A Stochastic Control Approach to a Robust Utility Maximization Problem

Abstract

Summary.We study a stochastic control problem arising in the context of utility maximization under model uncertainty. The latter is formulated as a sup-inf problem over strategies π and models (measures) Q, and we treat the inner problem of minimizing over Q the sum of a Q-expected utility term and a penalty term based on the relative entropy of Q with respect to a reference measure P . We prove in general that there exists a unique optimal measure Q * and show that Q * is equivalent to P . For a continuous filtration, we characterize the dynamic value process of our stochastic control problem as the unique solution of a generalized backward stochastic differential equation with a quadratic driver. Our results extend earlier work in [21] and are based on a different approach.Mathematics Subject Classification (2000): 93E20, 91B16, 60H10, 46N10

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

“…coherent) monetary utility functional; see [31, Theorem 3.1]. Recall that a concave monetary utility functional is a quasiconcave utility functional which is also translation invariant; 8 if it is further positively homogeneous, 9 it is coherent. For the concave case, representation results akin to Theorem 2.4 were established in [29,34] (see [19] for the coherent case).…”

“…Homothetic preferences It is shown in [57] that a preorder satisfies full certainty independence if and only if it satisfies weak certainty independence along with the following axiom of homotheticity: 8 Translation invariance is also referred to as cash additivity and means that for m ∈ R, φ(X + m) = φ(X) + m. Any translation invariant quasiconcave utility functional is concave, i.e., φ(λX…”

“…For this choice, the problem is naturally formulated in terms of utility from consumption (or stochastic differential utilities) and the natural tool is the theory of BSDEs. While a systematic study was initiated in [75], these results have been considerably extended in a number of articles; see [8,15,26,40,56]. For the case of utility from terminal wealth and general variational criteria, stochastic control methods have been successfully applied for the choice of logarithmic utility, for stochastic factor models as well as for the non-Markovian case; see, respectively, [38,39] and [53].…”

Robust Portfolio Choice and Indifference Valuation

“…The next theorem shows that the solution to the optimization problem (4.7) can be obtained directly from the solution to a BSDE: 9) withḡ i defined in (4.8) for i = 1, 2. Furthermore, the optimal strategy π * s is a predictable process that attains the infimum in (4.9) for (z,z) = (Z s ,Z s ).…”

“…Bordigoni, Matoussi and Schweizer [9] study ambiguity using the relative entropy and Jeanblanc, Matoussi and Ngoupeyou [41] generalize this work to a noncontinuous filtration, assuming a one-point jump distribution. Björk and Slinko [8] study asset prices with jumps in a non-utility framework, using different possible pricing kernels to obtain good-deal bound prices.…”