Minimal Hellinger martingale measures of order q

This paper proposes an extension of the minimal Hellinger martingale measure (MHM hereafter) concept to any order q not equal 1 and to the general semimartingale framework. This extension allows us to provide a unified formulation for many optimal martingale measures, including the minimal martingale measure of Follmer and Schweizer (here q = 2). Under some mild conditions of integrability and the absence of arbitrage, we show the existence of the MHM measure of order q and describe it explicitly in terms of pointwise equations in R-d. Applications to the maximization of expected power utility at stopping times are given. We prove that, for an agent to be indifferent with respect to the liquidation time of her assets (which is the market's exit time, supposed to be a stopping time, not any general random time), she is forced to consider a habit formation utility function instead of the original utility, or equivalently she is forced to consider a time-separable preference with a stochastic discount factor.