I am grateful to Martin Schweizer and an anonymous referee for helpful comments. Financial support by the National Centre of Competence in Research “Financial Valuation and Risk Management” (NCCR FINRISK) is gratefully acknowledged.

ARTICLE

# BEHAVIORAL PORTFOLIO SELECTION: ASYMPTOTICS AND STABILITY ALONG A SEQUENCE OF MODELS

Version of Record online: 11 OCT 2013

DOI: 10.1111/mafi.12053

© 2013 Wiley Periodicals, Inc.

Additional Information

#### How to Cite

Reichlin, C. (2016), BEHAVIORAL PORTFOLIO SELECTION: ASYMPTOTICS AND STABILITY ALONG A SEQUENCE OF MODELS. Mathematical Finance, 26: 51–85. doi: 10.1111/mafi.12053

#### Publication History

- Issue online: 11 JAN 2016
- Version of Record online: 11 OCT 2013
- Manuscript Revised: JUN 2013
- Manuscript Received: JUL 2012

- Abstract
- Article
- References
- Cited By

### Keywords:

- portfolio selection;
- nonconcave utility;
- Choquet integral;
- stability;
- convergence;
- behavioral finance

### Abstract

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. PROBLEM FORMULATION AND MAIN RESULTS
- 3. APPLICATIONS
- 4. STABILITY OF THE DEMAND PROBLEM FOR RDEU
- 5. STABILITY OF THE DEMAND PROBLEM FOR ENCU
- 6. CONCLUSION
- APPENDIX
- REFERENCES

We consider a sequence of financial markets that converges weakly in a suitable sense and maximize a behavioral preference functional in each market. For expected *concave* utilities, it is well known that the maximal expected utilities and the corresponding final positions converge to the corresponding quantities in the limit model. We prove similar results for nonconcave utilities and distorted expectations as employed in behavioral finance, and we illustrate by a counterexample that these results require a stronger notion of convergence of the underlying models compared to the *concave* utility maximization. We use the results to analyze the stability of behavioral portfolio selection problems and to provide numerically tractable methods to solve such problems in complete continuous-time models.

### 1. INTRODUCTION

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. PROBLEM FORMULATION AND MAIN RESULTS
- 3. APPLICATIONS
- 4. STABILITY OF THE DEMAND PROBLEM FOR RDEU
- 5. STABILITY OF THE DEMAND PROBLEM FOR ENCU
- 6. CONCLUSION
- APPENDIX
- REFERENCES

Portfolio optimization constitutes a fundamental problem in economics. For classical preference functionals defined by expected concave utility, this problem and its solution are well known; see Biagini (2010) for an overview. In practical applications, however, these classical functionals are often too restrictive. Nonstandard incentives for risk-averse agents such as option compensation or performance-based salary systems lead to nonconcave demand problems. In addition, the behavioral finance literature suggests using nonlinear expectations to account for the observation that people tend to overweight extreme events with small probabilities; see for instance Tversky and Kahneman (1992) and references therein.

The demand problem for nonconcave preference functionals (with or without nonlinear expectations) is less standard; a rigorous mathematical analysis has started only recently. Jin and Zhou (2008), Carlier and Dana (2011), He and Zhou (2011a), Jin Zhang, and Zhou (2011), and Rásonyi and Rodrigues (2013) analyze demand problems for nonconcave utility functions and nonlinear expectations in continuous-time models. De Giorgi and Hens (2006), Bernard and Ghossoub (2010), He and Zhou (2011b), Carassus and Rásonyi (2015), and Pirvu and Schulze (2012) study similar problems in discrete time. Larsen (2005), Carassus and Pham (2009), Rieger (2012), Bichuch and Sturm (2011), and Muraviev and Rogers (2013) consider related problems with linear expectations. All these general results lead to several applications: Sung et al. (2011) and Bernard et al. (2015) analyze the consequences of behavioral demand on the optimal insurance design; Jin and Zhou (2013) quantify the notion of greed in the context of behavioral demand problems; and Xu and Zhou (2013) study optimal stopping for behavioral preference functionals. A detailed overview on recent developments in mathematical behavioral finance can be found in Zhou (2010).

All the work mentioned above studies the demand problem for a fixed underlying model. Since one is never exactly sure of the accuracy of a proposed model, it is important to know whether the behavioral predictions generated by a model change drastically if one slightly perturbs the model. To the best of our knowledge, results on the stability of behavioral portfolio selection problems have not been available in the literature so far, and the main purpose of this paper is to study this issue in detail. Formally, we consider a sequence of models, each represented by some probability space and some pricing measure , and we assume that this sequence converges weakly in a suitable sense (to be made precise later) to a limit model . For each model, we are interested in the demand problem

- (1.1)

where the functional is defined by

- (1.2)

for a *nonconcave* and *nonsmooth* utility function *U* on and a strictly increasing function representing the *probability distortion* of the beliefs. We are then interested in the asymptotics of the value (indirect utility) and its maximizer , and we want to compare them with the analogous quantities in the limit model.

Functionals of the form (1.2) as well as the demand problem (1.1) have well-established economic interpretations. From a theoretical point of view, (1.2) arises naturally as a representation for preference functionals satisfying a certain comonotonicity condition; see for instance Schmeidler (1986). In applications, they also serve as the main building block for several behavioral theories such as *rank-dependent expected utility* (RDEU); see Quiggin (1993) or *cumulative prospect theory* (CPT); see Tversky and Kahneman (1992). If we set , then (1.2) covers the classical expected utility functional.

The problem (1.1) can be seen as portfolio optimization problem in a complete market. More precisely, consider an agent in a complete financial market who is dynamically trading in the underlying (discounted) assets *S* with filtration and time horizon *T*. The agent invests the initial capital *x* in self-financing strategies with nonnegative associated wealth process in order to maximize his/her preference functional that only depends on the terminal wealth. Because the market is complete, any fixed -measurable nonnegative random variable *f* is the terminal wealth associated to a self-financing strategy if and only if , where denotes the unique martingale measure for *S*. The agent is thus brought back to solving a static problem of type (1.1).

The main ingredients of the model are described by . The assumption that the sequence of models converges (in a suitable sense) to a limit model means that the economic situation described by the *n*th model is for sufficiently large *n* close (in a suitable sense) to the one described by the limit model. Our main contribution is to give easily verifiable assumptions such that similar economic situations also imply similar behavioral predictions for the agent, in the sense that the values as well as (along a subsequence) the optimal final positions converge to the corresponding quantities in the limit model.

In *concave* utility maximization, the (essentially) sufficient condition for these stability results is the weak convergence of the *pricing density* (or *pricing kernel*) to ; see, for instance, He (1991) and Prigent (2003). However, in our nonconcave setting, we present an example of a sequence of financial markets for which converges weakly to , but where the limit and as well as the corresponding final positions differ substantially. We discuss these new effects in detail and give sufficient conditions to prevent such unpleasant phenomena.

In order to illustrate the main results, we provide several applications. First, we consider a sequence of binomial models approximating the Black–Scholes model; this is the typical example for the transition from discrete- to continuous-time models. Apart from its purely theoretical interest, this example is also of practical relevance since the discrete-time analysis provides numerical procedures for the explicit computation of the optimal consumption. This allows one to numerically determine the value function for (computationally difficult) continuous-time models via the value functions for (computationally tractable) discrete-time models. The second application is motivated by the practical difficulties to calibrate an underlying model. As one example, we therefore study whether a (small) misspecification of the drift in the Black–Scholes model significantly influences the optimal behavior of the agent. In both examples above, we use a fixed time horizon *T* for the portfolio optimization problem. In practical applications, however, the time horizon might be uncertain or changing. In the third application, we therefore analyze whether or not a (marginal) misspecification of the investment horizon significantly influences the optimal behavior of the agent.

