Utility maximization problem with random endowment and transaction costs: when wealth may become negative

Abstract

Abstract. In this paper we study the problem of maximizing expected utility from the terminal wealth with proportional transaction costs and random endowment. In the context of the existence of consistent price systems, we consider the duality between the primal utility maximization problem and the dual one, which is set up on the domain of finitely additive measures. In particular, we prove duality results for utility functions supporting possibly negative values. Moreover, we construct the shadow market by the dual optimal process and consider the utility based pricing for random endowment.

Robust utility maximisation in markets with transaction costs

“…The same argument works for t = T , too. Now it extends to arbitrary t ∈ [0, T ] using Fatou's lemma and (23). Finally, it extends to arbitrary s ∈ [0, T ] by the backward martingale convergence theorem and by right-continuity of t → V x t (θ, H * ).…”

“…Condition (11) is mild and so is (12): as shown in Corollary 4.2(i) of [32], for every utility function U with reasonable asymptotic elasticity, its conjugate V satisfies (12). The studies [11], [23] assumed a smooth U which is strictly concave on its entire domain, we do not need either smoothness or strict concavity of U .…”

“…Assumption 4.4. For each θ ∈ Θ, the price process S θ admits a λ-SCPS [11,12,23] and in Section 3, in the present section we do not impose the existence of consistent price systems for every transaction cost coefficient 0 < µ < λ, we only stipulate Assumption 4.4. The following example shows that it is quite possible to have CPSs for relatively large λ, without having them for arbitrarily small µ.…”

On the existence of shadow prices for optimal investment with random endowment

“…In particular, this can be ensured if S is an exponential fractional Brownian motion. In addition, we refer to [39] for the generalization of the results in [14] with random endowment.…”

A note on utility maximization with transaction costs and random endoment: numéraire-based model and convex duality

“…We remark that only utility functions supporting the positive halfplane are concerned in [1] as well as in this note. For the results on utility functions allowing for negative wealth, we refer the reader to [14] (submission in preparation, draft available on request).…”