Conditional preference orders and their numerical representations

Drapeau, Samuel; Jamneshan, Asgar

doi:10.1016/j.jmateco.2015.12.004

Cited by 10 publications

(19 citation statements)

References 45 publications

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“…In case of (conditional) expected utility, the generators are closely related with dynamic and conditional risk measures; see [9][10][11][12][13][14]. The preferences which underly conditional expected utility functionals were studied in [15].…”

Section: Definition 21mentioning

confidence: 99%

Parameter-Dependent Stochastic Optimal Control in Finite Discrete Time

Jamneshan¹,

Kupper

Zapata-García

2020

J Optim Theory Appl

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We prove a general existence result in stochastic optimal control in discrete time, where controls, taking values in conditional metric spaces, depend on the current information and past decisions. The general form of the problem lies beyond the scope of standard techniques in stochastic control theory, the main novelty is a formalization in conditional metric space and the use of conditional analysis. We illustrate the existence result by several examples such as wealth-dependent utility maximization under risk constraints and utility maximization with a conditional dimension. We also provide a discussion as to how our methods compare to techniques based on random sets.

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Section: Definition 21mentioning

confidence: 99%

Parameter-Dependent Stochastic Optimal Control in Finite Discrete Time

Jamneshan¹,

Kupper

Zapata-García

2020

J Optim Theory Appl

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“…A conditional Fenchel-Moreau theorem is applied in Drapeau et al [17] in vector duality. A conditional Riesz representation theorem is used in Drapeau and Jamneshan [14] in decision theory. Remark 4.27 Separation and duality results in topological L 0 (R)-modules are established in Guo et al [29] and Filipovic et al [19] with respect to two types of module topologies respectively, their connection is discussed in Guo [27], see also the discussion in Jamneshan and Zapata [38].…”

Section: Banach Space Theorymentioning

confidence: 99%

“…In this subsection, we will elaborate on basic results in separable Hilbert spaces 14 where our focus lies on the conditional L 2 -space An orthonormal sequence (x n ) n∈L 0 (N) is generating if for every x ∈ H there exists a sequence (r n ) n∈L 0 (N) in L 0 (G, R) such that x = n r n x n 15 . A generating orthonormal sequence is called an orthonormal basis.…”

Section: Hilbert Spacesmentioning

confidence: 99%

A transfer principle for second-order arithmetic, and applications

Carl

Jamneshan

2018

JLA

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In the theory of conditional sets, many classical theorems from areas such as functional analysis, probability theory or measure theory are lifted to a conditional framework, often to be applied in areas such as mathematical economics or optimization. The frequent experience that such theorems can be proved by 'conditionalizations' of the classical proofs suggests that a general transfer principle is in the background, and that formulating and proving such a transfer principle would yield a wealth of useful further conditional versions of classical results, in addition to providing a uniform approach to the results already known. In this paper, we formulate and prove such a transfer principle based on second-order arithmetic, which, by the results of reverse mathematics, suffices for the bulk of classical mathematics, including real analysis, measure theory and countable algebra, and excluding only more remote realms like category theory, set-theoretical topology or uncountable set theory, see e.g. the introduction of Simpson [47]. This transfer principle is then employed to give short and easy proofs of conditional versions of central results in various areas of mathematics, including theorems which have not been proven by hand previously such as Peano existence theorem, Urysohn's lemma and the Markov-Kakutani fixed point theorem. Moreover, we compare the interpretation of certain structures in a conditional model with their meaning in a standard model. 03C90,03F35; 28B20,54C65

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“…In case of (conditional) expected utility, the generators are closely related with dynamic and conditional risk measures, see [1,6,14,19,20]. The preferences which underly conditional expected utility functionals were studied in [15] under the name of conditional preference orders. In decision theory, there is an extensive literature on recursive utilities starting with the seminal work [30,31].…”

Section: And L0mentioning

confidence: 99%

“…This perspective is implicitly present in [25,29], however without a systematic treatment. A conditional version of basic results in functional analysis were established in [13,18,16], and applied to financial mathematics in [2,6,11,12,15,19,20,21,37]. In [16], conditional versions of classical theorems were studied systematically and related to a conditional variant of set theory.…”

Section: Introductionmentioning

confidence: 99%

Parameter-dependent Stochastic Optimal Control in Finite Discrete Time

Jamneshan¹,

Kupper²,

Zapata³

2017

Preprint

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We prove a general existence result in stochastic optimal control in discrete time where controls take values in conditional metric spaces, and depend on the current state and the information of past decisions through the evolution of a recursively defined forward process. The generality of the problem lies beyond the scope of standard techniques in stochastic control theory such as random sets, normal integrands and measurable selection theory. The main novelty is a formalization in conditional metric space and the use of techniques in conditional analysis. We illustrate the existence result by several examples including wealth-dependent utility maximization under risk constraints with bounded and unbounded wealth-dependent control sets, utility maximization with a measurable dimension, and dynamic risk sharing. Finally, we discuss how conditional analysis relates to random set theory.

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Conditional preference orders and their numerical representations

Cited by 10 publications

References 45 publications

Parameter-Dependent Stochastic Optimal Control in Finite Discrete Time

Parameter-Dependent Stochastic Optimal Control in Finite Discrete Time

A transfer principle for second-order arithmetic, and applications

Parameter-dependent Stochastic Optimal Control in Finite Discrete Time

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