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Keywords:

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

Abstract

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. PROBLEM FORMULATION AND MAIN RESULTS
  5. 3. APPLICATIONS
  6. 4. STABILITY OF THE DEMAND PROBLEM FOR RDEU
  7. 5. STABILITY OF THE DEMAND PROBLEM FOR ENCU
  8. 6. CONCLUSION
  9. APPENDIX
  10. 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

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. PROBLEM FORMULATION AND MAIN RESULTS
  5. 3. APPLICATIONS
  6. 4. STABILITY OF THE DEMAND PROBLEM FOR RDEU
  7. 5. STABILITY OF THE DEMAND PROBLEM FOR ENCU
  8. 6. CONCLUSION
  9. APPENDIX
  10. 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 inline image and some pricing measure inline image, and we assume that this sequence converges weakly in a suitable sense (to be made precise later) to a limit model inline image. For each model, we are interested in the demand problem

  • display math(1.1)

where the functional inline image is defined by

  • display math(1.2)

for a nonconcave and nonsmooth utility function U on inline image and a strictly increasing function inline image representing the probability distortion of the beliefs. We are then interested in the asymptotics of the value (indirect utility) inline image and its maximizer inline image, 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 inline image, 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 inline image 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 inline image that only depends on the terminal wealth. Because the market is complete, any fixed inline image-measurable nonnegative random variable f is the terminal wealth associated to a self-financing strategy if and only if inline image, where inline image 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 inline image. The assumption that the sequence of models converges (in a suitable sense) to a limit model means that the economic situation described by the nth 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 inline image as well as (along a subsequence) the optimal final positions inline image 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) inline image to inline image; 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 inline image converges weakly to inline image, but where the limit inline image and inline image 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

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

The following notation is used. If inline image, denote inline image, inline image and inline image. For a function G and a random variable X, we write inline image for the positive/negative parts inline image. For a sequence inline image of random variables, we denote weak convergence of inline image to f0 by inline image. A quantile function inline image of a distribution function F is a generalized inverse of F, i.e., a function inline image satisfying

  • display math

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 inline image of a random variable f is understood to be a quantile function inline image of the distribution F of the random variable f. If the sequence inline image converges weakly to f0, then any corresponding sequence inline image of quantile functions converges a.e. on (0,1) to inline image; 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 inline image, where the probability space inline image 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 inline image equivalent to inline image with density inline image. We refer to inline image as pricing measure and to inline image as pricing density (or pricing kernel). We assume that the sequence inline image converges weakly to the pricing density in the atomless model, i.e., inline image. To ensure that the atomic structure tend to the atomless structure, we assume that the atoms disappear in the following sense. Let inline image be the set of atoms in inline image (with respect to inline image).

Assumption 2.1. inline image.

We impose the following integrability condition on inline image.

Assumption 2.2. The family inline image is uniformly integrable for all inline image.

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 inline image, which is strictly increasing, continuous and satisfies the growth condition

  • display math(2.1)

We consider only nonconcave utility functions defined on inline image. To avoid any ambiguity, we set inline image for inline image and define inline image and inline image. Without loss of generality, we may assume that inline image. 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 inline image.

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

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

  • display math

Because of the nonconcavity of U, the concave envelope inline image 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 inline image for the convex function J and by inline image for the concave function inline image. The right- and left-hand derivatives of J are denoted by inline image and inline image. Our proofs (mainly in the Appendix) use the classical duality relations between inline image, J, inline image, and inline image. 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 inline image. For a nonconcave utility function, we impose a similar condition via the AE of the conjugate J,

  • display math

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 inline image that is strictly increasing and satisfies inline image and inline image.

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

  • display math(2.2)

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

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

  • display math(2.3)

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

  • display math(2.4)

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

The functional inline image defined in (2.3) can be seen as a Choquet integral inline image with respect to the monotone set function inline image; see Chapter 5 of Denneberg (1994) for an exposition of this concept. In the case inline image, the functional inline image in (2.3) coincides with the classical expected utility inline image 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) inline image, the (budget) set inline image in the nth model is

  • display math

For each model, we are interested in the demand problem

  • display math(2.5)

An element inline image is optimal if inline image. By a maximizer for inline image, 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

  • display math(2.6)
  • display math(2.7)
  • display math(2.8)

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

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, inline image, (2.7) is satisfied for inline image. A sufficient condition for (2.6) is inline image (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 inline image be defined as in (2.3). In this case, we suppose that the distribution of φ0 is continuous and that Assumption 2.7 is satisfied.