These examples show the necessity of our analysis: Our models are at best approximations to the reality, so if we perturb one model slightly in a reasonable way and the behavioral predictions generated by the model change drastically, we may suspect that the model cannot tell us much about the real world behavior. Our convergence results demonstrate that for a fairly broad class of preference functionals and models, the optimal behavior is stable with respect to such small perturbations.

The paper is structured as follows. In Section 'PROBLEM FORMULATION AND MAIN RESULTS', we abstractly describe the sequence of models, preference functionals, and optimization problems. We also formulate and discuss the main result. In Section 'APPLICATIONS', we present three applications of the main result. This also allows us to discuss the connections to the existing literature in more detail. In addition, we provide a concrete numerical example to illustrate the results. We prove the main result in Sections 'STABILITY OF THE DEMAND PROBLEM FOR RDEU' and 'STABILITY OF THE DEMAND PROBLEM FOR ENCU'.

### 2. PROBLEM FORMULATION AND MAIN RESULTS

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. PROBLEM FORMULATION AND MAIN RESULTS
- 3. APPLICATIONS
- 4. STABILITY OF THE DEMAND PROBLEM FOR RDEU
- 5. STABILITY OF THE DEMAND PROBLEM FOR ENCU
- 6. CONCLUSION
- APPENDIX
- REFERENCES

The following notation is used. If , denote , and . For a function *G* and a random variable *X*, we write for the positive/negative parts . For a sequence of random variables, we denote weak convergence of to *f*^{0} by . A *quantile function* of a distribution function *F* is a generalized inverse of *F*, i.e., a function satisfying

Quantile functions are not unique, but any two for a given *F* coincide a.e. on (0, 1). Thus, we sometimes blur the distinction between “the” and “a” quantile function. A quantile function of a random variable *f* is understood to be a quantile function of the distribution *F* of the random variable *f*. If the sequence converges weakly to *f*^{0}, then any corresponding sequence of quantile functions converges a.e. on (0,1) to ; see, for instance, Theorem 25.6 of Billingsley (1986). More properties of quantile functions can be found in Appendix A.3 of Föllmer and Schied (2011).

#### 2.1. Sequence of Models and Optimization Problems

We consider a sequence of probability spaces , where the probability space is atomless; see Definition A.26 and Proposition A.27 in Föllmer and Schied (2011) for a precise definition and equivalent formulations. On each probability space, there is a probability measure equivalent to with density . We refer to as *pricing measure* and to as *pricing density* (or *pricing kernel*). We assume that the sequence converges weakly to the pricing density in the atomless model, i.e., . To ensure that the atomic structure tend to the atomless structure, we assume that the atoms disappear in the following sense. Let be the set of atoms in (with respect to ).

Assumption 2.1. .

We impose the following integrability condition on .

Assumption 2.2. The family is uniformly integrable for all .

Having specified the sequence of models, we turn to the preference functionals. One cornerstone is the concept of a nonconcave utility function.

Definition 2.3. A *nonconcave utility* is a function , which is strictly increasing, continuous and satisfies the growth condition

- (2.1)

We consider only nonconcave utility functions defined on . To avoid any ambiguity, we set for and define and . Without loss of generality, we may assume that . Observe that we do not assume that *U* is concave. In the concave case, the growth condition (2.1) not only implies, but is even equivalent to, the Inada condition at ∞ that .

Definition 2.4. The *concave envelope* of *U* is the smallest concave function such that holds for all .

Some properties of as well as of can be found in Lemma 2.11 of Reichlin (2013). A key tool to study the relation between *U* and is the conjugate of *U* defined by

Because of the nonconcavity of *U*, the concave envelope is not necessarily strictly concave and the latter implies that *J* is no longer smooth; we therefore work with the subdifferential that is denoted by for the convex function *J* and by for the concave function . The right- and left-hand derivatives of *J* are denoted by and . Our proofs (mainly in the Appendix) use the classical duality relations between , *J*, , and . Precise statements and proofs can be found in Lemma 2.12 of Reichlin (2013).

In classical concave utility maximization, the asymptotic elasticity (AE) of the utility function is of importance. In particular, many results impose an upper bound on . For a nonconcave utility function, we impose a similar condition via the AE of the conjugate *J*,

In order to define our preference functionals, we introduce an additional function *w* that represents the distortion of the distribution of the beliefs.

Definition 2.5. A *distortion* is a function that is strictly increasing and satisfies and .

In the literature, one can find several explicit functional forms for *w*. The most prominent example is

- (2.2)

suggested by Kahneman and Tversky (1979); they use the parameter . For each model, we now define a preference functional on .

Definition 2.6. We consider one of the following cases: **Case 1**: The preference functional is defined by

- (2.3)

for a distortion *w* and a nonconcave utility *U* satisfying . We refer to this case as *rank-dependent expected utility* (RDEU). **Case 2**: The preference functional is defined by

- (2.4)

for a nonconcave utility *U*, where we set if . We refer to this case as *expected nonconcave utility* (ENCU).

The functional defined in (2.3) can be seen as a Choquet integral with respect to the monotone set function ; see Chapter 5 of Denneberg (1994) for an exposition of this concept. In the case , the functional in (2.3) coincides with the classical expected utility in (2.4) for a positive nonconcave utility *U*. We distinguish the two cases since the conditions for their treatments will be different.

Finally, we formulate the sequence of optimization problems. For a fixed (initial capital) , the (budget) set in the *n*th model is

For each model, we are interested in the *demand problem*

- (2.5)

An element is *optimal* if . By a *maximizer* for , we mean an optimal element for the optimization problem (2.5).

#### 2.2 Main Results

Even in the classical case of expected concave utility, the stability of the utility maximization problem is only obtained under suitable growth conditions on *U* (or its conjugate *J*). In the case of the RDEU functional in (2.3), the corresponding assumption has to be imposed jointly on *U* and *w*.

Assumption 2.7. We suppose that

- (2.6)

- (2.7)

- (2.8)

with and . This allows us to find and fix λ such that and .

This assumption is inspired by Assumption 4.1 in Carassus and Rásonyi (2015). For the example distortion in (2.2), condition (2.7) is satisfied. In the case without distortion, , (2.7) is satisfied for . A sufficient condition for (2.6) is (see Lemma A.3). For later reference, we summarize the case-dependent assumptions.

Assumption 2.8. We assume that we have one of the following cases:

**RDEU**: Let be defined as in (2.3). In this case, we suppose that the distribution of φ^{0} is continuous and that Assumption 2.7 is satisfied.

**ENCU**: Let be defined as in (2.4). In this case, we suppose that Assumption 2.1 and are satisfied.

We are now in a position to formulate the main result of this paper. Note that this covers simultaneously both cases.

