Convergence of optimal expected utility for a sequence of discrete‐time markets

Abstract We examine Kreps' conjecture that optimal expected utility in the classic Black–Scholes–Merton (BSM) economy is the limit of optimal expected utility for a sequence of discrete‐time economies that “approach” the BSM economy in a natural sense: The nth discrete‐time economy is generated by a scaled n‐step random walk, based on an unscaled random variable ζ with mean 0, variance 1, and bounded support. We confirm Kreps' conjecture if the consumer's utility function U has asymptotic elasticity strictly less than one, and we provide a counterexample to the conjecture for a utility function U with asymptotic elasticity equal to 1, for ζ such that E[ζ3]>0.


INTRODUCTION
Fix a random variable with mean 0, variance 1, and bounded support. For = 1, 2, …, construct a financial-market economy with two securities, a riskless bond, which serves as numeraire (hence, has interest rate 0) and a risky security, called the stock, which trades against the bond in frictionless markets at time 0, 1∕ , 2∕ , … , ( − 1)∕ . The price process for the stock is generated as follows: For an i.i.d. sequence { ; = 1, 2, …}, where each has the distribution of , the law for the price of the stock at time ∕ is ( ∕ ) ∶= ( ∕ ) , where ( ∕ ) ∶= ∑ =1 √ .
Donsker's theorem tells us that  ⇒ , where  is Wiener measure on 0 [0, 1]; that is, under  is a standard Brownian motion, starting at (0) = 0, and ( ) under  is geometric Brownian motion, so that , together with the riskless bond, prescribes the simple continuous-time economy of Black and Scholes (1973) and Merton (1973) (hereafter, the BSM economy or model).
We imagine an expected-utility-maximizing consumer who is endowed with initial wealth , with which she purchases an initial portfolio of stock and bond. Thereafter, she trades in nonanticipatory and self-financing fashion in the stock and bond (i.e., (a) the information she possesses at time ∕ , on which basis she trades, is [only] the history of the stock price up to and including time ∕ , and (b) any purchase of stock after time 0 is financed by the sale of bonds, and the proceeds of any sale of stock are invested in bonds), seeking to maximize the expectation of a utility function ∶ (0, ∞) → 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 consumer in the th discretetime economy (where the stock and bond trade (only) at times 0, 1∕ , 2∕ , … , ( − 1)∕ ), does the optimal expected utility she can attain approach, as → ∞, what she can optimally attain in the continuous-time BSM economy?
Let ( ) be the supremal expected utility she can attain in the th discrete-time economy if her initial wealth is , and let ( ) be her supremal expected utility in the BSM economy. Kreps (2019) obtains partial one-sided results, showing that lim inf ( ) ≥ ( ). And he proves lim ( ) = ( ) in the very special cases of having either constant absolute or relative risk aversion. But he only conjectures that the second "half," or lim sup ( ) ≤ ( ) is true for general (concave and differentiable) .
Employing the notion of asymptotic elasticity of utility from Kramkov and Schachermayer (1999) (and making extensive use of their analysis), we verify that lim ( ) = ( ) if the utility function has asymptotic elasticity less than 1. However, we show by example that if the asymptotic elasticity of is 1, it is possible that ( ) is finite while lim ( ) = ∞, both for all > 0.