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

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

Theorem 2.9. Let Assumptions 2.2 and 2.8 be satisfied. Then

  • display math

and for any sequence of maximizers inline image for inline image, there are a subsequence inline image and a maximizer inline image for inline image such that inline image as inline image.

The maximizers for inline image 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 inline image for inline image. For the ENCU functional in (2.4), the existence of a maximizer inline image for inline image 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 inline image has to be verified in any given setting. One sufficient criterion is that inline image (or inline image due to the equivalence of inline image and inline image) is atomless (see Remark 4.5 below). Another sufficient criterion is that inline image consists of finitely many atoms. The latter, in fact, implies that any maximizing sequence inline image for inline image is bounded; this allows us to extract a subsequence a.s. converging to some limit inline image, and arguments similar to the ones in Proposition 4.4 show that inline image is a maximizer for inline image. 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 inline image that describes the goal-reaching problem initiated by Kulldorff (1993) and investigated extensively by Browne (1999; 2000). But under the assumption that φ0 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) inline image-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 inline image-measurable. Additional information (if available) is used by the agent to avoid the nonconcave part inline image 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 inline image. More concretely, Assumption 2.1 excludes the (pathological) behavior illustrated in the next example.

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

  • display math

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

3. APPLICATIONS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. PROBLEM FORMULATION AND MAIN RESULTS
  5. 3. APPLICATIONS
  6. 4. STABILITY OF THE DEMAND PROBLEM FOR RDEU
  7. 5. STABILITY OF THE DEMAND PROBLEM FOR ENCU
  8. 6. CONCLUSION
  9. APPENDIX
  10. 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 inline image as a function of inline image. 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 inline image, a probability space inline image on which there is a standard Brownian motion inline image, and a (discounted) market consisting of a savings account inline image and one stock S described by

  • display math

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

  • display math

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

  • display math

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

  • display math

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

  • display math

for some inline image (which is the number of up moves in the path). For n large, we have inline image and we see that inline image. Taking the limit inline image gives inline image, which means that Assumption 2.1 is satisfied. The distribution of φ0 is continuous if inline image. Finally, a proof of uniform integrability of inline image for inline image 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 inline image in the binomial models converges to the value function v0 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 v0 in the Black–Scholes model. Note that for RDEU, we have no dynamic programming and hence no description of v0 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 inline image and a probability space inline image on which there is a Brownian motion inline image. We introduce the sequence of probability spaces by inline image for inline image. In order to define a sequence of price processes, we consider a sequence inline image converging to some drift parameter inline image in the limit model. For each n, we consider a (discounted) market consisting of a savings account inline image and one stock inline image described by

  • display math

in the filtration generated by W. The pricing density for the nth model is then given by

  • display math

In this example, each model is atomless. Assumption 2.1 is therefore trivially satisfied. Moreover, looking at the explicit form of inline image shows that inline image for all inline image, so in particular inline image. Finally, it is straightforward to check the uniform integrability of inline image for any inline image. 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 inline image converge to the corresponding quantities in the model with drift μ0. 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 inline image to inline image. If we consider more generally a stochastic market price of risk inline image, then assuming inline image gives weak convergence of the stochastic exponential inline image to inline image; see Proposition A.1 in Larsen and Žitković (2007). In addition, one then needs some integrability condition on inline image to ensure that the family inline image is uniformly integrable for inline image; for instance, a nonrandom upper bound for all the inline image 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 inline image 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 inline image on which there is a Brownian motion inline image, and we introduce the sequence of probability spaces by setting inline image for inline image. We now fix a sequence inline image representing the time horizons. For each n, we consider the Black–Scholes model with time horizon inline image as described in Section '(Numerical) Computation of the Value Function'. The pricing density for the nth model is therefore given by

  • display math

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 inline image 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 inline image 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

  • display math

Every maximizer inline image for inline image satisfies inline image for some inline image; see Proposition A.1. If inline image has a continuous distribution, then we have inline image; (see Lemma 5.7 in Reichlin 2013 for details) and it follows that inline image. In this way, the existence of a maximizer as well as several properties of inline image can be derived directly via the concavified problem. If the limit model is atomless but the distribution of the pricing density φ0 is not continuous, then this reduction does not follow directly.