The maximizers for are not necessarily unique; see Example 3.7 of Reichlin (2013). Weak convergence along a subsequence of maximizers is therefore the best we can hope for. Moreover, note that for the second statement in Theorem 2.9, we start with a sequence of maximizers for . For the ENCU functional in (2.4), the existence of a maximizer for is guaranteed under the present assumptions; see Theorem 3.4 of Reichlin (2013). For the RDEU functional in (2.3), on the other hand, the existence of a maximizer for has to be verified in any given setting. One sufficient criterion is that (or due to the equivalence of and ) is atomless (see Remark 4.5 below). Another sufficient criterion is that consists of finitely many atoms. The latter, in fact, implies that any maximizing sequence for is bounded; this allows us to extract a subsequence a.s. converging to some limit , and arguments similar to the ones in Proposition 4.4 show that is a maximizer for . These two criteria cover all the examples discussed in Section 'APPLICATIONS'.

The assumption that *U* is strictly increasing and continuous is not strictly necessary; it avoids some (more) technical details. Let us shortly discuss a relevant excluded special case.

Remark 2.10. The ENCU functional defined in (2.4) does not cover the piecewise constant function that describes the *goal-reaching problem* initiated by Kulldorff (1993) and investigated extensively by Browne (1999; 2000). But under the assumption that φ^{0} has a continuous distribution, one can adapt the arguments in its proof to show that the results of Theorem 2.9 also hold for the goal-reaching problem. We provide a detailed argument at the end of Section 'STABILITY OF THE DEMAND PROBLEM FOR ENCU'.

#### 2.3. The Need for Assumption 2.1

For expected *concave* utilities, Assumption 2.1 is not necessary to obtain Theorem 2.9; see Proposition 5.4 below. However, for nonconcave utilities, Assumption 2.1 cannot be dropped. The difference between these cases can be explained as follows. For a risk-averse agent with a concave *U*, the optimal final position is (essentially) -measurable, and so it is enough to have convergence in distribution of the sequence of pricing densities. For risk-seeking agents, the optimal final position is not necessarily -measurable. Additional information (if available) is used by the agent to avoid the nonconcave part of *U*. In the atomless limit model, every payoff distribution can be supported, and Assumption 2.1 ensures that also the models along the sequence become sufficiently rich as . More concretely, Assumption 2.1 excludes the (pathological) behavior illustrated in the next example.

Example 2.11. Consider a nonconcave utility *U* with that is strictly concave on (0, *a*) and . The initial capital *x* is in , but not exactly in the middle of the interval . The probability spaces are all given by the same probability space consisting of two states with ; and is an arbitrary atomless probability space. Set for all . Jensen's inequality and Theorem 5.1 of Reichlin (2013) give . On the other hand, we have for every and we now show that for *x* chosen above. First, note that admits a maximizer since the model consists of two atoms (see the discussion following Theorem 2.9). The maximizer satisfies since *U* is strictly increasing, so we can replace by . We therefore get the inequality

The first inequality is an equality if and only if both values and are not in ; the second inequality is an equality if and only if the two values and are in . But these two conditions cannot be satisfied at the same time by our choice of *x*. This shows that . We conclude that converges as (it is constant), but the limit is not .

### 3. APPLICATIONS

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. PROBLEM FORMULATION AND MAIN RESULTS
- 3. APPLICATIONS
- 4. STABILITY OF THE DEMAND PROBLEM FOR RDEU
- 5. STABILITY OF THE DEMAND PROBLEM FOR ENCU
- 6. CONCLUSION
- APPENDIX
- REFERENCES

So far, our analysis has been conducted for an abstract sequence of models. We now present three types of application to illustrate the main results. We also provide a numerical example in Section 'A Numerical Illustration' to visualize our results.

#### 3.1. (Numerical) Computation of the Value Function

In recent years, there has been remarkable progress in the problem of behavioral portfolio selection. In particular, there are several new results for complete markets in continuous time. Most of them give the existence of a solution and describe the structure of the optimal final position as a decreasing function of the pricing density. While these results are interesting from a theoretical point of view, they are less helpful for explicit computations. In this section, we show how Theorem 2.9 can be used to determine the value function numerically for a complete model in continuous time.

The idea is to approximate the (computationally difficult) continuous-time model by a sequence of (computationally tractable) discrete-time models. We illustrate this for the Black–Scholes model that can be approximated by a sequence of binomial models. This is the typical example for the transition from a discrete- to a continuous-time setting. In this example, the limit model is atomless, while the approximating models are not. He (1991) and Prigent (2003) analyze the stability for expected *concave* utilities for this setting by directly analyzing the sequence of optimal terminal wealths as a function of . This is possible due to the concavity of their utility function, but cannot be used here.

To fix ideas, we briefly recall the (classical) binomial approximation of the Black–Scholes model to verify that our assumptions in Section 'PROBLEM FORMULATION AND MAIN RESULTS' are satisfied. We consider a time horizon , a probability space on which there is a standard Brownian motion , and a (discounted) market consisting of a savings account and one stock *S* described by

in the filtration generated by *W*. The pricing density is then given by

For the construction of the *n*th approximation, we start with a probability space on which we have independent and identically distributed random variables taking values 1 and −1, both with probability . For any *n*, we consider the *n*-step market consisting of the savings account and the stock given by for and

This process is right-continuous with left limits. The filtration generated by is denoted by and we take . The market is active at the times . It is well known that this market is complete, and we denote by the unique martingale measure. The martingale condition implies that

and solving gives . This is positive for *n* large enough and we only consider such *n* from now on. The measures and are equivalent on and we denote the pricing density by . It is shown in Theorem 1 of He (1990) that ; this is a consequence of the central limit theorem. The set of atoms in can be identified with the paths of . The -probability for a path is of the form

for some (which is the number of up moves in the path). For *n* large, we have and we see that . Taking the limit gives , which means that Assumption 2.1 is satisfied. The distribution of φ^{0} is continuous if . Finally, a proof of uniform integrability of for can be found in Prigent (2003, p. 172, Lemma c).

We conclude that all the assumptions of Section 'PROBLEM FORMULATION AND MAIN RESULTS' are satisfied. We can therefore apply Theorem 2.9 to relate the optimization problem in the Black–Scholes model with the sequence of optimization problems in the sequence of binomial models. More precisely, Theorem 2.9 shows that the sequence of value functions in the binomial models converges to the value function *v*^{0} in the Black–Scholes model and that the sequence of maximizers converges along a subsequence. In particular, in the case of the preference functional (2.3), this turns out to be useful for computational purposes: While there are (abstract) results on the existence of a maximizer in the Black–Scholes model in the literature, these results are less helpful to determine a maximizer and the corresponding value explicitly. In the binomial model, however, the (numerical) computation of the value function and its maximizers is straightforward since the model consists of a finite number of atoms. In this context, Theorem 2.9 provides the insight that we can use the value functions in the binomial model to approximate the value function in the Black–Scholes model. This gives a method to determine numerically the value function *v*_{0} in the Black–Scholes model. Note that for *RDEU*, we have no dynamic programming and hence no description of *v*_{0} by a (HJB) PDE we could solve numerically.

#### 3.2. Stability Results

In this section, we use Theorem 2.9 to show, as explained in Section 'INTRODUCTION', the stability of the portfolio choice results for a fixed model with respect to small perturbations. While Section '(Numerical) Computation of the Value Function' can be seen as perturbation of the underlying model itself, we are interested here in perturbations of a model's parameters.

##### 3.2.1. Misspecifications of the Market Model

The first example is motivated by the practical difficulties one encounters when trying to calibrate an underlying model. In this section, we analyze how the optimal final position and the corresponding value are affected by a (small) misspecification of the underlying market model.

