled. Wedo this by showing the following two uniform bounds:For every y > (cid:15) > , there exists M > E P n (cid:2) | V ( yZ ) | · { V ( yZ ) < − M } (cid:3) < (cid:15), uniformly in n. (8.7)For every y > (cid:15) > , there exists M > E P n (cid:2) V ( yZ ) · { V ( yZ ) >M } (cid:3) < (cid:15), uniformly in n. (8.8)Begin with (8.7). If U (0) is finite, it follows that V ( y ) ≥ V ( ∞ ) = U (0) for all y , sotaking M = − U (0) immediately works. The (slightly) harder case is where U (0) = −∞ .In this case, recall that, by the Inada conditions, lim y →∞ V (cid:48) ( y ) = − lim y →∞ ( U (cid:48) ) − ( y ) = 0.Since V is convex, this implies that for large enough M , | V ( y ) | ≤ (cid:15)y , provided V ( y ) ≤− M . But then E P n (cid:2)(cid:12)(cid:12) V ( yZ ) (cid:12)(cid:12) · { V ( yZ ) ≤− M } (cid:3) ≤ E P n (cid:2) (cid:15)Z (cid:3) ≤ (cid:15) E P n (cid:2) Z (cid:3) . (8.9)By Lemma 1, lim n →∞ E P n [ Z ] = E P [ Z ] = 1, which shows (8.7).And to show the (uniform) inequality (8.8): For the parameters α and L that give thebound (8.2) and for fixed y > (cid:15) > 0, let B be large enough so that E P n (cid:2) e αω (1) / · { ω (1) ≥ B } (cid:3) ≤ (cid:15)Ly − α e α/ for all n. (8.10)The existence of such a B follows from Lemma 1 and, because P n ⇒ P ,(1) E P n (cid:2) min { e αω (1) , B } (cid:3) → E P (cid:2) min { e αω (1) , B } (cid:3) and(2) P n ( { ω (1) ≥ B } ) → P ( { ω (1) ≥ B } ), for all B > M = Ly − α exp (cid:18) αB α (cid:19) . (8.11)We have V ( y ) ≤ Ly − α for all y > 0, and so (cid:8) V ( yZ ) ≥ M (cid:9) ⊆ (cid:8) L ( yZ ) − α ≥ M (cid:9) = (cid:8) L ( yZ ) − α ≥ Ly − α e ( αB/ α/ (cid:9) = (cid:8) Z − α ≥ e αB/ α/ (cid:9) = (cid:8) ( e − ω (1) / − / ) − α ≥ e αB/ α/ (cid:9) = (cid:8) e αω (1) / ≥ e αB/ (cid:9) = (cid:8) ω (1) ≥ B (cid:9) . Hence, E P n (cid:2) V ( yZ ) · { V ( yZ ) ≥ M } (cid:3) ≤ E P n (cid:2) L ( yZ ) − α · { ω (1) ≥ B } (cid:3) = E P n (cid:2) Ly − α ( e − ω (1) / − / ) − α · { ω (1) ≥ B } (cid:3) = Ly − α e α/ E P n [ e αω (1) / · { ω (1) ≥ B } (cid:3) ≤ (cid:15), uniformly in n .Having shown the two uniform bounds (8.7) and (8.8), what remains is a standardargument. Fix y and (cid:15) both > 0, and find M such that (8.7) and (8.8) both hold.Let V M ( yZ ) := max (cid:8) − M, min { M, V ( yZ ) } (cid:9) ; that is, V M ( yZ ) is V ( yZ ) “truncated” at ± M . This truncated function is bounded and continuous, so P n ⇒ P implies E P n [ V M ] → E P [ V M ]. And the differences E P n [ V ( yZ ) − V M ( yZ )] are uniformly bounded by 2 (cid:15) . Hence, v n ( y ) = E P n [ V ( yZ )] → E P [ V ( yZ )] = v ( y ) for all y , which implies that u Zn ( x ) → u ( x ) forall x > Step 4. For every y > and (cid:15) > , there exists M > such that E P n (cid:2) | V ( yZ n ) | · { V ( yZ n ) < − M } (cid:3) < (cid:15), uniformly in n. (8.12)The parallel to the uniform inequality (8.7) is obvious: (8.7) uniformly controls the right-hand tail of the integral v Zn ( y ) = E P n [ V ( yZ )]; here we are uniformly controlling theright-hand tail of the integral v Z n n ( y ) = E P n [ V ( yZ n )]. And the same proof works; indeed,in this case we even have E P n [ Z n ] = 1 (as opposed to E P n [ Z ] → Step 5. There is a constant C > such that, for all n , C ≤ Z n ( ω ) Z ( ω ) ≤ C, P n − a.s. (8.13)(The reason for this step is to prove a uniform bound for E P n (cid:2) V ( yZ n ) (cid:3) analogous to (8.8),which is Step 6.)We have that Z n ( ω ) Z ( ω ) = e − a n ω (1) − b n e − ω (1) / − / , where a n = 1 / d/ √ n + o (1 / √ n ), where d = E [ ζ ] / 24, and b n = 1 / o (1). Hence, Z n ( ω ) Z ( ω ) = exp (cid:20) d ω (1) √ n + ω (1) · o (cid:18) √ n (cid:19) + o (1) (cid:21) . Since ζ has bounded support, there is some constant K such that | ζ | ≤ K with probability1, and so | ω (1) | ≤ K √ n , P n -a.s. The ability to find a constant C > tep 6. For every y > and (cid:15) > , there exists M > such that E P n (cid:2) V ( yZ n ) · { V ( yZ n ) >M } (cid:3) < (cid:15), uniformly in n. (8.14)Rewrite the left-hand inequality in (8.13) as Z ( ω ) /C ≤ Z n ( ω ), on the support of P n .Since V is a decreasing function, this implies that, for all y > V ( yZ ( ω ) /C ) ≥ V ( yZ n ( ω )) and, therefore, (cid:8) V ( yZ ( ω ) /C ) > M (cid:9) ⊇ (cid:8) V ( yZ n ( ω )) > M (cid:9) , both restricted to the support of P n . Therefore, for any M > E P n (cid:2) V ( yZ n ) · { V ( yZ n ) >M } (cid:3) ≤ E P n (cid:2) V ( yZ ( ω ) /C ) · { V ( yZ ( ω ) /C ) >M } (cid:3) . But then the proof of the existence of M such that (8.8) is satisfied can be applied to y (cid:48) = y/C , which completes this step. Step 7. For all y > , lim n (cid:12)(cid:12) E P n [ V ( yZ n )] − E P n [ V ( yZ )] (cid:12)(cid:12) = 0 . The argument for this step changes a bit when V (0) = U ( ∞ ) and/or V ( ∞ ) = U (0)are finite valued. So we first give the argument in the case where V (0) = U ( ∞ ) = ∞ and V ( ∞ ) = U (0) = −∞ , and then sketch how to handle the easier cases where one orthe other is finite.Fix (cid:15) > y > 0, and pick M > M (cid:48) = M + 2. Let w and w be the solutions, respectively,to V ( yZ ( w )) = M (cid:48) and V ( yZ ( w )) = − M (cid:48) , where by Z ( w ), we temporarily meanexp( − w/ − / ω is such that ω (1) ∈ [ w , w ], then V ( yZ ( ω )) ∈ [ − M (cid:48) , M (cid:48) ]. Moreover, by the continuity and monotonicity of V , Z , and Z n (the latter twoviewed as functions of ω (1)), and the fact that for fixed w , Z n ( w ) → Z ( w ), there exists n such that for all n > n , ω (1) ∈ [ w , w ] implies that V ( yZ n ( ω )) ∈ [ − M − , M + 1] . The functions V , Z , and Z n are all uniformly and even Lipschitz continuous on compactdomains (for V , that are strictly bounded away from 0), so there is a Lipschitz constant (cid:96) such that (cid:12)(cid:12) V ( yZ n ( w )) − V ( yZ ( w )) (cid:12)(cid:12) ≤ (cid:96) · (cid:12)(cid:12) Z n ( w ) − Z ( w ) (cid:12)(cid:12) , if n > n and w ∈ [ w , w ]. And, on the event ω (1) ∈ [ w , w ], there is n such that forall n > n , | Z n ( ω ) − Z ( ω ) | < (cid:15)/(cid:96) .Hence, on the event D = { ω (1) ∈ [ w , w ] } , and for n > max { n , n } , we have | V ( yZ ( ω )) − V ( yZ n ( ω )) | ≤ (cid:15) , and E P n (cid:2) | V ( yZ ( ω )) − V ( yZ n ( ω )) |· D (cid:3) < (cid:15). By construction,the complement of D is a subset of the union of the four events on which we haveuniformly controlled the integrals of V ( yZ ( ω )) and V ( yZ n ( ω )), so for n > max { n , n } , E P n (cid:2) | V ( yZ ( ω )) − V ( yZ n ( ω )) (cid:3) < (cid:15) , uniformly in n , which proves Step 7.When V (0) and/or V ( ∞ ) are finite, the argument needs a bit of modification. Suppose V (0) < ∞ . This is relevant when yZ and yZ n are both close to zero, which is for paths ω where ω (1) is large. And for those paths, Z n ( ω ) can be quite far from Z ( ω ). Howevereven if these terms are far apart, V ( yZ ( ω )) and V ( yZ n ( ω )) will be close