A dual characterization of self-generation and exponential forward performances

TL;DR

This paper introduces a dual framework for understanding forward performances in financial markets, characterizing their structure and providing explicit conditions and parametrizations for exponential cases in semimartingale models.

Contribution

It develops a dual characterization of self-generation for forward performances and offers explicit parametrizations for exponential cases in Itô-process driven markets.

Findings

Dual formulation of self-generation established

Necessary and sufficient conditions for exponential forward performances derived

Explicit parametrization of exponential forward performances in Itô markets provided

Abstract

We propose a mathematical framework for the study of a family of random fields--called forward performances--which arise as numerical representation of certain rational preference relations in mathematical finance. Their spatial structure corresponds to that of utility functions, while the temporal one reflects a Nisio-type semigroup property, referred to as self-generation. In the setting of semimartingale financial markets, we provide a dual formulation of self-generation in addition to the original one, and show equivalence between the two, thus giving a dual characterization of forward performances. Then we focus on random fields with an exponential structure and provide necessary and sufficient conditions for self-generation in that case. Finally, we illustrate our methods in financial markets driven by It\^o-processes, where we obtain an explicit parametrization of all exponential forward performances.