This question is well studied for expected *concave* utilities; see, for instance, Larsen and Žitković (2007) and Kardaras and Žitković (2011). Their sequence of model classes is more general in the sense that they need not restrict the setup to a single pricing density. However, the key to solving their problem is the classical duality theory that can be applied since their utilities are (strictly) concave. In our setting, this is not possible.

As one example in our framework, we can think of the Black–Scholes model where it is generally difficult to measure the drift. To formalize this situation, we fix some time horizon and a probability space on which there is a Brownian motion . We introduce the sequence of probability spaces by for . In order to define a sequence of price processes, we consider a sequence converging to some drift parameter in the limit model. For each *n*, we consider a (discounted) market consisting of a savings account and one stock described by

in the filtration generated by *W*. The pricing density for the *n*th model is then given by

In this example, each model is atomless. Assumption 2.1 is therefore trivially satisfied. Moreover, looking at the explicit form of shows that for all , so in particular . Finally, it is straightforward to check the uniform integrability of for any . Hence, Assumption 2.2 is satisfied.

We conclude that the assumptions of Section 'PROBLEM FORMULATION AND MAIN RESULTS' are satisfied and we can apply the results there. Theorem 2.9 tells us that the value functions (as well as the corresponding maximizers along a subsequence) for the model with drift converge to the corresponding quantities in the model with drift μ^{0}. The economic interpretation of this result is that the behavioral prediction does not change drastically if we slightly perturb the drift.

It is also worth mentioning that the above arguments only use convergence of the market price of risk to . If we consider more generally a stochastic market price of risk , then assuming gives weak convergence of the stochastic exponential to ; see Proposition A.1 in Larsen and Žitković (2007). In addition, one then needs some integrability condition on to ensure that the family is uniformly integrable for ; for instance, a nonrandom upper bound for all the is sufficient.

In the present setting, the limit model as well as the approximating sequence are given by atomless models. For this class of models and for the ENCU functional (2.4), the optimization problem can be reduced to the concavified utility maximization problem; see Theorem 5.1 of Reichlin (2013). In this way, the stability result can also be obtained via stability results for expected *concave* utilities. For the RDEU functional (2.3) with distortion, however, the results are new.

##### 3.2.2. Horizon Dependence

In Section '(Numerical) Computation of the Value Function' as well as in the first example in this section, we have started with a fixed time horizon *T*. In practical applications, however, the time horizon might be uncertain or changing. The goal of this section is to use Theorem 2.9 to study whether a (marginal) misspecification of the investment horizon significantly influences the optimal behavior of the agent. For expected *concave* utilities, Larsen and Yu (2012) analyze this question in an incomplete Brownian setting. The key to solving their problem is again the duality theory that cannot be used in our setup.

In order to formalize a similar situation in our framework, we start again with a probability space on which there is a Brownian motion , and we introduce the sequence of probability spaces by setting for . We now fix a sequence representing the time horizons. For each *n*, we consider the Black–Scholes model with time horizon as described in Section '(Numerical) Computation of the Value Function'. The pricing density for the *n*th model is therefore given by

Assumption 2.1 is again trivially satisfied since each model is atomless. Moreover, adapting the arguments from Section 'Misspecifications of the Market Model' shows that and Assumption 2.2 are satisfied as well. As in Section 'Misspecifications of the Market Model', we can therefore use Theorem 2.9 to conclude that behavioral predictions of the model are stable with respect to small misspecifications in the time horizon.

#### 3.3. Theoretical Applications

For the ENCU functional defined in (2.4) without distortion, the problem in the limit model turns out to be tractable if has a continuous distribution. In this section, we apply Theorem 2.9 to approximate a pricing density with a general distribution by a pricing density with a continuous distribution. To explain the idea in more detail, we use the notation

Every maximizer for satisfies for some ; see Proposition A.1. If has a continuous distribution, then we have ; (see Lemma 5.7 in Reichlin 2013 for details) and it follows that . In this way, the existence of a maximizer as well as several properties of can be derived directly via the concavified problem. If the limit model is atomless but the distribution of the pricing density φ^{0} is not continuous, then this reduction does not follow directly.

The idea now is to construct a sequence weakly converging to φ^{0} for which each has a continuous distribution. For this approach, we assume that for all . Since is atomless, we can find a uniformly distributed random variable such that *P*^{0}-a.s.; see Lemma A.28 in Föllmer and Schied (2011). Moreover, we choose another random variable with having a continuous distribution (e.g., ). Now we define the sequence by

Every element satisfies by construction. Moreover, the function converges pointwise to the function . The set of all points where *h* is not continuous is at most countable since *h* is increasing; and has a continuous distribution. So, it follows that and we obtain ; see Theorem 5.1 of Billingsley (1968).

With the arguments so far, we have a sequence of probability spaces defined by for together with a sequence of pricing measures weakly converging to φ^{0}. To verify Assumption 2.2, note that gives for every , which yields a uniformly integrable upper bound due to our assumption that for all .

It remains to show that the distribution of is continuous. Since the function is increasing, it follows that if and only if and . But is strictly increasing since *Y* has a continuous distribution, so we infer that for .

Theorem 2.9 now gives as . Since the distribution of is continuous for each *n*, we have that for all *n* and we also get in the limit. In this way, we recover Theorem 5.1 in Reichlin (2013) under less general assumptions, but with completely different techniques. Instead of rearrangement techniques as in Reichlin (2013), we here approximate the mass points in the distribution of φ^{0} by continuous distributions and apply Theorem 2.9.

#### 3.4. A Numerical Illustration

The goal of this section is to illustrate the convergence result numerically. We consider the functional (for a specific nonconcave utility) in the framework presented in Section '(Numerical) Computation of the Value Function' where we can derive explicitly so that we can compare with the value functions in the approximating models. As in Section 'Theoretical Applications', we use the notation .

The utility function in this example is given by

This function is strictly increasing, continuous, in *C*^{1} and satisfies the Inada conditions at 0 and ∞. Its concave envelope is given by

and the conjugate of *U* (and ) is

The conjugate satisfies . On , the conjugate is differentiable and is a singleton. More precisely, we have

Figure 3.1 shows *U* and as well as the conjugate *J*.

Let us now determine . Recall from Section '(Numerical) Computation of the Value Function' that

where and *T* are fixed. For simplicity, we assume that . We now consider some for some . Plugging in the above particular form of , using the fact that has *P*^{0}-measure 0 for any , and doing some elementary calculations gives

In the next step, we rewrite the set in a suitable way and use that is a *Q*^{0}-Brownian motion (by Girsanov's theorem) to obtain

where denotes the cumulative distribution function of the standard normal distribution. From this explicit form, we see that is a continuous and decreasing function of λ with limits 1 and 0 at 0 and ∞, respectively. The equation

therefore has a unique solution . Fix . By definition of , satisfies , which means that . Moreover, satisfies

and this gives . The conjugacy relation between *U* and *J* and the explicit form of give

for all which, together with , gives optimality of for . In order to determine , we recall the explicit expression for and use the fact that has measure 0 to get

Elementary calculations show that and

We conclude that

In order to illustrate the convergence result, we determine the parameter for . For comparison purposes, we compute numerically for particular by backward recursion. Figure 3.2 shows the value functions for some approximations as well as the value function for the Black–Scholes model.