PREVIOUS AND CONTEMPORANEOUS LITERATURE
A substantial body of literature concerns utility maximization problems in financial markets, going back to seminal work by Merton (1969Merton ( , 1971) and continuing, for example, in Biagini and Frittelli (2008); Cox and Huang (1989); Cvitanić, Schachermayer, and Wang (2001); Hu, Imkeller, and Müller (2005); Karatzas, Lehoczy, Shreve, and Xu (1991); Kramkov and Schachermayer (1999); Rásonyi and Stettner (2005); Backhoff and Silva (2018); Dolinsky and Neufeld (2018); Mostovyi and Sirbu (2019). With regard to the continuity of utility maximization under weak convergence of financial markets, positive results are obtained in He (1991); Prigent (2003); Reichlin (2016); these results all assume that, in each discrete-time model, markets are complete. Our interest, motivated by the discussions in Kreps (2019), is in cases where the discrete-time markets are incomplete. Since the seminal paper of Cox, Ross, and Rubinstein (1979), financial economists have believed that, if has two-element support (the so-called binomial case), and so markets are complete in each discrete-time economy, then these discrete-time economy for large behave (in economic terms) like the continuous-time limit, at least for the BSM continuous-time limit. But what if has support of, say, size three, but there are only the two securities? Markets are incomplete for any finite ; does this incompleteness mean very different economic outcomes? Or, if the probability laws  that govern the discrete-time security-price processes converge weakly to , is it then true that lim ( ) = ( )? It is already known that weak convergence is insufficient. Merton (1971) observes that if has constant relative risk aversion with risk-aversion parameter less than 1/2, the optimal 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 scaled copies of a single random variable , has support that is unbounded below. Trying to achieve such a leverage strategy in any of the finite-time economies would give a positive probability of bankruptcy, which is incompatible with these utility functions. The best an investor can do for large enough in these circumstances is to hold 100% of her wealth in the risky asset, which results in lim ( ) < ( ). On the other hand, weak convergence of  to  alone does not preclude the possibility of asymptotic arbitrage (Kabanov & Kramkov, 1994;Klein & Schachermayer, 1996), in which case lim ( ) = ∞, even when ( ) is finite valued (and is very well behaved; see Kreps, 2019, Chapter 7).
By assuming in our setting that has bounded support, we avoid the first problem. And, in our setting, asymptotic arbitrage is precluded (see Kreps, 2019, Proposition 7.1). Still, ill-behaved can pose problems: Within our setting, we show that lim ( ) = ( ) for all > 0 if has asymptotic elasticity less than 1. But if has asymptotic elasticity of 1, even if markets are complete for each , convergence can fail, and fail in spectacular fashion.
The incomplete-market case on which we focus has recently been treated in a setting of greater generality by Bayraktar, Dolinsky, and Guo (2018).
1 Their paper assumes that the financial markets ( ) ∞ =1 are general semi-martingales and the limiting market is a continuous semi-martingale. Also, the utility function in Bayraktar et al. (2018) may measurably depend on the observed trajectory of the stock price. Hence, their model includes our special and paradigmatic case, where ( ) is induced by a single (scaled) random variable and is geometric Brownian motion. In this more general setting, they make assumptions sufficient to show that lim ( ) = ( ). The key assumptions in Bayraktar et al. (2018) are Assumption 2.3(ii), that a certain family of random variables is uniformly integrable, and Assumption 2.5, which effectively assumes away the possibility of asymptotic arbitrage. Lemma 2.2 of Bayraktar et al. (2018) 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 variables (see Equations 12 and 18) from the assumption that has asymptotic elasticity less than 1. And, in our more limited setting, the impossibility of asymptotic arbitrage is a conclusion.

THE UTILITY FUNCTION, ITS CONJUGATE FUNCTION, AND ASYMPTOTIC ELASTICITY
We always assume the following Assumption 3.1. The utility function is strictly increasing, strictly concave, and continuously differentiable, and satisfies the Inada conditions that lim →0 ′ ( ) = ∞ and lim →∞ ′ ( ) = 0.
Moreover, without loss of generality and for notational convenience later, we assume unless otherwise specified that is normalized so that lim →∞ ( ) > 0, without precluding the possibility that lim →∞ ( ) = ∞. (Of course, lim →0 ( ) can be either finite or −∞.) We let denote the conjugate function to : The following results are standard (see, e.g., Kramkov & Schachermayer, 1999) and follow from Assumption (3.1).