together, sinceeach is close to the finite V (0). A similar argument works for cases where V ( ∞ ) is finite.14 tep 8. Combine Steps 7 and 3 to conclude that lim n →∞ u Z n n = u ( z ) for all z > . Step 3 shows that v Zn ( y ) → v ( y ) for all y > u Zn ( x ) → u ( x )). Step 7 then implies that v Z n n ( y ) → v ( y ) for all y > 0. This, in turn, implieslim n →∞ u Z n n ( x ) → u ( x ) by standard arguments on conjugate functions. Step 9. Combine Steps 8 and Proposition 2 to finish the proof. The argument has already been given.This proof clarifies why we introduced the analogous problem, where a consumer withprobability assessment P n faces complete markets and prices given by Z : Comparingthis with the BSM model, the conjugates v n and v to optimal expected utility functions u n and u are the expectations of a fixed function for different probability measures. So,after controlling the tails of the integrals that define these conjugate functions, we have amore or less standard consequence-of-weak-convergence result in Step 3. In Step 7, both the probability assessments and the prices (for one of the two problems being compared)change with n . While the pairs of problems being compared differ only in the prices,because both the integrand and the integrating measure P n change with n , a level offinicky care is required. Theorem 1 guarantees that for utility functions U that satisfy the conditions (3 . 1) andhave asymptotic elasticity less than one, everything works out nicely within the contextof the BSM model and the discrete-time approximations to BSM that we have posited. It is natural to ask, then, what can be said if we maintain (3 . 1) and these specificmodels of the financial markets, but we look at utility functions U for which AE( U ) = 1.In such cases, it may be that things work out in the sense of Theorem 1. But it is alsopossible that lim sup n →∞ u n ( x ) > u ( x ) . That is, when AE( U ) = 1, Kreps’ conjecture canfail. In this section, we provide an example to illustrate this failure in stark fashion: Inthis example, u ( x ) < ∞ while lim sup n →∞ u n ( x ) = ∞ , both for all x > . In this example (and also in Section 9, where we finish the proof of Proposition 1), weconstruct conjugate functions V taking the form V ( y ) := ∞ (cid:88) k =1 β k y − α k , where α k , β k > { α k } and { β k } are chosen so that the sum defining V ( y ) is finite for all y > U and V , when V has theform V ( y ) = βy − α . Lemma 2. For α > and β > , denote by V α,β ( y ) the function V α,β ( y ) := βy − α for y > . For the utility function U α,β that is conjucate to V α,β , if x = − V (cid:48) α,β ( y ) = βαy − α − for given y , then U α,β ( x ) = (1 + α ) V α,β ( y ) . (9.1)15 he resulting utility function U α,β is U α,β ( x ) = 1 + αα α/ (1+ α ) β / (1+ α ) x α/ (1+ α ) , for x > . Since α > , α/ (1 + α ) ∈ (0 , , and AE ( U α,β ) = α/ (1 + α ) < . Lemma 3 provides some analysis of the consumer’s maximization problem in the con-text of the BSM model, when the conjugate to her utility function has the form V α,β . Lemma 3. Imagine a consumer in the BSM economy whose utility function U α,β is givenby (9.1) . The (dual) value function corresponding to U α,β and V α,β in the BSM economyis v α,β ( y ) = E P (cid:2) V α,β (cid:0) yZ (cid:1)(cid:3) = βe ( α + α ) / y − α = e ( α + α ) / V α,β ( y ) . (9.2) And the primal expected-utility function, giving the supremal expected utility that theconsumer can