### 4. STABILITY OF THE DEMAND PROBLEM FOR RDEU

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. PROBLEM FORMULATION AND MAIN RESULTS
- 3. APPLICATIONS
- 4. STABILITY OF THE DEMAND PROBLEM FOR RDEU
- 5. STABILITY OF THE DEMAND PROBLEM FOR ENCU
- 6. CONCLUSION
- APPENDIX
- REFERENCES

In this section, we analyze the stability for the *RDEU* for which the functional is defined in (2.3) by

The goal is to prove Theorem 2.9 for this case, that is, to prove that

- (4.1)

and to show that any given sequence of maximizers for contains a subsequence that converges weakly to a maximizer for .

#### 4.1. Weak Convergence of Maximizers

We start with a convergence result for the maximizers. For later purposes, we prove a slightly more general statement; in particular, our proof needs no assumption on the distribution of φ^{0} so that we can use Proposition 4.1 also in Section 'STABILITY OF THE DEMAND PROBLEM FOR ENCU'.

Proposition 4.1. For every sequence with , there are a subsequence and some such that as .

Let us outline the main ideas of the proof. We first use Helly's selection principle to get a limit distribution . In order to find a final position with distribution , we then follow the path of Jin and Zhou (2008), He and Zhou (2011a), and Carlier and Dana (2011) and define the candidate payoff in the limit model as a quantile function applied to a uniformly distributed random variable. This ensures that the distribution of this final position is . In order to find the cheapest final position with the given distribution , one has to choose the “right” uniformly distributed random variable. If the pricing density φ^{0} has a continuous distribution (as assumed in Jin and Zhou 2008, He and Zhou 2011a, and Carlier and Dana 2011 mentioned above), then turns out to be the good choice. In the general case where the distribution of φ^{0} is not necessarily continuous, one can work with a uniformly distributed random variable satisfying *P*^{0}-a.s. and then proceed similarly as in the first case. We also make use of the Hardy–Littlewood inequality, which states that any two random variables satisfy

- (4.2)

see Theorem A.24 of Föllmer and Schied (2011) for a proof.

In the proof of Proposition 4.1, we use the following tightness result to apply Helly's selection principle. Its proof is given at the end of this subsection.

Lemma 4.2. Let be the distribution of . Then, is tight, i.e.,

We are now in a position to prove Proposition 4.1.

Proof of Proposition 4.1. Let be the distribution function of . Since the sequence is tight (Lemma 4.2), we may apply Helly's selection theorem (Billingsley 1968, Theorem 6.1 and p. 227) to get a subsequence and a distribution function such that holds for all continuity points *a* of .

Since is atomless, it is possible to find on a random variable uniformly distributed on (0, 1) such that *P*^{0}-a.s.; see Lemma A.28 in Föllmer and Schied (2011). Define Since is again uniformly distributed on (0, 1), the candidate has distribution ; see Lemma A.19 in Föllmer and Schied (2011). This gives as .

The proof is completed by showing that , as follows. We rewrite φ^{0} and in terms of , and combine Fatou's lemma and the fact that weak convergence implies convergence of any quantile functions to get a first inequality. A second one follows by applying the Hardy–Littlewood inequality (4.2). Finally, we make use of . These steps together give

which proves that .

It remains to give the

Proof of Lemma 4.2. We show below that

- (4.3)

This allows us for every to choose *c*_{0} and in such a way that for . For any and any , we then obtain the inequality , and therefore

By increasing *c*_{0} to *c*_{1} to account for the finitely many , we get

Because was arbitrary, this means that the family is tight.

We now show (4.3). First, note that the assumption implies and by the definition of a quantile function, that is positive and satisfies for every . Thus, must be strictly positive on (0, ε) for which implies that is strictly positive for . Assume by way of contradiction that . For small enough, choose a constant *c*_{0} in such a way that and . Weak convergence gives convergence of the quantile functions and so we have on for sufficiently large *n*, so dominated convergence gives . Because the limit is strictly positive, this and the choice of *c*_{0} allow us to choose *n*_{0} in such a way that

- (4.4)

and . The latter implies that can be used to control on . Indeed, the last inequality and the definition of a quantile give for any that implies that

- (4.5)

on . Finally, we use the Hardy–Littlewood inequality (4.2) to rewrite in terms of quantiles, plug in (4.5) and use (4.4) to obtain

which contradicts .

#### 4.2. Upper Semicontinuity of

In this section, we prove the first inequality of (4.1), namely, that

- (4.6)

Having proved weak convergence along a subsequence for any sequence with , the remaining step is to show that the corresponding sequence of values converges as well. For this, we use the growth condition imposed on *U* and *w* as well as of the integrability condition imposed on .

**Throughout this section, we assume that Assumptions** 2.2 **and** 2.7 **hold true**.

Lemma 4.3. Let . Then, the family is uniformly integrable.

Proof. Since is nonnegative for every , it is sufficient to find an integrable upper bound independent of *n*. We first apply (2.7), the Chebyshev inequality, and (2.6), and then use that for some constant to obtain

- (4.7)

where λ is the one fixed in Assumption 2.7. In the next step, we estimate the term . Recall that by Assumption 2.7, so the conjugate of the function is for some constant *c*_{1}. Since , this gives

- (4.8)

Recall that by assumption, so also . Since the family is uniformly integrable by Assumption 2.2, we therefore obtain as . Together with (4.8), this gives

- (4.9)

for sufficiently large *n*. Combining (4.7) and (4.9) finally yields

which gives an integrable upper bound since by Assumption 2.7.

We now combine Proposition 4.1 and Lemma 4.3 to prove the upper-semicontinuity of .

Proposition 4.4. Let be a sequence with and , and the subsequence and its limit constructed in Proposition 4.1. Then, we have

- (4.10)

Consequently, we have

Proof. Starting from an arbitrary sequence with , Proposition 4.1 gives a subsequence and a weak limit such that . The function *U* is continuous, hence , and therefore we get for all points *y* where is continuous. But is a decreasing function of *y*; hence, there are at most countably many points where is not continuous, and we deduce that for a.e. *y*. Moreover, *w* is increasing; hence, it is continuous a.e. and we infer that we have for a.e. *y*. By Lemma 4.3, the family is uniformly integrable and we arrive at

For the proof of upper semicontinuity of (in *n*), assume by way of contradiction that . This allows us to choose a sequence with and . We can then pass to a subsequence realizing the lim sup and apply the first part of proof to the subsequence to get a further subsequence and a weak limit with

which gives the required contradiction.

Remark 4.5. Proposition 4.4 can also be used to prove the existence of a maximizer for , as follows. We formally introduce a sequence of models by setting for all and fix a maximizing sequence for . Proposition 4.4 then shows that the limit constructed in Proposition 4.1 is a maximizer. As a by-product, we also see that . Note that so far, we have not used the assumption that φ^{0} has a continuous distribution. Jin and Zhou (2008) and Carlier and Dana (2011) prove the existence of a maximizer for under the assumption that φ^{0} has a continuous distribution. Proposition 4.4 (together with Proposition 4.1) shows how to extend their results to an atomless underlying model with a pricing density that is not necessarily continuous.

#### 4.3. Lower Semicontinuity of

The purpose of this section is to show the second inequality “⩾” of (4.1). The natural idea is to approximate payoffs in the limit model by a sequence of payoffs in the approximating models. For a generic payoff, this might be difficult; but we argue in the first step that it suffices to consider payoffs of the form for a bounded function *h*. Those elements can be approximated by the sequence . Since *h* is bounded, the sequence as well as have nice integrability properties so that one obtains the desired convergence results for as well as for .

