Forward–backward systems for expected utility maximization

Abstract

In this paper we deal with the utility maximization problem with a general utility function. We derive a new approach in which we reduce the utility maximization problem with general utility to the study of a fully-coupled Forward-Backward Stochastic Differential Equation (FBSDE).

Forward Backward Semimartingale Systems for Utility Maximization

“…The proof under assumption (A2) (Proposition 3.2) is similar to that of Horst et al [7], but (A2) is weaker than condition (H5) used in their paper. Under (A1) (Proposition 3.1), we avoid the uniform integrability assumption by using the same analytical property of concave functions as that in Proposition 2.1.…”

“…We can now state the main result of this section, whose proof follows along the lines of the proofs of Theorems 4.1 and 4.2 in Horst et al [7].…”

“…The first set of hypotheses is new and refers only to the basic object of the optimization, that is, the utility function in the pure investment problem. The second set is similar, but is weaker than the one considered in Horst et al [7]. As in section 2, for the pure investment problem with deterministic mean-variance tradeoff, we characterize the solution to the system and the optimal strategy through a PDE, while displaying that the optimal allocation is a function of the current wealth.…”

“…Since the process h is bounded, one has π + n h ∈ Π x . Owing to the integrability of U (X π T + H) and the optimality of strategy π * , it follows that We are now in a position to state the main result of this section, whose proof follows along the lines of the proofs of Theorems 3.1 and 3.2 in Horst et al [7].…”

Robust Portfolio Choice and Indifference Valuation

“…Applications of BSDEs to utility maximization problems in incomplete markets in a Brownian setting include (with exponential, logarithmic or power utility) Hu, Imkeller and Müller [38], Cheridito and Hu [12], and (with a general utility function) Horst et al [37]; for a setting with continuous filtration or non-continuous filtration (but with exponential utility), see Mania and Schweizer [53], Morlais [57] and Becherer [4]. Morlais [58] generalizes some of these results adopting an exponential utility function and allowing for infinite activity jumps in the asset price processes, in a purely risk-based setting without ambiguity.…”