Robust utility maximization in non-dominated models with 2BSDEs

TL;DR

This paper addresses robust utility maximization under volatility uncertainty in incomplete markets using 2BSDEs, establishing the value function representation and optimal strategies for various utilities.

Contribution

It introduces a 2BSDE framework for non-dominated models with volatility bounds, providing new insights into utility maximization under uncertainty.

Findings

Value function expressed via 2BSDEs for exponential, power, and logarithmic utilities.

Existence of optimal strategies proven within the 2BSDE framework.

The upper volatility bound significantly influences the problem, akin to uncertain volatility models.

Abstract

The problem of robust utility maximization in an incomplete market with volatility uncertainty is considered, in the sense that the volatility of the market is only assumed to lie between two given bounds. The set of all possible models (probability measures) considered here is non-dominated. We propose studying this problem in the framework of second-order backward stochastic differential equations (2BSDEs for short) with quadratic growth generators. We show for exponential, power and logarithmic utilities that the value function of the problem can be written as the initial value of a particular 2BSDE and prove existence of an optimal strategy. Finally several examples which shed more light on the problem and its links with the classical utility maximization one are provided. In particular, we show that in some cases, the upper bound of the volatility interval plays a central role, exactly as in the option pricing problem with uncertain volatility models of [2].