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As a matter of fact, scale invariance is the key property of acceptability indices that distinguishes them from risk measures, and, typically, acceptability indices are not cash-additive.
From now on, we will use as synonyms the terms measures of performance, performance measures, and acceptability indices.
In particular, they generalize the work done in (Bielecki et al. 2014b) regarding duality between cash-additive risk measures and acceptability indices.
In (Bielecki et al. 2014b), the authors introduce a new form of time consistency that is suitable for dynamic coherent acceptability indices.
Dynamic acceptability indices and performance measures are complementary concepts to dynamic (coherent) risk measures, for which good deal valuation bounds are a typical example (sign conventions aside).
The dynamic version of coherent acceptability indices was introduced by Bielecki et al. (2014b), for the case of stochastic processes, finite probability space, and discrete time.
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Let α t, t=0,1,…,T, be a dynamic coherent acceptability index acting on L ∞ (i.e., discounted terminal cash flows).
It was shown in (Bielecki et al. 2014b) that φ is a dynamic acceptability index for processes.
Surface mine reclamation is to be considered as third best solution, obtaining an acceptability index of 0.74.
For example, time consistency of a dynamic coherent acceptability index cannot be expressed in terms of a single benchmark family.
Among them, Sengupta and Pal [38] proposed the acceptability index for ranking any two interval numbers based on decision maker's satisfaction.
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