## Banach Journal of Mathematical Analysis

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

#### 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
Accepted: 19 June 2017
First available in Project Euclid: 8 September 2017

https://projecteuclid.org/euclid.bjma/1504857611

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

Mathematical Reviews number (MathSciNet)
MR3858750

Zentralblatt MATH identifier
06946292

#### 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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