Necessary and sufficient conditions for the existence of the q-optimal measure

TL;DR

This paper characterizes the conditions for the existence of the q-optimal measure, extending previous work to signed measures and continuous asset prices, with implications for financial mathematics.

Contribution

It provides a necessary and sufficient condition for the q-optimal measure's existence, including for signed measures, and updates its characterization for continuous asset prices.

Findings

Established the general form and properties of the q-optimal measure.

Proved the existence of the q-optimal measure under mild conditions.

Provided an updated characterization considering counterexamples in continuous asset models.

Abstract

This paper presents the general form and essential properties of the q-optimal measure following the approach of Delbaen and Schachermayer (1996) and proves its existence under mild conditions. Most importantly, it states a necessary and sufficient condition for a candidate measure to be the q-optimal measure in the case even of signed measures. Finally, an updated characterization of the q-optimal measure for continuous asset price processes is presented in the light of the counterexample appearing in Cerny and Kallsen (2006) concerning Hobson's (2004) approach.