Optimal Investment With Intermediate Consumption and Random Endowment

Abstract

We consider an optimal investment problem with intermediate consumption and random endowment, in an incomplete semimartingale model of the financial market. We establish the key assertions of the utility maximization theory, assuming that both primal and dual value functions are finite in the interiors of their domains and that the random endowment at maturity can be dominated by the terminal value of a self-financing wealth process. In order to facilitate the verification of these conditions, we present alternative, but equivalent conditions, under which the conclusions of the theory hold.

Sensitivity of optimal consumption streams

“…[6,17]). In general, the first-order condition is only guaranteed to hold for a dual variable from a larger class of supermartingale densities, compare [18,19,31]. However, it is often satisfied in concrete examples, cf., e.g., [5,11].…”

“…The dual considerations in the proof of Theorem 4.1 also show thatẐ is first-order optimal for the minimization problem dual to 2.3 (cf. [18,19,31,14] for more details. )…”

“…1 Existence, uniqueness, and duality for problems of this kind are well understood, even in more general settings that also allow for investment in a financial market. See, e.g., [18,19,31] and the references therein. In contrast, beyond standard utilities and very particular endowment streams (see, e.g.…”

On utility maximization without passing by the dual problem

“…• In the theory of large financial markets, our approach is the first to tackle utility maximization for U finite on R (the case of U defined on (0,∞) was first considered in [11]; subsequently [20] treated random endowments in the same setting), see Section 5.…”

Optimal Investment with Random Endowments and Transaction Costs: Duality Theory and Shadow Prices

“…In particular, to build the convex duality theory treating the problem with unhedgeable random endowment demands new techniques, especially if the endowment unbounded, see for example [4], [12], [19]. The references [18], [29], [23] and [27] also study this problem when the intermediate consumption is considered.…”