achieve in the BSM economy as a function of her initial wealth x , is u α,β ( x ) = e α/ U α,β ( x ) . (9.3) Proof. Equation (9.3) is easily derived from (9.2), so we only give the proof of (9.2).Let Y be N (0 , Y − / , / = − Y / − / d P ∗ /d P ) =ln( Z ). Hence, the random variable β (cid:2) y exp( − Y / − / (cid:3) − α has the law of V α,β ( y Z ), andso v α,β ( y ) = E P (cid:20) β (cid:18) y exp (cid:18) − Y − (cid:19)(cid:19) − α (cid:21) = βy − α e α/ E P (cid:20) exp (cid:18) αY (cid:19)(cid:21) = βe ( α + α ) / y − α . The factor e ( α + α ) / recurs occasionally, so to save on keystrokes, let φ ( α ) := e ( α + α ) / .Denote by L ζ ( λ ) the Laplace transform of the law of ζ ; that is L ζ ( λ ) := E [exp( λζ )] . As above, denote by Y a standard (mean 0, variance 1) Normal variate and write L Y ( λ ) := E [exp( λY )] = e λ / . Letting Y n be the scaled sum of n independent copies of ζ , Y n = ζ + . . . + ζ n n / , we have L Y n ( λ ) = L ζ (cid:18) λn / (cid:19) n . L Y n ( λ ) convergesto L Y ( λ ). On the other hand, if E [ ζ ] > 0, by considering — similarly as in the proofof Lemma 1 — the Taylor series expansion up to degree 3 of exp( λY ) around λ = 0, itfollows that, for small enough λ > L ζ ( λ ) > L Y ( λ ) . (9.4)Now consider a conjugate utility function V α,β ( y ) = βy − α as above, its conjugate U α,β , the corresponding value functions for the BSM economy u α,β and its conjugate v α,β , as in Lemma 3, which we compare to the value functions for the various discrete-time economies where the consumer faces complete markets and prices given by Z n . Inthis section, we do not require value functions for discrete-time economies in which theconsumer faces prices Z , so to simplify notation, in this section we write v nα,β for theconjugate-to-the-value-function for the n th discrete-time economy — that is, v nα,β ( y ) := E P n (cid:2) β (cid:0) yZ n (cid:1) − α (cid:3) for all y > . (9.5)And we write u nα,β to denote that primal value function (the conjugate to v nα,β ).Consider, for integer k , the ratio v nα,β ( k ) /v α,β (1 /k ). From (9.2) and (9.5), it is evidentthat this ratio is independent of the value of β , and so, for integers k and n , and α > β > M ( k, n, α ) := v nα,β ( k ) v α,β (1 /k ) . (9.6) Lemma 4. For each integer k , there exists n large enough so that, for α := 2 λ n / , M ( k, n, α ) ≥ k . Proof. Fix k . Without loss of generality, set β = 1. For given n and α = 2 λ n / ,calculate the denominator and numerator in the M ( k, n, α ) separately.For the denominator, we have v α, (cid:18) k (cid:19) = E P (cid:20)(cid:18) k exp (cid:18) − ω (1)2 − (cid:19)(cid:19) − α (cid:21) = (cid:18) k (cid:19) − /α e α/ E P (cid:2) λ n / ω (1) (cid:3) = (cid:18) k (cid:19) − α e α/ E (cid:2) exp( λ n / Y ) (cid:3) = (cid:18) k (cid:19) − α e α/ L Y ( λ n / )= (cid:18) k (cid:19) − α e α/ ( e λ / ) n = (cid:18) k (cid:19) − α e α/ L Y ( λ ) n , where Y is a standard (mean 0, variance 1) Normal variate and E denotes the expectationwith respect to Y .And for the numerator, which we calculate for general y and β before specializing to y = k and β = 1: v nα,β ( y ) = E P n (cid:2) β ( yZ n ) − α (cid:3) = E P n (cid:2) β (cid:0) y exp( − a n ω (1) − b n ) (cid:1) − α (cid:3) = βy − α e αb n E (cid:20) exp (cid:18) λ a n n / (cid:18) ζ n / + . . . + ζ n n / (cid:19)(cid:21) = βy − α e αb n (cid:0) E (cid:2) exp(2 a n λ ζ ) (cid:3)(cid:1) n = βy − α e αb n L ζ (2 a n