05719927

j

05719927 Hara\'nczyk, Grzegorz S{\l}omczy\'nski, Wojciech Zastawniak, Tomasz Relative and discrete utility maximising entropy. Open Syst. Inf. Dyn. 15, No. 4, 303-327 (2008). 2008 World Scientific, Singapore EN

doi:10.1142/S1230161208000213

Summary: The notion of utility maximising entropy ($u$-entropy) of a probability density, which was introduced and studied in [37], is extended in two directions. First, the relative $u$-entropy of two probability measures in arbitrary probability spaces is defined. Then, specialising to discrete probability spaces, we also introduce the absolute u-entropy of a probability measure. Both notions are based on the idea, borrowed from mathematical finance, of maximising the expected utility of the terminal wealth of an investor. Moreover, u-entropy is also relevant in thermodynamics, as it can replace the standard Boltzmann-Shannon entropy in the Second Law. If the utility function is logarithmic or isoelastic (a power function), then the well-known notions of Boltzmann-Shannon and R\'enyi relative entropy are recovered. We establish the principal properties of relative and discrete u-entropy and discuss the links with several related approaches in the literature.