The idea now is to construct a sequence inline image weakly converging to φ0 for which each inline image has a continuous distribution. For this approach, we assume that inline image for all inline image. Since inline image is atomless, we can find a uniformly distributed random variable inline image such that inline image P0-a.s.; see Lemma A.28 in Föllmer and Schied (2011). Moreover, we choose another random variable inline image with inline image having a continuous distribution (e.g., inline image). Now we define the sequence inline image by

  • display math

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

With the arguments so far, we have a sequence of probability spaces defined by inline image for inline image together with a sequence of pricing measures inline image weakly converging to φ0. To verify Assumption 2.2, note that inline image gives inline image for every inline image, which yields a uniformly integrable upper bound due to our assumption that inline image for all inline image.

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

Theorem 2.9 now gives inline image as inline image. Since the distribution of inline image is continuous for each n, we have that inline image for all n and we also get inline image 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 φ0 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 inline image (for a specific nonconcave utility) in the framework presented in Section '(Numerical) Computation of the Value Function' where we can derive inline image explicitly so that we can compare inline image with the value functions inline image in the approximating models. As in Section 'Theoretical Applications', we use the notation inline image.

The utility function in this example is given by

  • display math

This function is strictly increasing, continuous, in C1 and satisfies the Inada conditions at 0 and ∞. Its concave envelope is given by

  • display math

and the conjugate of U (and inline image) is

  • display math

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

  • display math

Figure 3.1 shows U and inline image as well as the conjugate J.

image

Figure 3.1. Panel A shows the nonconcave utility U and its concave envelope inline image (the dotted line). Panel B shows the conjugate J of U and inline image. The conjugate J has a kink in 1 that is the slope of the affine part in inline image in Panel A.

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

  • display math

where inline image and T are fixed. For simplicity, we assume that inline image. We now consider some inline image for some inline image. Plugging in the above particular form of inline image, using the fact that inline image has P0-measure 0 for any inline image, and doing some elementary calculations gives

  • display math

In the next step, we rewrite the set inline image in a suitable way and use that inline image is a Q0-Brownian motion (by Girsanov's theorem) to obtain

  • display math

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

  • display math

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

  • display math

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

  • display math

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

  • display math

Elementary calculations show that inline image and

  • display math

We conclude that

  • display math

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

image

Figure 3.2. Value functions inline image for parameters inline image, inline image, inline image, and inline image. Recall that inline image is the limit case.

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4. STABILITY OF THE DEMAND PROBLEM FOR RDEU

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

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

  • display math

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

  • display math(4.1)

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

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 φ0 so that we can use Proposition 4.1 also in Section 'STABILITY OF THE DEMAND PROBLEM FOR ENCU'.

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

Let us outline the main ideas of the proof. We first use Helly's selection principle to get a limit distribution inline image. In order to find a final position with distribution inline image, 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 inline image applied to a uniformly distributed random variable. This ensures that the distribution of this final position is inline image. In order to find the cheapest final position with the given distribution inline image, one has to choose the “right” uniformly distributed random variable. If the pricing density φ0 has a continuous distribution (as assumed in Jin and Zhou 2008, He and Zhou 2011a, and Carlier and Dana 2011 mentioned above), then inline image turns out to be the good choice. In the general case where the distribution of φ0 is not necessarily continuous, one can work with a uniformly distributed random variable inline image satisfying inline image P0-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 inline image satisfy

  • display math(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 inline image be the distribution of inline image. Then, inline image is tight, i.e.,

  • display math

We are now in a position to prove Proposition 4.1.

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

Since inline image is atomless, it is possible to find on inline image a random variable inline image uniformly distributed on (0, 1) such that inline imageP0-a.s.; see Lemma A.28 in Föllmer and Schied (2011). Define inline image Since inline image is again uniformly distributed on (0, 1), the candidate inline image has distribution inline image; see Lemma A.19 in Föllmer and Schied (2011). This gives inline image as inline image.