λ ) n = y − α H ( n, α, β ) for H ( n, α, β ) := βe αb n L ζ (2 a n λ ) n , (9.7)17here the ζ j ’s are i.i.d. copies of ζ , and E denotes expectation with respect to theserandom variables.We therefore have that M ( k, n, α ) = v nα, ( k ) v α, (1 /k ) = k − α e αb n (cid:0) L ζ (2 a n λ ) (cid:1) n (1 /k ) − α e α/ (cid:0) L Y ( λ ) (cid:1) n = (cid:20) k − λ e λ ( b n − / (cid:21) n / (cid:20) L ζ (2 a n λ ) L Y ( λ ) (cid:21) n . (9.8)The term within the first square brackets on the rhs of (9.7) has a finite limit (since b n → / n → ∞ , this term, raised to the power n / is bounded above by G n / for some constant G . And the term within the second set of square brackets convergesto L ζ ( λ ) / L Y ( λ ), which, per (9.4), is a constant strictly greater than 1. This term israised to the power n . Hence, for fixed k , the second term overwhelms the first term forlarge enough n , proving Lemma 4.For each k = 0 , , , . . . , , let n k and α k = 2 λ n / k be the values of n and α guaranteedby Lemma 4. That is, for each k (and for all β > M ( k, n k , α k ) = v n k α k ,β k ( k ) v α k ,β k (1 /k ) > k . (9.9)It is clear that we can add the requirements that n k ≥ k and n k /k is increasing, and wedo so. Since α k = 2 λ n / k , this implies that lim k →∞ α k = ∞ .Let β k := 12 k v α, (1 /k ) so that β k v α k , (1 /k ) = v α k ,β k (1 /k ) = 12 k . (9.10)By (9.9), v n k α k ,β k ( k )2 k v α k ,β k (1 /k ) = v n k α k ,β k ( k ) > k . (9.11)Define V ( y ) := ∞ (cid:88) k =1 β k y − α k = ∞ (cid:88) k =1 V α k ,β k ( y ) , for all y > . (9.12)Clearly, the sum is well defined for all y > 0, and the function has all the propertiesrequired to be conjugate to a utility function U that satisfies Condition (3 . Indeed,(9.10) ensures that v ( y ) = E P (cid:2) V (cid:0) yZ ) (cid:1)(cid:3) = E P (cid:20) ∞ (cid:88) k =1 V α k ,β k ( yZ ) (cid:21) = ∞ (cid:88) k =1 v α k ,β k ( y ) < ∞ , for all y > , which implies that, for U the conjugate (utility) function to V , and for u the (maximalexpected) utility-of-wealth function corresponding to this U within the BSM economy, u ( x ) < ∞ for all x . It may be worth pointing out, however, that this utility function U is not (cid:80) k U α k ,β k . V α k ,β k ( y ) ≤ V ( y ). This implies that, for each k , v n k α k ,β k ( y ) ≤ v n k ( y ) for all y, and therefore u n k α k ,β k ( x ) ≤ u n k ( x ) for all x > . (9.13)Now we enlist Lemma 2. Using the notation from (9.7), for arbitrary y k > 0, let x k := − (cid:18) dv n k α k ,β k dy (cid:19) ( y k ) = α k y − α k − k H ( n k , α k , β k ) = α k y k · v n k α k ,β k (cid:0) y k (cid:1) . Lemma 2 tells us that u n k α k ,β k ( x k ) = v n k α k ,β k (cid:0) y k (cid:1)(cid:0) α k (cid:1) . Hence, u n k α k ,β k ( x k ) x k = (cid:0) α k (cid:1) v n k α k ,β k ( y k ) (cid:0) α k /y k ) v n k α k ,β k ( y k ) = 1 + α k α k y k > y k . (9.14)Choose y k = k / . Since v n k α k ,β k ( k ) > k and v n k α k ,β k ( y ) is decreasing in y , we know that v n k α k ,β k (cid:0) k / (cid:1) > k . Since α k = 2 λ n / k and n k /k is, by construction, nondecreasing, weknow that α k /k / is nondecreasing. Putting these two observations together, we knowthat lim k →∞ x k = lim k →∞ α k k / · v n k α k ,β k (cid:0) k / (cid:1) = ∞ . (9.15)And, for this choice of y k , (9.14) tells us thatlim k →∞ u n k α k ,β k ( x k ) x k = lim k →∞ α k α k k / = ∞ . (9.16)Each function u n k α k ,β k is concave and has value 0 at x = 