Proposition 4.6. Suppose that φ^{0} has a continuous distribution. Then,

Proof. 1) *Reduction to a bounded payoff*: Suppose by way of contradiction that In Lemma 4.7 below, we show lower semicontinuity of the function . This allows us to choose such that . Therefore, we can find and fix satisfying . Next, we define an additional sequence by . By construction, this sequence is increasing to , and this gives for all *y*. The function *w* is increasing, hence continuous a.e., and we thus have for a.e. *y*. Monotone convergence then yields . This allows us to find and fix *m*_{0} such that .

2) *Reduction to a payoff* : We define By the definition of a quantile, is increasing, so *h* is decreasing. Moreover, since is bounded by *m*_{0}, the quantile is bounded by *m*_{0} as well. Recall now that the distribution of φ^{0} is assumed to be continuous. Thus, as well as are uniformly distributed on (0, 1) and therefore has the same distribution as . But the preference functional *V*_{0} only depends on the distribution of its argument, and so we get

- (4.11)

Finally, we use the monotonicity of *h* together with (4.2), and to obtain

This gives .

3) *Convergence of* : Let denote the set of all points where *h* is not continuous. The function *h* is decreasing and so is at most countable; but φ^{0} has a continuous distribution and it follows that . Hence, we get that then implies for every *y*. By construction, *h* is positive and bounded by *m*_{0}; hence, is an integrable upper bound for and dominated convergence gives .

4) *Convergence of* : We define and note that the set of points where *g* is not continuous is again . As in step 3), we therefore have . Recall now that and . This gives uniform integrability of that, in turn, implies uniform integrability of since . Together with as proved in part 3), we obtain

as . This implies (for sufficiently large *n*). But on the other hand, (4.11) and part 3) give

which gives the required contradiction.

Lemma 4.7. Suppose that φ^{0} has a continuous distribution. Then,

Proof. We assume to the contrary that . Fix with and define . It follows that and a.s. The latter implies for a.e. *y* as , and since *w* is increasing, we have a.e. Monotone convergence then yields as , which gives a contradiction.

### 5. STABILITY OF THE DEMAND PROBLEM FOR ENCU

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. PROBLEM FORMULATION AND MAIN RESULTS
- 3. APPLICATIONS
- 4. STABILITY OF THE DEMAND PROBLEM FOR RDEU
- 5. STABILITY OF THE DEMAND PROBLEM FOR ENCU
- 6. CONCLUSION
- APPENDIX
- REFERENCES

In this section, we analyze the case that the functional is defined by

for a nonconcave utility function *U*. Except for nonconcavity, this coincides with the classical expected utility where the value function is usually denoted by instead of . We follow that tradition and switch to from now on. Moreover, the analysis in Reichlin (2013), in particular sections 'STABILITY OF THE DEMAND PROBLEM FOR RDEU' and 'STABILITY OF THE DEMAND PROBLEM FOR ENCU', shows that the optimization problem for the nonconcave utility function *U* is closely linked to the optimization problem for its concave envelope , and both of them are useful for the analysis in this section. Therefore, we use in this section the notation

for the value function. The goal is to prove Theorem 2.9 for the present case, that is, to prove

- (5.1)

and to show that a given sequence of maximizers for contains a subsequence that converges weakly to a maximizer for . In contrast to Section 'STABILITY OF THE DEMAND PROBLEM FOR RDEU', we prove the stability results here without further assumptions on the distribution of φ^{0}. This is necessary for the theoretical applications described in Section 'Theoretical Applications'. An example of an atomless limit model with a unique (nontrivial) pricing density that does not have a continuous distribution can be found in Example 5.10 of Reichlin (2013).

**Throughout this section, we assume that** **and that Assumption** 2.2 **is satisfied**.

#### 5.1. Upper Semicontinuity of

The main idea is similar to the proof of upper semicontinuity of in Section 'Upper Semicontinuity of '. Starting from a sequence with , we use the results of Section 'Weak Convergence of Maximizers' to obtain weak convergence along a subsequence to an element . Using Fatou's lemma for , the remaining step is then to show that the corresponding sequence converges as well. This requires uniform integrability of the family that can be proved with the help of the following lemma.

Lemma 5.1. Assumption 2.2 and imply the uniform integrability of the family for any .

Proof. We show below that Assumption 2.2 and ensure that

- (5.2)

The statement is then clear for since *J* is decreasing. For , the assumption that (in combination with Lemma A.3) can be used to obtain constants and such that for all and . Applying this estimate on the set and using monotonicity of *J* on the complement gives

The second term in the last line is constant, and uniform integrability of the first one is due to (5.2).

It remains to prove (5.2). The second part of Lemma A.3 shows that implies the existence of constants *k*_{1}, *k*_{2} and such that for . Plugging this inequality into the definition of *J* and doing some elementary computations gives

for some constant *C*. But then, it follows that , and Assumption 2.2 yields the uniform integrability of .

We now describe, as outlined above, the limit behavior of . This gives upper semicontinuity in *n* for and it can also be used (later) to deduce the optimality of constructed in Proposition 4.1.

Proposition 5.2. The sequence of maximizers for contains a subsequence weakly converging to some limit , and it satisfies Consequently, we have

Proof. We consider a (relabeled) subsequence realizing the lim sup γ, say, for (or equivalently for , since the are maximizers). Proposition 4.1 gives a further subsequence with weak limit . Since *U*, , and are continuous, we infer that Fatou's lemma then gives

We show below that is uniformly integrable, which implies that converges to as . Combining this with the inequality for the negative parts yields

where we use in the last step that the form a subsequence of the sequence for which we have from above.

It remains to show uniform integrability of . This family is, by the definition of *J*, dominated by . Uniform integrability of the first summand family follows from Lemma 5.1, and since is bounded in *L*^{1}, the sequence can be made arbitrarily small in expectation by choosing ε small. So, uniform integrability of follows and the proof is complete.

#### 5.2. Lower Semicontinuity of

The goal of this section is to show lower semicontinuity in *n* for .

Theorem 5.3. *Suppose that Assumption* 2.1 *holds true. Then*,

The approach to prove this statement is as follows. Observe that

If one shows (as we do below in Section 'Continuity in *n* of ') that

- (5.3)

it only remains to show that

- (5.4)

While the proof of (5.3) follows (essentially) from nonsmooth versions of known stability results on *concave* utility maximization, the proof of (5.4) requires a careful analysis of the *nonconcave* problem that will be explained in detail in Section 'Controlling the Difference '. Note that the additional Assumption 2.1 is only used to prove (5.4). We start with the proof of (5.3).

##### 5.2.1. Continuity in *n* of

Instead of lower semicontinuity, we prove more than needed, namely,

Proposition 5.4. .

In the case of strictly concave utility functions, this result follows by directly analyzing the sequence of optimal terminal wealths as a function of . In the nonconcave framework, is not strictly concave; hence, its conjugate *J* is nonsmooth and cannot be written as a function of ( only lies in the subgradient of at ). Instead, we use the fact that can be written (see Lemma 5.7 below) in a dual form as

for some dual minimizer . Continuity in *n* of can then be shown by proving that the sequence converges (along a subsequence) to a dual minimizer in the limit model and that the sequence converges to the corresponding value in the limit model. The latter requires uniform integrability of the family . For the positive parts, this can be proved via Lemma 5.1. We now show that the family of negative parts is uniformly integrable as well.

