Minimal sufficient conditions for a primal optimizer in nonsmooth utility maximization

Westray, Nicholas; Zheng, Harry

doi:10.1007/s00780-010-0128-6

Cited by 11 publications

(8 citation statements)

References 13 publications

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“…As in part 2), the solution f * for U c satisfies f * ∈− ∂ J (λ * ϕ) and E[ϕ f * ]=x for a particular dual object ϕ. However, Westray and Zheng [42] show that these conditions are in general not sufficient for optimality of f * in the concavified problem. This is in contrast to the case with one pricing measure where these assumptions are sufficient for optimality and this is the reason why the proof of Proposition 5.3 cannot be extended directly to the case with many pricing measures.…”

Section: ⊓ ⊔mentioning

confidence: 99%

Utility maximization with a given pricing measure when the utility is not necessarily concave

Reichlin

2013

Math Finan Econ

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We study the problem of maximizing expected utility from terminal wealth for a not necessarily concave utility function U and for a budget set given by one fixed pricing measure. We prove the existence and several fundamental properties of a maximizer. We analyze the (not necessarily concave) value function (indirect utility) u(x, U). In particular, we show that the concave envelope of u(x, U) is the value function u(x, U c) of the utility maximization problem for the concave envelope U c of the utility function U. The two value functions are shown to coincide if the underlying probability space is atomless. This allows us to characterize the maximizers for several model classes explicitly

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Section: ⊓ ⊔mentioning

confidence: 99%

Utility maximization with a given pricing measure when the utility is not necessarily concave

Reichlin

2013

Math Finan Econ

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“…Almost all work in literature on utility maximization are for continuously differentiable and strictly concave utility functions. The main references for nonsmooth utility maximization are [4,5,16,17].…”

Section: Dual Control Problemmentioning

confidence: 99%

Smooth Value Functions for a Class of Nonsmooth Utility Maximization Problems

Bian¹,

Miao²,

Zheng³

2011

SIAM J. Finan. Math.

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Abstract. In this paper we prove that there exists a smooth classical solution to the HJB equation for a large class of constrained problems with utility functions that are not necessarily differentiable or strictly concave. The value function is smooth if admissible controls satisfy an integrability condition or if it is continuous on the closure of its domain. The key idea is to work on the dual control problem and the dual HJB equation. We construct a smooth, strictly convex solution to the dual HJB equation and show that its conjugate function is a smooth, strictly concave solution to the primal HJB equation satisfying the terminal and boundary conditions.

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“…Westray and Zheng [30] show that when B ≡ 0 the conditions on budget equality, subdifferential relation and feasibility are the minimal sufficient conditions for X * being a primal optimizer if the utility function U is not strictly concave. In this sense, Theorem 4.1 is almost a necessary and sufficient optimality condition if a dual optimizer is known to exist.…”

Section: Main Results and Its Proofmentioning

confidence: 99%

Constrained nonsmooth utility maximization on the positive real line

Westray¹,

Zheng²

2015

Mathematical Control &Amp; Related Fields

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We maximize the expected utility of terminal wealth in an incomplete market where there are cone constraints on the investor's portfolio process and the utility function is not assumed to be strictly concave or differentiable. We establish the existence of the optimal solutions to the primal and dual problems and their dual relationship. We simplify the present proofs in this area and extend the existing duality theory to the constrained nonsmooth setting.

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Minimal sufficient conditions for a primal optimizer in nonsmooth utility maximization

Cited by 11 publications

References 13 publications

Utility maximization with a given pricing measure when the utility is not necessarily concave

Utility maximization with a given pricing measure when the utility is not necessarily concave

Smooth Value Functions for a Class of Nonsmooth Utility Maximization Problems

Constrained nonsmooth utility maximization on the positive real line

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