Robust utility maximization with unbounded random endowment

Abstract

We study the convex duality method for robust utility maximization in the presence of a random endowment. When the underlying price process is a locally bounded semimartingale, we show that the fundamental duality relation holds true for a wide class of utility functions on the whole real line and unbounded random endowment. To obtain this duality, we prove a robust version of Rockafellar's theorem on convex integral functionals and apply Fenchel's general duality theorem.

The robust Merton problem of an ambiguity averse investor

“…On the contrary, in the dominated priors case there is a rich literature. We content ourselves with citing Chen and Epstein [3], Garlappi et alii [11], Maenhout [13], Föllmer et alii [10] for a comprehensive review and references, and, more recently, the work by Owari [16].…”

On admissible strategies in robust utility maximization

“…In the robust case, the Q-supermartingale property for all Q ∈ M V (P) (hence all Q ∈ M 0 V (Q,P) sinceθ · S is aQ-martingale) is shown by [8] (see also [17] for a slight extension). We emphasize that the difference between M V (P) and M V is essential here.…”

“…In the general robust case with P containing (infinitely) many elements, [8] (see also [17] for a slight generalization) provides a partial analogue of the above result which states that, under certain stronger assumptions, an optimal strategy is obtained in the class of θ with θ · S being a supermartingale under all local martingale measures Q having a finite entropy w.r.t. a certain elementP ∈ P called a least favorable measure, i.e., in the class Θ V (P).…”

“…As for the equivalence, we still haveQ ∼P whenever (λ ,Q,P) is a dual optimizer (see [17], Theorem 2.7). Also, by an exhaustion argument, there exists a maximal solution (λ ,Q,P) in the sense that if (λ , Q, P) is another dual optimizer, then P P (hence Q Q ) and λ dQ/dP =λ dQ/dP, P-a.s., where the density dQ/dP is defined P-a.s. in the sense of Lebesgue decomposition.…”

“…[23], [22], [8]). In the case of utility taking finite values for all x ∈ R, [18] shows the key duality, while [8] and [17] give a partial result on the existence of optimal strategy which we shall complete in this paper. See also [9] for more comprehensive reference and the background of the robust utility maximization problem.A key subtlety intrinsic to the case of utility on R is the "good definition" of admissible strategies θ , which will constitute the central theme of this paper.…”