0. So (9.16) implies that, overthe interval x ∈ (0 , x k ], u n k α k ,β k ( x ) > k / x . Hence, from (9.13), the same is true for u n k ( x ).But as k increases toward ∞ , the intervals [0 , x k ] over which this is true expand to allof (0 , ∞ ] — this is (9.15) — and the underestimate of u n k on this interval approachesinfinity. This implies that lim k →∞ u n k ( x ) = ∞ , for all x > . (9.17)The limit established in (9.17) doesn’t quite accomplish what we set out to do. Wewant to show that, in the n th discrete-time economy, where the consumer faces prices Z n and the constraint that she must be above to synthesize her consumption claim , she can(at least, along a subsequence) asymptotically generate infinite expected utility, althoughshe can only generate finite expected utility in the BSM economy. The limit in (9.17)concerns what expected utility she can generate facing prices Z n and complete markets. But this final step is easy. The properties of ζ that are used to get to (9.16) (incontrast to the finiteness of supremal expected utility in the limit BSM economy) are (1) ζ has mean zero, (2) ζ has variance 1, (3) ζ has finite support, and (4) E [ ζ ] > 0. Forexample, suppose ζ is the asymmetric binomial ζ = (cid:40) , with probability 1/5 , and − / , with with probability 4/5 . 19t is straightforward to verify that all four required properties are satisfied. And, since ζ has two-element support, for any n , it gives complete markets. For this asymmetricbinomial ζ , u n k ( x ) is precisely what she can attain in the n k th discrete-time economy,even with the synthesizability constraint imposed.We therefore have the desired counterexample to Kreps’ conjecture.We come to the same conclusion for any asymmetric binomial with mean zero and an“uptick” greater in absolute value than the “downtick”, as this gives E [ ζ ] > It isnatural to ask, then, what happens in the case of the symmetric binomial, where ζ = ± E [ ζ ] ≤ ζ with mean zero, bounded support, and E [ ζ ] ≤ . For such ζ , the above reasoning does not apply. To the contrary, in the specific caseof the symmetric random walk, we have L ζ ( λ ) = cosh( λ ) ≤ e λ / = L Y ( λ ) , for all λ ∈ R. (9.18)This is most easily seen by comparing the Taylor seriescosh( λ ) = ∞ (cid:88) k =0 λ k (2 k ) ! versus e λ / = ∞ (cid:88) k =0 ( λ / k k ! = ∞ (cid:88) k =0 λ k k k ! . Hence, the logic of the counterexample constructed above, which requires L ζ ( λ ) > L Y ( λ ) for some λ > 0, fails.It may still be true for the symmetric binomial that lim sup n u n ( x ) > u ( x ), for someutility function U (necessarily, in view of Theorem 1, satisfying AE( U ) = 1). Or it maybe that equality holds true, in the case of the symmetric binomial (and, perhaps, in thecase of all symmetric ζ or even ζ such that E [ ζ ] ≤ 10 Proof of Proposition 1 b Because Z = d P ∗ /d P = e − ω (1) / − / , the law of Z is that of exp( Y − / , / ), where Y − / , / is Normal variate with mean − / / 4. By standard calculations, then, thelaw of Z has density function f ( y ) = (cid:114) π y exp (cid:20) − (cid:18) ln( y ) + 18 (cid:19) (cid:21) . Consider the function V ( y ) = 1 f ( y ) = (cid:114) π y exp (cid:20) (cid:18) ln( y ) + 18 (cid:19) (cid:21) . (10.1)Differentiating V shows that it is decreasing in y as long as ln( y ) < / 8, and asimiliar computation shows that V is also convex on (0 , z ) for small enough z > V : R ++ → R ++ that coincides with V on the interval(0 , z ) and is extended to all of (0 , ∞ ) and that is convex, decreasing, differentiable, and Having a variance of 1 changes the formulas but not the basic conclusion. V (cid:48) ( ∞ ) = 0. The conjugate function to this V , denoted U , therefore satisfies theconditions (1.1).We want to determine the range of values of strictly positive y for which the valuefunction v ( y ) = E P (cid:2) V (cid:0) y Z ) (cid:1)(cid:3) (10.2)is finite and for which values it is infinite. Clearly, this depends only on the small values of Z = d P ∗ /d P . Write the expectation (10.2) over the interval where Z = d P ∗ /d P ∈ (0 , z )as E P (cid:2) V (cid:0) yZ ( ω ) (cid:1) ; Z ( ω ) ∈ (0 , z ) (cid:3) = (cid:90) z V ( yz ) f ( z ) dz = (cid:90) z f ( yz ) f ( z ) dz = 1 y (cid:90) yz f ( w/y ) f ( w ) dw, where the last step involves the change of variable w = yz .By straightforward calculation, we find that, for some constant K depending on y , f ( w/y ) yf ( w ) = Kw ( y ) ; hence, E P (cid:2) V ( yz ); Z ∈ (0 , z ) (cid:3) = K (cid:90) yz w ( y ) dw. This integral diverges if 4 ln( y ) ≤ − 1; that is, if y ≤ e − / .By the duality between u and v , this demonstrates, for y = e − / , the possibility that v ( y ) = ∞ for y ≤ y and is finite for y > y ; moreover, as y (cid:38) e − / = y , v ( y ) (cid:37) ∞ .This is possibility b(i) in Proposition 1. And to extend this result to a general y > 0, itsuffices to pass from the function V ( y ) to V y ( y ) = V (cid:18) e − / y y (cid:19) . For possibility b(iii) in Proposition 1: Lemma 2 shows that E P (cid:2)(cid:0) y Z (cid:1) − α (cid:3) = e ( α + α ) / y − α . Recall that φ ( α ) := e ( α + α ) / . Begin by defining, for y > V ( y ) := ∞ (cid:88) k =1 β k y − α k , (10.3)where α k and β k are given by α k = 2 k − β k = 12 k α k φ ( α k ) . The sum (10.3) converges for all y > k (that depends on y ).Moreover, it is evident that V is strictly positive, convex, and twice (and more) con-tinuously differentiable. And from Lemma 3, we have that v ( y ) = E P (cid:2) V (cid:0) y Z (cid:1)(cid:3) = ∞ (cid:88) k =1 β k E P (cid:2)(cid:0) y Z (cid:1) − α k (cid:3) = ∞ (cid:88) k =1 β k φ ( α k ) y − α k (10.4)21nd v (cid:48) ( y ) = − ∞ (cid:88) k =1 β k φ ( α k ) α k y − α k − . (10.5)Substituting in the formulas for α k and β k , (10.4) and (10.5) become v ( y ) = ∞ (cid:88) k =1 y − k +1 k (2 k − 1) and v (cid:48) ( y ) = − ∞ (cid:88) k =1 y − k k . (10.6)By inspection, v ( y ) = ∞ for y < y ≥ 1. And v (cid:48) ( y ) is finite for y ≥ b(iii) in Proposition 1, for y = 1.We leave to the reader the construction of an example of possibility b ( ii ) and exampleswhere the pole is y (cid:54) = 1. References [1] Backhoff, J., and Silva, F. J. (2018). “Sensitivity analysis for expected utility maxi-mization in incomplete Brownian market models,” Mathematics and Financial Eco-nomics , Vol. 12, 387–411.[2] Bayraktar, E., Dolinsky, Y., and Guo, J. “Continuity of Utility Maximization underWeak Convergence,” arXiv: 1811.01420 (Nov. 2018).[3] Bayraktar, E., Dolinskyi, L., and Dolinsky, Y. “Extended Weak Convergence andUtility Maximization with Proportional Transaction Costs,” arXiv: 1912.08863(Dec. 2019).[4] Biagini, S., and Frittelli, M. (2008). “A unified framework for utility maximizationproblems: An Orlicz space approach ,” The Annals of Applied Probability , Vol. 18,929–966.[5] Black, F., and Scholes, M. (1973). “The Pricing of Options and Corporate Liabili-ties,” Journal of Political Economy , Vol. 81, 637–59.