Lemma 5.5. For each , the family is uniformly integrable.

Proof. The idea for this result goes back to Kramkov and Schachermayer (1999); the extension to the nonsmooth case is proved in Lemma 6.1 of Bouchard, Touzi, and Zeghal (2004). A modified version of their proof works for our setup, as follows.

Since the conjugate *J* is decreasing, it is enough to check uniform integrability of If , all the are bounded by a uniform constant and the statement is clear. So, assume . To use the de la Vallée–Poussin characterization of uniform integrability, we need to find a convex increasing function such that and . The function *J* is convex, decreasing and finite on (0, ∞); see Lemma 2.12 of Reichlin (2013). So for , *J* is strictly decreasing and *J* as well as have a classical inverse. Let be the inverse of . Since is increasing and concave, its inverse Φ is increasing and convex. In order to prove that

- (5.5)

note first that (see Lemma A.3 in Reichlin 2013) implies

Hence, for all *M*, there is *y*_{0} such that for all . Fix some *y*_{1} and *y*_{2} satisfying and set The mean value theorem gives such that . This implies by the definition of the subdifferential that

Taking the lim inf as gives . The proof of (5.5) is complete since the constant *M* is arbitrary.

It remains to prove that . Recall that *J* is convex and finite on (0, ∞) and hence continuous, and that by the assumption on *U*. Moreover, in the present case, so there is with and this implies . By a direct computation, we see that for ,

which completes the proof.

We now show that weak convergence of to indeed implies convergence of to .

Lemma 5.6. Let be given. Then, it holds that

Proof. The continuity of *J* together with and implies as . Since the limit λ is in (0, ∞), the lie eventually in a compact set *B* of the form with , and so it is enough to show the uniform integrability of . For the negative parts , this is a consequence of Lemma 5.5, and for the positive parts, it follows by Lemma 5.1.

For the *n*th model, the classical dual representation of for our setting with a fixed pricing density gives a dual minimizer . The sequence does not necessarily converge; however, every cluster point yields a dual minimizer in the limit model.

Lemma 5.7. Given any , the problem admits a maximizer , where is a minimizer of

- (5.6)

Any cluster point of the sequence is a minimizer of and satisfies .

Proof. Lemmas 5.5 and 5.1 give for all and all . Existence and structure of the solution for and the dual representation then follow by Proposition A.1.

For the second part, we use the notation

for . Convexity of *J* implies convexity of . Fix a minimizer λ(0) for and a cluster point of . We show below that any values between and λ(0) are minimizers for . Since by Proposition A.1, the minimizers of are bounded away from 0 and ∞, we therefore must have , and continuity of *H*^{0} then implies that is also a minimizer.

We now argue that implies

By way of contradiction, we assume that holds for some . Lemma 5.6 with implies that as . Thus, for ε small enough, there is a constant *k*_{0} such that

for all . From the definition of the minimizer , it holds that . Putting the two inequalities together gives

- (5.7)

for . Since , the number λ is between and λ(0) for large enough values of *k*. Thus, (5.7) contradicts the convexity of .

We finally have all the ingredients to prove the convergence of .

Proof of Proposition 5.4. To obtain , we apply Proposition 5.2 to . For the other inequality, fix a relabeled sequence of maximizers with . We use Lemma 5.7 to fix for each a corresponding dual minimizer of (5.6). By classical duality theory and Lemma 5.7, any cluster point of satisfies

and . Fix one cluster point and a converging subsequence . It follows from Lemma 5.6 that and we conclude again from the dual representation for that But the full sequence converges to γ; so we finally obtain . This completes the proof.

##### 5.2.2. Controlling the Difference

Let us now turn to (5.4) and prove that . The idea here is as follows. In general, is smaller than since dominates *U*. For some initial values *x*, however, the maximizer for does not have probability mass in , i.e., , and thus also maximizes . Consequently, the values and coincide for such “good” initial values, and the key is to analyze the complement of these *x* more carefully. For the *n*th model, the “good” initial values induce a (*n*-dependent) partition of (0, ∞) and its (*n*-dependent) mesh size, and the maximal distance between two successive partition points goes to 0 as due to Assumption 2.1. The next result formalizes this idea.

Proposition 5.8. Let Assumption 2.1 be satisfied and let and be fixed. For every , there is a set such that

- i)for and
- ii)there is
*n*_{0}such that is nonempty for .

As a consequence, we have .

Let us first outline the two main ideas. The problem admits (under our conditions) a maximizer for some . The right- and left-hand derivatives satisfy ; see Lemma A.2 of Reichlin (2013). So, in order to have no probability mass in the area , it is sufficient if the maximizer value is equal to or . Therefore, the initial values given by

- (5.8)

for are good candidates for initial values satisfying property i).

In order to also have property ii), we need to control the distance between any two points defined by (5.8). This boils down to controlling terms of the form . These are nonzero if *y* is the slope of an affine part of . The distance between the points defined by (5.8) is therefore dominated by the product of the length of the longest affine part and the -probability of the biggest atom in . In the case of a single affine part in , this goes to 0 by Assumption 2.1. In general, there is no upper bound for the length of the affine parts, but we can estimate the tails with Lemma 5.9 below. Recall that is the set of -atoms in and that Assumption 2.1 ensures that the maximal -probability of all elements in goes to 0.

Proof of Proposition 5.8. In order to define the set for Proposition 5.8, we start with some preliminary definitions and remarks. For all , fix a maximizer for and the corresponding minimizer given in Lemma 5.7. This lemma also yields . So, fix such that for all *n*. Using Lemma 5.9 below, we obtain

Hence, we may choose α_{0} such that

- (5.9)

Define the set

Now we are in a position to define the set by

We claim that this satisfies the assumptions of Proposition 5.8.

1) *Property i)*: For any and , there is some such that

Note that by definition and fix some . Applying the definition of *J* together with gives

where the equality follows from the classical duality relation between and *J*. Taking the sup over all gives optimality of for . Since do not take values in (see Lemma A.2 of Reichlin 2013), satisfies and it follows that

because .

2) *Property ii)*: For this part, we use Assumption 2.1 to choose *n*_{0} large enough such that for . Fix some and define the map by

Monotonicity of (see Lemma A.1 of Reichlin 2013) implies . Moreover, recall that and are fixed in such a way that

satisfies . This gives .

We first consider the case . In order to construct a grid contained in , we decompose into disjoint subsets such that and ; this uses that for , the largest atom in has -probability at most . The values , are contained in , and since on and , these values satisfy

for . We deduce that and , form a grid with starting point and endpoint whose mesh size is smaller than δ.

It remains to consider the case . Since , it is sufficient to show . Observe first that

We rewrite in terms of and and use to obtain

where the definition of α_{0} in (5.9) is used in the last step.

3) *Proof of* : Fix . Because of the continuity of in *x* and Proposition 5.4, we can fix and *n*_{1} such that implies for all . Applying the first part of this proof for δ gives *n*_{0} such that for all , there is some set with properties i) and ii). So, for each , there is some . By definition of , the relation holds for all . Moreover, is increasing in *x*, so adding and subtracting and using that yields

With the arguments so far, we have shown that for every , we have

The result follows since for each .

It remains to state and prove

Lemma 5.9. Let *B* be a compact set of the form for . Then, is uniformly integrable.