The concavity of implies that AE( ) ≤ 1 in all cases; if is bounded above and if (∞) > 0, then AE( ) = 0. But if (∞) = ∞, AE( ) can equal 1; an example is where ( ) = ∕ ln( ) for sufficiently large . 2 Many of our results depend on the assumption that AE( ) < 1, which derives from a comparison of the average and marginal utilities provided by as the argument of approaches ∞: AE( ) < 1 is equivalent to: for some < 1, ′ ( ) < ( ) , for all large enough .
The economic interpretation of this condition can be sharpened if we think of the consumer comparing her marginal utility from very large consumption levels with the average utility she has accrued over some base level 0 . As where "large enough" depends on the value of 0 . As noted in Schachermayer (2004), the concept of asymptotic elasticity connects to the limiting behavior of relative risk aversion by an application of de l'Hôpital's rule as follows: If the limit of the coefficient of relative risk aversion, lim →∞ − ′′ ( )∕ ′ ( ), exists and is strictly positive, then lim →∞ ′ ( )∕ ( ) exists and is less than 1; that is, has asymptotic elasticity less than 1.
As it is believed to be "common" for economic agents to have nonincreasing relative risk aversion, this belief implies that agents with this common property have asymptotic elasticity less than one.

SOLUTIONS FOR THE CONTINUOUS-TIME ECONOMY
As is well known, the continuous-time BSM economy admits a unique equivalent martingale measure (emm) denoted by  * ; that is, a probability measure on Ω that is probabilistically equivalent to  and such that { ( , ); ∈ [0, 1]} is a martingale (over the natural filtration { }). This measure  * has Radon-Nikodym derivative with respect to  given by And, as is well known, this economy has "complete markets." That is, the consumer can construct (as a stochastic integral) any measurable positive contingent claim that she can afford, where what she can afford is given by the single budget constraint  * [ ] ≤ , where  * [⋅] denotes expectation with respect to  * . Hence, with wealth , the consumer's problem is to That is, ( ) is the supremum of expected-utility level that the consumer can achieve in the BSM economy, starting with wealth . It is convenient for later purposes to define the density function ∶ 0 [0, 1] → (0, ∞) by That is, is the unique continuous (in ) version of the random variable  * ∕ . Of course, for any random variable such that (at least) one of the expectations makes sense. And, in this notation We have the following from Cox and Huang (1989), Karatzas, Lehoczy, and Shreve (1987), and Kramkov and Schachermayer (1999). (See, in particular, Kramkov and Schachermayer, 1999 If the consumer's problem has a solution at wealth level > 0, then it has a solution for all wealth levels ′ > 0 such that ′ < .
However, it is possible that, at least for some , ( ) < ∞ and yet the supremum that defines ( ) is not attained by any contingent claim . (An example is given in Kramkov & Schachermayer, 1999, Section 5; we produce examples below.) If (for the given utility function ) this is true for some finite , let be the infimum of all for which there is no solution (but ( ) < ∞); there is a solution at , and so the range of for which there is a solution is the interval (0, ]. The function → ( ) is continuously differentiable and the "Lagrange multiplier function" → ( ) = ′ ( ) is continuous and strictly decreasing on (0, ).

4.5.
Let be the conjugate function to . That is, Then The function → ( ) is convex and nonincreasing, and it is strictly decreasing and continuously differentiable where it is finite.
Of course, it may be that ( ) ≡ ∞, in which case ( ) ≡ ∞. But suppose ( ) < ∞ for some, and therefore for all, > 0. While ( ) is necessarily concave, differentiable, and strictly increasing, it is not in general true that lim →∞ ′ ( ) = 0. That is, the marginal (maximal expected) utility of wealth need not approach zero as the wealth level goes to ∞. Roughly speaking, this can happen when a consumer can purchase ever larger amounts of consumption on events of ever smaller probability, but where the ratio of the amount purchased to the probability of the event approaches infinity at a rapid enough rate. This idea was exploited by Kramkov and Schachermayer (1999) for any utility function that satisfies AE( ) = 1, by choosing, based on , specific measures that are different from but play an analogous role to  * and . Here,  * and  are fixedthey come from BSM-so we show this sort of possibility through the selection of specific utility functions .
