Decision Theory with Imperfect Information by Rafik A Aliev & Oleg H Huseynov

Author:Rafik A Aliev & Oleg H Huseynov
Language: eng
Format: epub
ISBN: 9789814611053
Publisher: World Scientific Publishing Company
Published: 2014-08-08T00:00:00+00:00

(i) Comonotonic additivity: for comonotonic , holds;

(ii) Monotonicity: if for all then .

Under these conditions, defining for all , where H * denotes the indicator function of H, we have

such that

Note that comonotonically additive and monotonic Ĩ on satisfies Ĩ(λã) = λĨ (ã) for λ > 0. Indeed, α-cut of Ĩ(λã) is defined as , where because λ > 0 So,

Thus, we will have:

So, Ĩ(λã) = λĨ (ã), λ > 0

In order to prove Theorem 6.3 we need to use the following remark.

Remark 6.1. The integrand in (6.2) can be compactly expressed as follows:

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