The proof is completed by showing that inline image, as follows. We rewrite φ0 and inline image in terms of inline image, 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 inline image. These steps together give

  • display math

which proves that inline image. inline image

It remains to give the

Proof of Lemma 4.2. We show below that

  • display math(4.3)

This allows us for every inline image to choose c0 and inline image in such a way that inline image for inline image. For any inline image and any inline image, we then obtain the inequality inline image, and therefore

  • display math

By increasing c0 to c1 to account for the finitely many inline image, we get

  • display math

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

We now show (4.3). First, note that the assumption inline image implies inline image and by the definition of a quantile function, that inline image is positive and satisfies inline image for every inline image. Thus, inline image must be strictly positive on (0, ε) for inline image which implies that inline image is strictly positive for inline image. Assume by way of contradiction that inline image. For inline image small enough, choose a constant c0 in such a way that inline image and inline image. Weak convergence gives convergence of the quantile functions and so we have inline image on inline image for sufficiently large n, so dominated convergence gives inline image. Because the limit is strictly positive, this and the choice of c0 allow us to choose n0 in such a way that

  • display math(4.4)

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

  • display math(4.5)

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

  • display math

which contradicts inline image. inline image

4.2. Upper Semicontinuity of inline image

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

  • display math(4.6)

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

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

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

Proof. Since inline image is nonnegative for every inline image, 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 inline image for some constant inline image to obtain

  • display math(4.7)

where λ is the one fixed in Assumption 2.7. In the next step, we estimate the term inline image. Recall that inline image by Assumption 2.7, so the conjugate of the function inline image is inline image for some constant c1. Since inline image, this gives

  • display math(4.8)

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

  • display math(4.9)

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

  • display math

which gives an integrable upper bound since inline image by Assumption 2.7. inline image

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

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

  • display math(4.10)

Consequently, we have inline image

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

  • display math

For the proof of upper semicontinuity of inline image (in n), assume by way of contradiction that inline image. This allows us to choose a sequence inline image with inline image and inline image. 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 inline image and a weak limit inline image with

  • display math

which gives the required contradiction. inline image

Remark 4.5. Proposition 4.4 can also be used to prove the existence of a maximizer for inline image, as follows. We formally introduce a sequence of models by setting inline image for all inline image and fix a maximizing sequence inline image for inline image. Proposition 4.4 then shows that the limit inline image constructed in Proposition 4.1 is a maximizer. As a by-product, we also see that inline image. Note that so far, we have not used the assumption that φ0 has a continuous distribution. Jin and Zhou (2008) and Carlier and Dana (2011) prove the existence of a maximizer for inline image under the assumption that φ0 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. inline image

4.3. Lower Semicontinuity of inline image

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 inline image for a bounded function h. Those elements can be approximated by the sequence inline image. Since h is bounded, the sequence inline image as well as inline image have nice integrability properties so that one obtains the desired convergence results for inline image as well as for inline image.

Proposition 4.6. Suppose that φ0 has a continuous distribution. Then,

  • display math

Proof. 1) Reduction to a bounded payoff: Suppose by way of contradiction that inline image In Lemma 4.7 below, we show lower semicontinuity of the function inline image. This allows us to choose inline image such that inline image. Therefore, we can find and fix inline image satisfying inline image. Next, we define an additional sequence inline image by inline image. By construction, this sequence is increasing to inline image, and this gives inline image for all y. The function w is increasing, hence continuous a.e., and we thus have inline image for a.e. y. Monotone convergence then yields inline image. This allows us to find and fix m0 such that inline image.

2) Reduction to a payoff inline image: We define inline image By the definition of a quantile, inline image is increasing, so h is decreasing. Moreover, since inline image is bounded by m0, the quantile inline image is bounded by m0 as well. Recall now that the distribution of φ0 is assumed to be continuous. Thus, inline image as well as inline image are uniformly distributed on (0, 1) and inline image therefore has the same distribution as inline image. But the preference functional V0 only depends on the distribution of its argument, and so we get

  • display math(4.11)

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

  • display math

This gives inline image.

3) Convergence of inline image: Let inline image denote the set of all points where h is not continuous. The function h is decreasing and so inline image is at most countable; but φ0 has a continuous distribution and it follows that inline image. Hence, we get inline image that then implies inline image for every y. By construction, h is positive and bounded by m0; hence, inline image is an integrable upper bound for inline image and dominated convergence gives inline image.

