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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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

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

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

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Zentralblatt MATH identifier

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

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


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.

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