Proposition 4.1. Assume that satisfies the conditions (3.1). It is possible that ( ) = ∞ (for all ≥ 0). But if ( ) < ∞ for some , hence for all , it must be that → ( ) is strictly increasing. Moreover, we have the following possibilities.
(a) For some utility functions , lim →∞ ′ ( ) = 0, in which case ( ) is finite for all > 0. (As lim →0 ( ) = lim →∞ ( ) and lim →∞ ( ) = lim →0 ( ), the function can have limit ∞ or a finite limit as approaches 0; and can have limit −∞ or a finite limit as → ∞.) As for the behavior of as ↘ 0 , we have the following possibilities: Moreover, all are possible for any value of 0 > 0.
That is, asymptotic elasticity less than 1 removes the cases given by part b.
The possibility outlined in part a is simple to show: Take utility functions with constant relative risk aversion, for which solutions are well known and fit case a. And the final assertion needs no proof; it derives from (Kramkov & Schachermayer, 1999, Theorem 2.2). To give examples of the three possibilities outlined in part b requires some calculations. As this is a diversion from our main message, we leave this to Section 10. In fact, while part b seems to be the most intriguing aspect of the proposition, we note that for the proof of Theorem 8.1 we only rely on the final assertion of Proposition 4.1.

DISCRETE TIME IS ASYMPTOTICALLY NO WORSE THAN CONTINUOUS TIME
Proposition 5.1. For the sequence of discrete-time economies as described in Section 1, and a utility function that satisfies conditions (3.1), If ( ) = ∞, this proposition still applies, implying that lim →∞ ( ) = ∞.
Proof. Kreps (2019, Proposition 5.2) states that if, in the BSM economy, a bounded and continuous contingent claim satisfies  [ ( )] = and  * [ ] = (so that ( ) ≥ ), then for every > 0, there exists such that, for all > , the consumer in the th discrete-time economy can synthesize a claim for an initial investment of such that  [ ( )] ≥ − . 3 Suppose we know that ( ) < ∞ and, for the given , a solution to the consumer's problem exists (that is, the sup that defines ( ) is a max). We then know, as the solution is of the form = ( ) for some multiplier > 0 (see (4.4) above), that the solution ∶ Ω → (0, ∞) is a continuous function of . By truncating the solution , we get approximately ( ), with what is a bounded and continuous claim. Hence, we conclude that The cases where ( ) < ∞ but no solution exists and where ( ) = ∞ are a bit more delicate, because we don't know, a priori, that we approach the upper bound (finite in the first case, ∞ in the second) with bounded and continuous contingent claims. But we can show this is so. Suppose for some level , there is a measurable contingent claim such that  [ ( )] = and  * [ ] = .
In this context, of course ≥ 0.
Fix > 0. We first replace with a bounded claim ′ , bounded away from ∞ above and away from 0 below, in two steps. First, for < 1 but close to 1, let ∶= Of course, is bounded below by (1 − ) . As for the upper bound, cap at some large . That is, let , be ∧ . For large enough , this is bounded above and will satisfy  [ ( , ] > − ∕2, while capping can only relax the budget constraint. So, it is without loss of generality to assume that our original (that gives expected utility close to and satisfies the budget constraint for ) is bounded above and bounded away from zero. Now apply a combination of Luzin's theorem and Tietze's Extension theorem: We can approximate with a continuous function ′ that differs from on a set of arbitrarily small measure and that satisfies the same upper and lower bounds as ; this allows the choice of ′ to satisfy  [ ( ′ )] > − 3 ∕4. It may be that  * [ ′ ] > , but the last ∕4 is used to replace ′ with ′ − (  * [ ′ ] − ), giving a bounded and continuous contingent claim that costs (or less) and provides expected utility − , at which point Proposition 5.2 of Kreps (2019) can be applied to prove (in general) Proposition 5.1. □

THE "RELAXED" PROBLEM
To tackle the reverse inequality of the one in Proposition 5.1, we need some preparation. For the th discrete-time economy, the consumer faces three types of constraints.