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

  • display math

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

  • display math

which gives the required contradiction. inline image

Lemma 4.7. Suppose that φ0 has a continuous distribution. Then,

  • display math

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

Proof of Theorem 2.9. for RDEU. It follows from Propositions 4.4 and 4.6 that inline image. For the second part, we apply Proposition 4.1 to inline image. This gives a subsequence inline image weakly converging to some inline image, and Proposition 4.4 implies that inline image is a maximizer for inline image. inline image

5. STABILITY OF THE DEMAND PROBLEM FOR ENCU

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

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

  • display math

for a nonconcave utility function U. Except for nonconcavity, this coincides with the classical expected utility where the value function is usually denoted by inline image instead of inline image. We follow that tradition and switch to inline image 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 inline image, and both of them are useful for the analysis in this section. Therefore, we use in this section the notation

  • display math

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

  • display math(5.1)

and to show that a given sequence of maximizers inline image for inline image contains a subsequence that converges weakly to a maximizer for inline image. In contrast to Section 'STABILITY OF THE DEMAND PROBLEM FOR RDEU', we prove the stability results here without further assumptions on the distribution of φ0. 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 inline image and that Assumption 2.2 is satisfied.

5.1. Upper Semicontinuity of inline image

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

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

Proof. We show below that Assumption 2.2 and inline image ensure that

  • display math(5.2)

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

  • display math

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 inline image implies the existence of constants k1, k2 and inline image such that inline image for inline image. Plugging this inequality into the definition of J and doing some elementary computations gives

  • display math

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

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

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

  • display math

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

  • display math

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

  • display math

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

It remains to show uniform integrability of inline image. This family is, by the definition of J, dominated by inline image. Uniform integrability of the first summand family follows from Lemma 5.1, and since inline image is bounded in L1, the sequence inline image can be made arbitrarily small in expectation by choosing ε small. So, uniform integrability of inline image follows and the proof is complete. inline image

5.2. Lower Semicontinuity of inline image

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

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

  • display math

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

  • display math

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

  • display math(5.3)

it only remains to show that

  • display math(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 inline image'. 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 inline image

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

Proposition 5.4. inline image.

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

  • display math

for some dual minimizer inline image. Continuity in n of inline image can then be shown by proving that the sequence inline image converges (along a subsequence) to a dual minimizer in the limit model and that the sequence inline image converges to the corresponding value in the limit model. The latter requires uniform integrability of the family inline image. 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 inline image, the family inline image 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 inline image If inline image, all the inline image are bounded by a uniform constant and the statement is clear. So, assume inline image. To use the de la Vallée–Poussin characterization of uniform integrability, we need to find a convex increasing function inline image such that inline image and inline image. The function J is convex, decreasing and finite on (0, ∞); see Lemma 2.12 of Reichlin (2013). So for inline image, J is strictly decreasing and J as well as inline image have a classical inverse. Let inline image be the inverse of inline image. Since inline image is increasing and concave, its inverse Φ is increasing and convex. In order to prove that

  • display math(5.5)

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

  • display math

Hence, for all M, there is y0 such that inline image for all inline image. Fix some y1 and y2 satisfying inline image and set inline image The mean value theorem gives inline image such that inline image. This implies by the definition of the subdifferential inline image that

  • display math

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

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

  • display math

which completes the proof. inline image

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

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

  • display math

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

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

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

  • display math(5.6)

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

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

For the second part, we use the notation

  • display math

for inline image. Convexity of J implies convexity of inline image. Fix a minimizer λ(0) for inline image and a cluster point inline image of inline image. We show below that any values between inline image and λ(0) are minimizers for inline image. Since by Proposition A.1, the minimizers of inline image are bounded away from 0 and ∞, we therefore must have inline image, and continuity of H0 then implies that inline image is also a minimizer.

We now argue that inline image implies

  • display math

By way of contradiction, we assume that inline image holds for some inline image. Lemma 5.6 with inline image implies that inline image as inline image. Thus, for ε small enough, there is a constant k0 such that

  • display math

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

  • display math(5.7)

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

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

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

  • display math

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

5.2.2. Controlling the Difference inline image

Let us now turn to (5.4) and prove that inline image. The idea here is as follows. In general, inline image is smaller than inline image since inline image dominates U. For some initial values x, however, the maximizer for inline image does not have probability mass in inline image, i.e., inline image, and thus also maximizes inline image. Consequently, the values inline image and inline image coincide for such “good” initial values, and the key is to analyze the complement of these x more carefully. For the nth 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 inline image due to Assumption 2.1. The next result formalizes this idea.

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

  • i)
    inline image for inline image and
  • ii)
    there is n0 such that inline image is nonempty for inline image.

As a consequence, we have inline image.

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

  • display math(5.8)

for inline image 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 inline image. These are nonzero if y is the slope of an affine part of inline image. 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 inline image-probability of the biggest atom in inline image. In the case of a single affine part in inline image, 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 inline image is the set of inline image-atoms in inline image and that Assumption 2.1 ensures that the maximal inline image-probability of all elements in inline image goes to 0.