[6] Cox, J., and Huang, C. (1989). “Optimal Consumption and Portfolio Policies whenAsset Prices Follow a Diffusion Process,” Journal of Economic Theory , Vol. 49, 33–83.[7] Cox, J., Ross, S. A., and Rubinstein, M. (1979). “Option Pricing: A SimplifiedApproach,” Journal of Financial Economics , Vol. 7, 229–263.[8] Cvitani´c, J., Schachermayer, W., and Wang, H. (2001). “Utility maximization in in-complete markets with random endowment,” Finance and Stochastics , Vol. 5, 259 ˆA-272.[9] Dolinsky, Y., and Neufeld, A. (2018). “Super-replication in fully incomplete mar-kets,” Mathematical Finance , Vol. 28 ,483–515.2210] He, H. (1991) “Optimal consumption-portfolio policies: A convergence from discreteto continuous time models,” Journal of Economic Theory , Vol. 55, 340–363.[11] Hu, Y., Imkeller, P., and M¨uller, M. (2005). “Utility maximization in incompletemarkets,” The Annals of Applied Probability , Vol. 15, 1691–1712.[12] Kabanov, Y. M., and Kramkov, D. O. (1994). “Large financial markets: asymptoticarbitrage and contiguity,” Theory of Probability and Applications , Vol. 39, 222–228.[13] Karatzas, I., Lehoczy, J., Shreve, S., and Xu, G. (1991). “Martingale and Dual-ity Methods for Utility Maximization in an Incomplete Market,” SIAM Journal onControl and Optimization , Vol. 29, 702–730.[14] Karatzas, I., Lehoczy, J., and Shreve, S. (1987). “Optimal portfolio and consumptiondecisions for a ‘small investor’ on a finite horizon,” SIAM Journal on Control andOptimization , Vol. 25, 1157–86.[15] Klein, I., and Schachermayer, W. (1996). “Asymptotic Arbitrage in Non-CompleteLarge Financial Markets,” Theory of Probability and its Applications , Vol. 41, 927–934.[16] Kramkov, D., and Schachermayer, W. (1999). “The Asymptotic Elasticity of Util-ity Functions and Optimal Investment in Incomplete Markets,” Annals of AppliedProbability , Vol. 3, 904–50.[17] Kreps, D. M. (2019). The Black–Scholes–Merton Model as an Idealization ofDiscrete-Time Economies , Econometric Society Monographs , Cambridge UniversityPress.[18] Kreps, D. M., and Schachermayer, W. (2019). “Asymptotic Synthesis ofContingent Claims in a Sequence of Discrete-Time Models,” Stanford Grad-uate School of Business Research Paper No. 3795. Available at SSRN:https://ssrn.com/abstract=3402645.[19] Merton, R. C. (1969). “Lifetime portfolio selection under uncertainty: The continu-ous time case,” The Review of Economics and Statistics , Vol. 51, 247–257.[20] Merton, R. C. (1971). “Optimum consumption and portfolio rules in a continuous-time model,” Journal of Economic Theory , Vol. 3, 373–413.[21] Merton, R. C. (1973). “The Theory of Rational Option Pricing,” Bell Journal ofEconomics and Management Science , Vol. 4, 141–83.[22] Mostovyi, O., and Sirbu, M. (2019). “Sensitivity analysis of the utility maximisationproblem with respect to model perturbations,” Finance and Stochastics , Vol. 23,595-640.[23] Prigent, J.-L. (2003). “Weak Convergence of Financial Markets”, Springer Finance.Springer Verlag, Berlin. 2324] R´asonyi, M., and Stettner, L. (2005). “On utility maximization in discrete-timefinancial market models,” The Annals of Applied Probability , Vol. 15, 1367–1395.[25] Reichlin, C. (2016). “Behavioral portfolio selection: asymptotics and stability alonga sequence of models,” Mathematical Finance , Vol. 26, 51–85.[26] Schachermayer, W. (2004). “Portfolio Optimization in Incomplete Financial Mar-kets,”