The minimax identity for a nondecreasing upper-semicontinuous utility function satisfying mild growth assumption is studied. In contrast to the classical setting, concavity of the utility function is not asumed. By considering the concave envelope of the utility function, equalities and inequalities between the robust utility functionals of an initial utility function and its concavification are obtained. Furthermore, similar equalities and inequalities are proved in the case of implementing an upper bound on the final endowment of the initial model.
Consider a complete market model framework with the unique equivalent local martingale measure
The set The set
Also, to Assumption
where
In this paper we study the minimax identity for the robust nonconcave utility functional in a complete market model, i.e.
the standard budget constraint:
an additional upper bound:
One of the key tasks of financial mathematics is proving the existence as well as the construction of optimal investment strategies, in other words, finding the utility-maximizing investment strategies. Mostly, this problem was studied under the assumption that the probability distribution of the value process is known.
However, in reality, along with the exact probabilities unknown, there are abundant aspects that can be considered in mentioned maximization problems such as the completeness of the market, the set of prior probability measures, the assumptions on investor’s utility function, the modeling of payoff and so on. That is why instead of a single measure it is sound to consider the set of probability measures with natural assumptions on it. Thus, the standard utility maximization problem becomes the robust utility maximization problem
In the case of a standard utility maximization problem it is possible to construct the optimal investment strategy given a strictly concave utility function, see Föllmer and Schied [
In this paper, we consider the robust maximization problem with the general nonconcave utility function, with and without budget constraints likewise. In the previous literature different approaches were used for robust portfolio optimization such as reducing the robust case to the standard one through proving the existence of the “worst-case scenario measure” or “the least favorable measure”, e.g., [
Besides, for solving the optimal investment problems one can make use of the following interim finding such as minimax identity and duality theory. Using the minimax identity for concave functions, see [
Neufeld and Šikić [
For more results concerning the robust utility maximization problem we refer to Bartl, Kupper and Neufeld [
The majority of articles on utility maximization assume that the investor’s utility function is strictly concave, strictly increasing, continuously differentiable, and satisfies the Inada conditions. While the assumption of monotonicity is natural, since an agent prefers more wealth to less, other assumptions can be omitted or relaxed. There is a wide class of models in which the maximization of the nonconcave and not necessarily continuously differentiable utility function has been studied by reducing the problem to the concave case. One of the most important works was done by Reichlin [
While considering two cases of admissible final endowments – the standard budget constraint and additional upper bound (which has not been considered before in such model setup) – we extend Reichlin’s results by proving new connections in the form of equalities and inequalities of the robust utility maximization functionals of initial nonconcave utility functions and its concavification. Furthermore, we proceed in proving the minimax identity for general nonconcave utility functions. The crucial step for obtaining the mentioned results with implementing an additional upper bound is the use of the regular conditional distribution which sheds new light on the possible approaches for solving the optimization problem.
The paper is organized as follows. In Section
Throughout the paper the measurability of real-valued functions is understood in the Borel sense.
This problem is already solved in [
To formulate the goal of this paper first let us remind some notations. For any initial capital
Moreover, we consider a utility function
It follows from [
This section aims to prove some equalities and inequalities related to the minimax identity for the robust nonconcave utility functionals:
We will assume that the probability space
Also, we need the finiteness of value functions, which we can write as follows.
Note that finiteness of
The proof of this theorem will be divided into several parts.
Now we are going to show that the minimax identity holds for
There is a lot of literature with proofs of the minimax identity for robust utility functionals, the most general case was considered in [
The function
The next lemma is almost the same as [ This lemma holds if we will consider utility function The proof can be found in [
In this section, we want to prove lemmas which will help us to complete the proof of Theorem
Note that the main argument in the proof of minimax identity for the robust utility maximization problem is the lop sided minimax theorem by Aubin and Ekeland, see [
The proof can be found in [
The above lemma also holds for
The proof of equality The proof can be found in [
To obtain Inequality Since
□
This section is in general similar to Section
Specifically, we assume that there is an upper bound on the endowment, given by a random variable
We keep all of the assumptions from Section
There exists a regular conditional distribution given
A simple sufficient condition for
The second assumption is verified if, for example, on
As in [
Note that the function
Our goal is to prove some equalities and inequalities related to the minimax identity for the robust nonconcave utility functionals
Introduce the following notation:
Since
It is natural to consider only the case where
The formulation of the next theorems and lemmas are the same as in Section
The proof of this theorem will be divided into several parts.
Now we are going to show that minimax identity holds for
Consider Take Then, noting that Take Define Moreover, in the proof of [ Noting that weak convergence follows from almost sure convergence, the conditions of the lop sided minimax theorem [ Hence, we arrive at
The last inequality follows from the fact that for all Sending From Assumption Hence,
This concludes the proof. □
□
In this section, we will establish auxiliary results which will allow us to complete the proof of Theorem
The proof is the same as in the nonconstrained case. See [
The main idea of the proof is to utilize the ideas of [
Fix
By Lemma
Apply the
Follows immediately from Lemma
To obtain Inequality Since
□
In what follows
We will show only measurability of
Note that
Further, since for For arbitrary Define By the definition of
Since
Assume first that the distribution of
For general
We will adapt the construction used in the proof of [
Define
For
Now set
O. Bahchedjioglou thanks Prof. Dr. Mitja Stadje and Dr. Thai Nguyen for their help and support during her work on the topic.
Authors thank the anonymous reviewer for careful reading of the manuscript and helpful suggestions.