Proof. implies by the definition of that there are a constant and such that we have

for . An application of this inequality for and on the set , some elementary calculations and yield

The family is uniformly integrable by Lemma 5.1, and so is the family by Lemma 5.5. With the arguments so far, we have shown that the family is uniformly integrable. Now fix some and recall that any for satisfies and thus also . The classical conjugacy relation between and gives

for . Applying this inequality for and on the set shows that is dominated by . This completes the proof since the latter family is uniformly integrable by Lemma 5.5.

#### 5.3. Putting Everything Together

On the way, we have separately proved the second case of Theorem 2.9. For completeness, we summarize the main steps.

Proof of Theorem 2.9. *for ENCU*. Theorem 5.3 and Proposition 5.2 give the convergence . For the second part, fix a maximizer for for every *n*. Proposition 4.1 shows that the sequence contains a subsequence weakly converging to some . It then follows from Proposition 5.2, the optimality of and that

This shows that is a maximizer for since .

It remains to give the proof for the stability of the goal-reaching problem. Recall from Remark 2.10 that this is the case where so that for . In particular, is strictly increasing on (0, 1) and uniformly bounded by 1.

Proof of Remark 2.10. The statement is clear for since there for each ; so, we assume that . In Section 'Continuity in *n* of ', strict monotonicity of is only used via Proposition A.1 to show the existence of the lower bound . A closer inspection of the argument there shows that we only need strict monotonicity of . But admits a maximizer (see the discussion following Theorem 2.9) and the constraint implies . This yields strict monotonicity of for since is strictly increasing on [0, 1) and so we can prove for as in Proposition 5.4. This implies . For the lim inf, we first fix for each a maximizer for and recall that implies and that holds for ; see Lemma A.2 of Reichlin (2013). This gives

- (5.10)

We now fix a subsequence realizing the and such that the associated sequence converges to . As in Lemma 5.7 (and again using the modified version of Proposition A.1), this gives . The assumptions that φ^{0} has a continuous distribution and imply then that . Moreover, as the function is uniformly bounded, the right-hand side of (5.10) converges to . But since φ^{0} has a continuous distribution, it follows that *P*-a.s. and applying similar arguments in the reverse order, we find that

With the arguments so far, we have proved that

which gives . But the limit model is atomless, so we have by Theorem 5.1 of Reichlin (2013) and the result follows. Finally, the convergence of the maximizers along a subsequence follows as in Proposition 5.2.

### 6. CONCLUSION

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. PROBLEM FORMULATION AND MAIN RESULTS
- 3. APPLICATIONS
- 4. STABILITY OF THE DEMAND PROBLEM FOR RDEU
- 5. STABILITY OF THE DEMAND PROBLEM FOR ENCU
- 6. CONCLUSION
- APPENDIX
- REFERENCES

In this paper, we study the stability along a sequence of models for a class of behavioral portfolio selection problems. The analyzed preference functionals allow for nonconcave and nonsmooth utility functions as well as for probability distortions. These features are motivated by several applications such as manager compensation, portfolio delegation, and behavioral finance. While there are several explicit results in the literature for behavioral portfolio selection problems in complete continuous-time markets, there are no comparable results for the discrete-time analog.

Our convergence results demonstrate that the explicit results from the continuous-time model are approximately valid also for the discrete-time setting if the latter is *sufficiently close* to the continuous-time setting. As illustrated by a counterexample, the required notion of *sufficiently close* is slightly but strictly stronger compared to the stability results for *concave* utility maximization problems. The convergence results can also be applied to other situations such as (marginal) drift misspecification or changing time horizons.

### APPENDIX

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. PROBLEM FORMULATION AND MAIN RESULTS
- 3. APPLICATIONS
- 4. STABILITY OF THE DEMAND PROBLEM FOR RDEU
- 5. STABILITY OF THE DEMAND PROBLEM FOR ENCU
- 6. CONCLUSION
- APPENDIX
- REFERENCES

#### A.1. Nonsmooth Utility Maximization

This appendix contains the results on nonsmooth (concave) utility maximization, which are relevant for the proofs in Section 'STABILITY OF THE DEMAND PROBLEM FOR ENCU'. Following the notation there, we use to denote the value function. Recall that *J* is the conjugate of *U* (as well as ).

Proposition A.1. Fix . Suppose that for all . Then, the concave problem has a solution for every . Every solution satisfies , where is a minimizer for

- (A.1)

and and are strictly positive constants.

Most of the statements contained in Proposition A.1 are proved in greater generality in Bouchard et al. (2004) and Westray and Zheng (2009). For completeness, we include a proof. We make use of Lemma 6.1 in Bouchard et al. (2004) which reads in our setup as follows.

Lemma A.2. There is a function that is convex and increasing with and

- (A.2)

Proof of Proposition A.1. 1) The existence of a maximizer in the present setting is shown in Theorem 3.4 of Reichlin (2013). Remark 3.3 there also shows that for all λ implies for some so that we can use Theorem 4.1 there to get

Moreover, is on (0, ∞) finite and concave, hence continuous. This implies that we also have In order to find the upper bound , we consider a minimizing sequence for (A.1) and show that it is bounded by some constant. Since is minimizing, it holds that

- (A.3)

for *k* large enough. We use the function introduced in Lemma A.2. Then for all , there is some such that for and then for all . Using (A.2), we compute that for some ,

Combining this inequality and (A.3) gives Choosing shows that is bounded by some constant.

In order to find the lower bound , we start with the case . Due to the existence of a maximizer for and the strict monotonicity of , we also deduce strict monotonicity of and we infer . Together with the continuity of the function in 0, we can find such that the minimization in (A.1) can be reduced to . In the case , we can again find since for .

2) With the arguments so far, we find a maximizer and some parameter satisfying

Suppose by way of contradiction that there exists a set satisfying and on the set *A*. The conjugacy relation between and *J* then implies which is the required contradiction.

#### A.2. Auxiliary Results

Lemma A.3. The AE condition is equivalent to the existence of two constants and such that

Moreover, if is satisfied, then there are constants *k*_{1} and *k*_{2} and such that for .

Proof of Lemma A.3. The equivalence is proved in Lemma 4.1 of Deelstra, Pham, and Touzi (2001). We only prove the last implication. Similarly to Lemma 4.1 of Deelstra et al. (2001), we argue that holds for and . Recall that

- (A.4)

and . Now choose some and observe that for all . We want to compare the functions and for . Let *F* be the concave function on [1, ∞) defined by . Fix some . By definition, this implies that b and therefore that . Thus, it follows from (A.4) that

- (A.5)

Set . In order to complete the proof, we have to check that for all Clearly, the function *G* satisfies the equation

- (A.6)

for all . Since , it follows from (A.5) and (A.6) that Hence, we have for all . Since *G* is continuously differentiable, there exists such that for all and . This gives

- (A.7)

for all and . To show that holds for all , let and suppose that . By the definition of and (A.7), we have on and . This implies that

- (A.8)

for some . On the other hand, combining (A.5) and (A.6) implies for all . The latter is equivalent to and gives the required contradiction to (A.8).

Above, it is proved that there exist constants and such that we have for and . This gives

for . Thus, choosing and gives that is the desired result.

### REFERENCES

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. PROBLEM FORMULATION AND MAIN RESULTS
- 3. APPLICATIONS
- 4. STABILITY OF THE DEMAND PROBLEM FOR RDEU
- 5. STABILITY OF THE DEMAND PROBLEM FOR ENCU
- 6. CONCLUSION
- APPENDIX
- REFERENCES

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