Robust Utility Maximization in Nondominated Models With 2bsde: The Uncertain Volatility Model

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 nondominated. 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.KEY WORDS: second-order backward stochastic differential equation, quadratic growth, robust utility maximization, volatility uncertainty.The authors would like to thank two anonymous referees and the associate editor for their helpful remarks and comments.

Quadratic backward stochastic differential equations driven by G-Brownian motion: Discrete solutions and approximation

“…To get rid of the measurability problem during the construction of solutions, Soner et al assume the technical condition that both ξ and F is uniformly continuous in ω, whereas this assumption is removed in the recent work of Possamaï et al [34]. In the framework of 2BSDEs, the results in [37,38] are generalized by Possamaï and Zhou [35] and by Lin [21] to the quadratic case and furthermore, Matoussi et al [22] applied quadratic 2BSDEs to solve the utility maximization problems from [14] in the context with non-dominated models. One could see that the GBSDE (1.3) actually corresponds to the 2BSDE defined with…”

Robust utility maximization for a diffusion market model with misspecified coefficients

“…In our case both coefficients (drift and volatility) are misspecified, (b) the volatility matrix √ a t satisfies the condition a ≤ a t ≤ a, where a and a are given matrices, which does not cover our "partially misspecified volatility" case since in our paper the matrices a t = Moreover, in the non-Markovian case the BSDE corresponding to our problem will not be a 2BSDE (see Remark 3.5). And, besides, we cannot even get our BSDE as a particular case of the 2BSDE given in [27]. So we can conclude that [27] has little in common with our paper.…”

“…And, besides, we cannot even get our BSDE as a particular case of the 2BSDE given in [27]. So we can conclude that [27] has little in common with our paper.…”

“…During the refereeing process, we have found the preprint of Matoussi et al [27] which is also devoted to a robust utility maximization problem. To study the exponential, power and logarithmic utility maximization, the authors use 2BSDE theory (this theory was thoroughly developed by Cheridito, Soner, Touzi, Victoir and Zhang in [8,31]).…”

On Relaxed Stochastic Optimal Control for Stochastic Differential Equations Driven by G-Brownian Motion

“…More precisely, we ask if there exists a stochastic control u ∈ U with values in an action set U such that f (t, x u (t), u(t))dt + h(x u (T )) (1.2) where x u is a G-sde given by (4.3), below. This problem has been studied in Hu et al (2014); Biagini et al (2014) and Matoussi et al (2015), where the authors suggest necessary and sufficient optimality conditions in terms of respectively a Pontryagin's type maximum principle and dynamic programming principle. The objective of this work is to investigate the problem of existence of an optimal control.…”