1. She has a level of initial wealth , and her initial portfolio cannot exceed 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, her final consumption bundle must be a synthesizable contingent claim.
The importance of 3 is that, for having support with more than two elements, the consumer does not face "complete markets." However, we know that any final consumption bundle that she constructs in the nth economy subject to these three constraints must satisfy * [ ] = , where * denotes expectation with respect to any probability measure * that is an emm for  . 4 We fix one particular emm for each  , namely, the emm, which we hereafter denote by  * , provided by the Esscher transform: for constants and , chosen such that  * is a martingale probability measure. Specifically, is fixed by the "martingale equation" that and is then fixed as a normalizing constant, given the value of . Moreover, it can be shown is the third moment of , and that lim = 1∕8.
(The notation [⋅] is used to denote expectations over .) Of course,  ⇒  (weakly on 0 [0, 1] endowed with the sup-norm topology) and, for this specific emm,  * ⇒  * . 5 So, suppose we pose the following problem for the consumer: where  [⋅] denotes expectation with respect to  and  * [⋅] denotes expectation with respect to the specific emm  * . In words, we allow the consumer any consumption claim she wishes to purchase, subject only to the constraint that she can afford at the "prices" given by  * ∕  . Let be the function on 0 [0, 1] given by ( ) = exp[− (1) − ]. Hence, is a specific version of the random variable  * ∕  and the constraint  * [ ] = can be rewritten as That is, we can think simply of a consumer facing complete markets with the "pricing kernel" for contingent claims. Using this interpretation, we denote the supremal utility the consumer can obtain in the problem (4.1) as The point of this is that the problem (1) relaxes the constraints that actually face the consumer in the th discrete-time economy; in problem (1) she faces "complete markets"; in her real problem, she faces further "synthesizability" constraints. Hence, we know that ( ) ≥ ( ) for all > 0 and = 1, 2, … If we can show that lim ( ) = ( ), we will know that lim sup ( ) ≤ ( ). This, together with Proposition 5.1, will establish that lim ( ) = ( ). So, this is what we set out to do.

AN ANALOGOUS PROBLEM
In fact, we add one more plot element. As we have stated above, is the function ( ) = exp(− (1)∕2 − 1∕8), which is a version (the unique continuous version) of  * ∕ . Define In words, ( ) is the supremal expected utility that the consumer can attain if she faces complete markets and "prices" in the th discrete-time economy. That is, moving from the consumer's problem in the BSM model to the problem described by Equation (4) changes the consumer's probability assessment from  to  but not the "prices" she faces. In moving from Equation (4) to Equation (2), we keep the probability assessment as  but change the prices from to . This "taking it one step at a time" is useful in the analysis to follow.

IF ASYMPTOTIC ELASTICITY IS LESS THAN 1, OPTIMAL EXPECTED UTILITIES ARE FINITE AND CONVERGE
In this section, we prove the following result.
The term on the right-hand side (rhs) converges to 2 ∕2 , which is (of course)  [exp( (1))]. □ Now we turn to the proof of Theorem 8.1.
This step rates a remark: For "general" price processes as investigated, for instance, in Kramkov and Schachermayer (1999), having asymptotic elasticity less than 1 does not guarantee that the optimal expected utility is finite. The result here strongly depends on the price processes being given by the BSM model.
The second key to Step 1 is the following bound.
Corollary 6.1 in Kramkov and Schachermayer (1999) establishes this bound as a consequence of AE( ) < 1, but only for 0 < ≤ 0 , for some 0 > 0. If we have that (∞) = (0) < 0, we get the bound for all > 0. And, for purposes of this theorem, it is without loss of generality to shift by a constant. So, if (0) ≥ 0, simply replace with ( ) − (0) − , for a suitable constant > 0. Then we have, and henceforth assume, (6) for all ∈ (0, ∞). Hence, we may estimate as the latter expectation is just an exponential moment of a Gaussian variable. Hence, by Kramkov and Schachermayer (1999, Theorem 2.0), the dual value function → ( ) as well as the primal value function → ( ) have finite values; and we deduce as well from part c of Proposition 4.1, that AE( ) ≤ AE( ) < 1.
