Banach Journal of Mathematical Analysis

On Banach spaces of vector-valued random variables and their duals motivated by risk measures

Thomas Kalmes and Alois Pichler

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Abstract

We introduce Banach spaces of vector-valued random variables motivated from mathematical finance. So-called risk functionals are defined in a natural way on these Banach spaces, and it is shown that these functionals are Lipschitz continuous. Since the risk functionals cannot be defined on strictly larger spaces of random variables, this creates an area of particular interest with regard to the spaces presented. We elaborate key properties of these Banach spaces and give representations of their dual spaces in terms of vector measures with values in the dual space of the state space.

Article information

Source
Banach J. Math. Anal., Volume 12, Number 4 (2018), 773-807.

Dates
Received: 29 March 2017
Accepted: 19 June 2017
First available in Project Euclid: 8 September 2017

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1504857611

Digital Object Identifier
doi:10.1215/17358787-2017-0026

Mathematical Reviews number (MathSciNet)
MR3858750

Zentralblatt MATH identifier
06946292

Subjects
Primary: 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Secondary: 46E40: Spaces of vector- and operator-valued functions 62P05: Applications to actuarial sciences and financial mathematics

Keywords
vector-valued random variables Banach spaces of random variables rearrangement invariant spaces dual representation risk measures

Citation

Kalmes, Thomas; Pichler, Alois. On Banach spaces of vector-valued random variables and their duals motivated by risk measures. Banach J. Math. Anal. 12 (2018), no. 4, 773--807. doi:10.1215/17358787-2017-0026. https://projecteuclid.org/euclid.bjma/1504857611


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