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

  • display math

Hence, we may choose α0 such that

  • display math(5.9)

Define the set

  • display math

Now we are in a position to define the set inline image by

  • display math

We claim that this inline image satisfies the assumptions of Proposition 5.8.

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

  • display math

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

  • display math

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

  • display math

because inline image.

2) Property ii): For this part, we use Assumption 2.1 to choose n0 large enough such that inline image for inline image. Fix some inline image and define the map inline image by

  • display math

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

  • display math

satisfies inline image. This gives inline image.

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

  • display math

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

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

  • display math

We rewrite inline image in terms of inline image and inline image and use inline image to obtain

  • display math

where the definition of α0 in (5.9) is used in the last step.

3) Proof of inline image: Fix inline image. Because of the continuity of inline image in x and Proposition 5.4, we can fix inline image and n1 such that inline image implies inline image for all inline image. Applying the first part of this proof for δ gives n0 such that for all inline image, there is some set inline image with properties i) and ii). So, for each inline image, there is some inline image. By definition of inline image, the relation inline image holds for all inline image. Moreover, inline image is increasing in x, so adding and subtracting inline image and using that inline image yields

  • display math

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

  • display math

The result follows since inline image for each inline image. inline image

It remains to state and prove

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

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

  • display math

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

  • display math

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

  • display math

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

The lower semicontinuity in n of inline image stated in Theorem 5.3 is now a straightforward consequence of Propositions 5.4 and 5.8. For completeness, we formally carry out the argument.

Proof of Theorem 5.3. Since inline image converges to 0 by Proposition 5.8, since inline image converges to inline image by Proposition 5.4 and because the inequality inline image holds true for all inline image, we deduce from inline image that

  • display math

This completes the proof. inline image

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 inline image. For the second part, fix a maximizer inline image for inline image for every n. Proposition 4.1 shows that the sequence inline image contains a subsequence weakly converging to some inline image. It then follows from Proposition 5.2, the optimality of inline image and inline image that

  • display math

This shows that inline image is a maximizer for inline image since inline image. inline image

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 inline image so that inline image for inline image. In particular, inline image is strictly increasing on (0, 1) and uniformly bounded by 1.

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

  • display math(5.10)

We now fix a subsequence inline image realizing the inline image and such that the associated sequence inline image converges to inline image. As in Lemma 5.7 (and again using the modified version of Proposition A.1), this gives inline image. The assumptions that φ0 has a continuous distribution and inline image imply then that inline image. Moreover, as the function inline image is uniformly bounded, the right-hand side of (5.10) converges to inline image. But since φ0 has a continuous distribution, it follows that inline image P-a.s. and applying similar arguments in the reverse order, we find that

  • display math

With the arguments so far, we have proved that

  • display math

which gives inline image. But the limit model is atomless, so we have inline image 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. inline image

6. CONCLUSION

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. PROBLEM FORMULATION AND MAIN RESULTS
  5. 3. APPLICATIONS
  6. 4. STABILITY OF THE DEMAND PROBLEM FOR RDEU
  7. 5. STABILITY OF THE DEMAND PROBLEM FOR ENCU
  8. 6. CONCLUSION
  9. APPENDIX
  10. 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

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. PROBLEM FORMULATION AND MAIN RESULTS
  5. 3. APPLICATIONS
  6. 4. STABILITY OF THE DEMAND PROBLEM FOR RDEU
  7. 5. STABILITY OF THE DEMAND PROBLEM FOR ENCU
  8. 6. CONCLUSION
  9. APPENDIX
  10. 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 inline image to denote the value function. Recall that J is the conjugate of U (as well as inline image).

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

  • display math(A.1)

and inline image and inline image 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 inline image that is convex and increasing with inline image and

  • display math(A.2)

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

  • display math

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

  • display math(A.3)

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

  • display math

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

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

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

  • display math

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

A.2. Auxiliary Results

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

  • display math

Moreover, if inline image is satisfied, then there are constants k1 and k2 and inline image such that inline image for inline image.

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 inline image holds for inline image and inline image. Recall that

  • display math(A.4)

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

  • display math(A.5)

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

  • display math(A.6)

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

  • display math(A.7)

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

  • display math(A.8)

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

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

  • display math

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

REFERENCES

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