Define for each the conjugate function , which is By the same argument that gave (7), we have Lemma 8.2 tells us that the expectation on the rhs of Inequality (10) converges, which implies that ( ) is uniformly bounded in for fixed , which, by standard arguments concerning conjugate functions, proves that ∞ ( ) is finite for each .
We show that lim ( ) = ( ) for all > 0, which proves Step 3, again using standard arguments concerning conjugate functions.
Compare ( ) and ( ): If were a bounded function (of course, is continuous), the conclusion would follow immediately from  ⇒ . But is typically not bounded, and so we must show that the contributions to the expectations from the "tails" can be uniformly controlled. We do this by showing the following two uniform bounds.
For every > 0 and > 0, there exists > 0 such that For every > 0 and > 0, there exists > 0 such that Begin with Assertion (11). If (0) is finite, it follows that ( ) ≥ (∞) = (0) for all , so taking = − (0) immediately works. The (slightly) harder case is where (0) = −∞. In this case, recall that, by the Inada conditions, lim →∞ ′ ( ) = − lim →∞ ( ′ ) −1 ( ) = 0. As is convex, this implies that for large enough , By Lemma 8.2, lim →∞  [ ] =  [ ] = 1, which shows Assertion (11). And to show the (uniform) Inequality (12): For the parameters and that give the bound (6) and for fixed > 0 and > 0, let be large enough, so that The existence of such a follows from Lemma 8.2 and, because  ⇒ , And let We have ( ) ≤ − for all > 0, and so Hence, uniformly in . Having shown the two uniform bounds (11) and (12), what remains is a standard argument. Fix and both > 0, and find such that (11) and (12)  Step 4. For every > 0 and > 0, there exists > 0 such that The parallel to the uniform inequality (11) is obvious: (11)  Step 5. There is a constant > 1 such that, for all , (The reason for this step is to prove a uniform bound for  [ ( )] analogous to the uniform bound (12), which is Step 6.) We have that .
As has bounded support, there is some constant such that | | ≤ with probability 1, and so | (1)| ≤ √ ,  -a.s. The ability to find a constant > 1 that gives the inequalities in (17) is now evident.
Step 6. For every > 0 and > 0, there exists > 0 such that Rewrite the left-hand inequality in (17) as ( )∕ ≤ ( ), on the support of  . As is a decreasing function, this implies that, for all > 0, ( ( )∕ ) ≥ ( ( )) and, therefore, both restricted to the support of  . Therefore, for any > 0, But then the proof of the existence of such that the inequality in (12) is satisfied can be applied to ′ = ∕ , which completes this step.
Step 7. For all > 0, The argument for this step changes a bit when (0) = (∞) and/or (∞) = (0) are finite valued. So we first give the argument in the case where (0) = (∞) = ∞ and (∞) = (0) = −∞, and then sketch how to handle the easier cases where one or the other is finite.
Step 9. Combine Steps 8 and Proposition 5.1 to finish the proof. The argument has already been given. □ This proof clarifies why we introduced the analogous problem, where a consumer with probability assessment  faces complete markets and prices given by : Comparing this with the BSM model, the conjugates and to optimal expected utility functions and 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 a more or less standard consequence-ofweak-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 . Although the pairs of problems being compared differ only in the prices, because both the integrand and the integrating measure  change with , a level of finicky care is required.

A COUNTEREXAMPLE TO KREPS' CONJECTURE
Theorem 1 guarantees that for utility functions that satisfy the conditions (3.1) and have asymptotic elasticity less than one, everything works out nicely within the context of 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 specific models of the financial markets, but we look at utility functions for which AE( ) = 1. In such cases, it may be that things work out in the sense of Theorem 1. But it is also possible that lim sup →∞ ( ) > ( ). That is, when AE( ) = 1, Kreps' conjecture can fail. In this section, we provide an example to illustrate this failure in stark fashion: In this example, ( ) < ∞ while lim sup →∞ ( ) = ∞, both for all > 0.
