Duality theory for optimal investments under model uncertainty

Abstract

Robust utility functionals arise as numerical representations of investor preferences, when the investor is uncertain about the underlying probabilistic model and averse against both risk and model uncertainty. In this paper, we study the duality theory for the problem of maximizing the robust utility of the terminal wealth in a general incomplete market model. We also allow for very general sets of prior models. In particular, we do not assume that all prior models are equivalent to each other, which allows us to handle many economically meaningful robust utility functionals such as those defined by AVaR λ , concave distortions, or convex capacities. We also show that dropping the equivalence of prior models may lead to new effects such as the existence of arbitrage strategies under the least favorable model.

Risk- and ambiguity-averse portfolio optimization with quasiconcave utility functionals

“…The results herein extend results established for variational preferences in [71] (see [68,72] for the coherent case). Our proofs are therefore inspired by those in the former articles.…”

“…Our results extend those established for variational preferences in [71] to the case of quasiconcave preferences; see [68,72] for the multiple-priors case. Following this previous literature, we prove our results by building on existing results for the classical utility maximization problem.…”

“…On the other hand, v Z (y) < ∞ implies that u Z (x) < ∞, x > 0 (cf. Lemma 3.5 in [72] along with the proof of Lemma 4.2). Hence,Q ⊆ Q f which yields the reverse inequality.…”

“…As argued above, when G corresponds to a coherent or concave utility functional, the criterion reduces to the multiple-priors and variational preferences studied in, among others, [68,71,72]. In the same way as the study of such preferences was motivated by the axiomatic results in [35,57], the study of the more general quasiconcave case relies on the axiomatic extensions in [11].…”

“…General existence and duality results have also been established for variational preferences. Specifically, by the use of methods similar to the ones introduced in [68,72], Schied [71] generalized these results to the concave case; see also [42,43].…”