Topological duals of locally convex function spaces

Pennanen, Teemu; Perkkiö, Ari-Pekka

doi:10.1007/s11117-022-00867-4

Cited by 4 publications

(4 citation statements)

References 24 publications

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“…The first claim then implies that Eh * is proper on Y. The second claim thus follows from [15,Corollary 6].…”

Section: Conjugates Of Integral Functionalsmentioning

confidence: 81%

“…in L p spaces with p > 1 or, more generally, in Orlicz spaces associated with finite Young functions Φ. These and more examples can be found in [15].…”

Section: Continuity Of Integral Functionalsmentioning

confidence: 99%

“…By Lemma 2, Y is necessarily the Köthe dual of U. Most familiar Banach spaces of random variables have a topological dual of the above form; see [15].…”

Section: The Strong Dualmentioning

confidence: 99%

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Dual solutions in convex stochastic optimization

Pennanen¹,

Perkkiö²

2022

Preprint

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This paper studies duality and optimality conditions for general convex stochastic optimization problems. The main result gives sufficient conditions for the absence of a duality gap and the existence of dual solutions in a locally convex space of random variables. It implies, in particular, the necessity of scenario-wise optimality conditions that are behind many fundamental results in operations research, stochastic optimal control and financial mathematics. Our analysis builds on the theory of Fréchet spaces of random variables whose topological dual can be identified with the direct sum of another space of random variables and a space of singular functionals. The results are illustrated by deriving sufficient and necessary optimality conditions for several more specific problem classes. We obtain significant extensions to earlier models e.g. on stochastic optimal control, portfolio optimization and mathematical programming.

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“…The first claim then implies that Eh * is proper on Y. The second claim thus follows from [15,Corollary 6].…”

Section: Conjugates Of Integral Functionalsmentioning

confidence: 81%

“…in L p spaces with p > 1 or, more generally, in Orlicz spaces associated with finite Young functions Φ. These and more examples can be found in [15].…”

Section: Continuity Of Integral Functionalsmentioning

confidence: 99%

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Dual solutions in convex stochastic optimization

Pennanen¹,

Perkkiö²

2022

Preprint

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“…Decomposable solid spaces of random variables in separating duality include Lebesgue spaces, Orlicz spaces, Marcinkiewich spaces paired with Lorentz spaces, spaces of finite moments u L p for all p ∈ (1, ∞) as well as the general class of Banach Function Spaces or, even more generally, locally convex function spaces; see [20] and its references. The spaces of continuous functions or various Sobolev spaces of functions on Ê n fail to be decomposable or solid.…”

Section: Dual Spaces Of Random Variablesmentioning

confidence: 99%

Duality in convex stochastic optimization

Pennanen¹,

Perkkiö²

2022

Preprint

Self Cite

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This paper studies duality and optimality conditions in general convex stochastic optimization problems introduced by Rockafellar and Wets in [28]. We derive an explicit dual problem in terms of two dual variables, one of which is the shadow price of information while the other one gives the marginal cost of a perturbation much like in classical Lagrangian duality. Existence of primal solutions and the absence of duality gap are obtained without compactness or boundedness assumptions. In the context of financial mathematics, the relaxed assumptions are satisfied under the well-known no-arbitrage condition and the reasonable asymptotic elasticity condition of the utility function. We extend classical portfolio optimization duality theory to problems of optimal semi-static hedging. Besides financial mathematics, we obtain several new frameworks in stochastic programming and stochastic optimal control.

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Dual spaces of cadlag processes

Pennanen

Perkkiö

2023

Stochastic Processes and their Applications

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Topological duals of locally convex function spaces

Cited by 4 publications

References 24 publications

Dual solutions in convex stochastic optimization

Dual solutions in convex stochastic optimization

Duality in convex stochastic optimization

Dual spaces of cadlag processes

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