In this example (and also in Section 9, where we finish the proof of Proposition 4.1), we construct conjugate functions taking the form where , > 0 and the sequences { } and { } are chosen so that the sum defining ( ) is finite for all > 0.
Lemma 9.2 provides some analysis of the consumer's maximization problem in the context of the BSM model, when the conjugate to her utility function has the form , .
Lemma 9.2. Imagine a consumer in the BSM economy whose utility function , is given by Equation (19). The (dual) value function corresponding to , and , in the BSM economy is And the primal expected-utility function, giving the supremal expected utility that the consumer can achieve in the BSM economy as a function of her initial wealth , is Proof. Equation (21) is easily derived from Equation (20), so we only give the proof of Equation (20). Let be (0, 1)-distributed, so that −1∕8,1∕4 = − ∕2 − 1∕8 has the law of ln(  * ∕ ) = ln( ). Hence, the random variable [ exp(− ∕2 − 1∕8)] − has the law of , ( ), and so The factor ( + 2 )∕8 recurs occasionally, so to save on keystrokes, let ( ) ∶= ( + 2 )∕8 . Denote by  ( ) the Laplace transform of the law of ; that is, As above, denote by a standard (mean 0, variance 1) Normal variate and write Letting be the scaled sum of independent copies of , The Central Limit theorem corresponds to the well-known fact that  ( ) converges to  ( ).
On the other hand, if [ 3 ] > 0, by considering-similarly as in the proof of Lemma 8.2-the Taylor series expansion up to degree 3 of exp( ) around = 0, it follows that, for small enough Now consider a conjugate utility function , ( ) = − as above, its conjugate , , the corresponding value functions for the BSM economy , and its conjugate , , as in Lemma 9.2, which we compare to the value functions for the various discrete-time economies where the consumer faces complete markets and prices given by . In this section, we do not require value functions for discrete-time economies in which the consumer faces prices , so to simplify notation, we write , for the conjugate-to-the-value-function for the th discrete-time economy-that is, And we write , to denote that primal value function (the conjugate to , ). Consider, for integer , the ratio , ( )∕ , (1∕ ). From Equations (20) and (23), it is evident that this ratio is independent of the value of , and so, for integers and , and > 0 (and any > 0), let . ( where is a standard (mean 0, variance 1) Normal variate and denotes the expectation with respect to . And for the numerator, which we calculate for general and before specializing to = and = 1: where the 's are i.i.d. copies of , and denotes expectation with respect to these random variables. We therefore have that .
The term within the first square brackets on the rhs of Equation (26) has a finite limit (as → 1∕8), so as → ∞, this term, raised to the power 1∕2 is bounded above by 1∕2 for some constant . And the term within the second set of square brackets converges to  ( 0 )∕ ( 0 ), which, per Equation (22), is a constant strictly greater than 1. This term is raised to the power . Hence, for fixed , the second term overwhelms the first term for large enough , proving Lemma 9.3. □ For each = 0, 1, 2, …, let and = 2 0 1∕2 be the values of and guaranteed by Lemma 9.3. That is, for each (and for all > 0), Choose = 1∕2 . As , ( ) > 2 and , ( ) is decreasing in , we know that , ( 1∕2 ) > 2 . As = 2 0 1∕2 and ∕ is, by construction, nondecreasing, we know that ∕ 1∕2 is nondecreasing. Putting these two observations together, we know that And, for the choice = 1∕2 , (32) and (33) Each function , is concave and has value 0 at = 0. So, Equation (34) implies that, over the interval ∈ (0, ], , ( ) > 1∕2 . Hence, from Inequality (31), the same is true for ( ). But as increases toward ∞, the intervals [0, ] over which this is true expand to all of (0, ∞]-this is Equation (33)-and the underestimate of on this interval approaches infinity. This implies that lim →∞ ( ) = ∞, for all > 0.
The limit established in Equation (35) does not quite accomplish what we set out to do. We want to show that, in the th discrete-time economy, where the consumer faces prices 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, although she can only generate finite expected utility in the BSM economy. The limit in Equation (35) concerns what expected utility she can generate facing prices and complete markets. But this final step is easy. The properties of that are used to get to Equation (34) (in contrast to the finiteness of supremal expected utility in the limit BSM economy) are (a) has mean 0, (b) has variance 1, (c) has finite support, and (d) [ 3 ] > 0. For example, suppose is the asymmetric binomial with probability 1∕5, and −1∕2, with probability 4∕5.
It is straightforward to verify that all four required properties are satisfied. And, as has twoelement support, for any , it gives complete markets. For this asymmetric binomial , ( ) is precisely what she can attain in the 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 0 and an "uptick" greater in absolute value than the "downtick," as this gives [ 3 ] > 0. 7 It is natural to ask, then, what happens in the case of the symmetric binomial, where = ±1, each with probability 1/2 or, more generally, for any asymmetric binomial with [ 3 ] ≤ 0 or, even more generally, any with mean 0, bounded support, and [ 3 ] ≤ 0. For such , the above reasoning does not apply. To the contrary, in the specific case of the symmetric random walk, we have This is most easily seen by comparing the Taylor series Hence, the logic of the counterexample constructed above, which requires  ( 0 ) >  ( 0 ) for some 0 > 0, fails. It may still be true for the symmetric binomial that lim sup ( ) > ( ), for some utility function (necessarily, in view of Theorem 1, satisfying AE( ) = 1). Or it may be that equality holds true, in the case of the symmetric binomial (and, perhaps, in the case of all symmetric or even such that [ 3 ] ≤ 0). We leave this question open.
We want to determine the range of values of strictly positive for which the value function where the last step involves the change of variable = .
The sum (39) converges for all > 0 and, in fact, does so faster than geometrically past some 0 (that depends on ). Moreover, it is evident that is strictly positive, convex, and twice (and more) continuously differentiable. And from Lemma 9.2, we have Substituting in the formulas for and , Equations (40) and (41) By inspection, ( ) = ∞ for < 1 and is finite for ≥ 1. And ′ ( ) is finite for ≥ 1. This, then, is the possibility b(iii) in Proposition 4.1, for 0 = 1. We leave to the reader the construction of an example of possibility b(ii) and examples where the pole is 0 ≠ 1.

A C K N O W L E D G M E N T S
Kreps' research is supported by the Graduate School of Business, Stanford University. Schachermayer gratefully acknowledges support by the Austrian Science Fund (FWF) under grant P28661 as well as 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 while this paper was being written.  (Bayraktar et al., 2018) was put on arXiv in November 2018; the authors kindly brought their paper to our attention after a first version of the current paper appeared on arXiv in July 2019. The references to their results refer to the arXiv version of Bayraktar et al. (2018) from September 2019. Compare also to the recent paper Bayraktar et al. (2019). 2 The conditions (3.1) on include lim →∞ ( ) > 0; this is solely so that AE( ) ≥ 0 in all cases. 3 The proof of this proposition relies on Theorem 1 in Kreps and Schachermayer (2019), which says that any bounded and continuous contingent claim can be synthesized with " -controlled risk" in the th discretetime economy for large enough , where "approximately synthesized" means: For given > 0 and large enough (depending on ), the synthesized claim, satisfies  (| − | > ) < ; and " -controlled risk" means that the synthesized claim satisfies ( ) ∈ (inf ′ ( ′ ), sup ′ ( ′ )) with  -probability 1. 4 Because we assume that has mean 0 and variance 1, we know that  admits emms. 5 For a detailed derivation, see Kreps (2019, Lemma 5.1). 6 It may be worth pointing out, however, that this utility function is not ∑ , . 7 Having a variance different from 1 changes the formulas but not the basic conclusion.