Some problems of portfolio optimization and hedging in a Lévy market via fictitious completions

The classical Merton’s problem of utility maximization was recently solved in [2] in a market consisting of a bond with constant interest rate, a stock that follows a geometric Lévy model, and certain “fictitious” stocks called powerjump assets. Using their previous work [3] on the completeness of such a market and the martingale method, it was proved there that for certain utility functions, it is possible to choose the fictitious stocks so that they are not needed to replicate the optimal final wealth. In this paper, we study other properties of these fictitious completions of the Lévy market. Problems that are considered here include optimal portfolios that maximize state-dependent utilities, or as a particular case, that minimize the “shortfall risk” in replicating a contingent claim. Also, we analyze conditions for replicating a contingent claim, possessing a discounted payoff of X̃, with an initial endowment w = supQ E [ X̃ ] , where the supremum is only taken over equivalent martingale measures associated to fictitious completions of the market, and not over all equivalent martingale measures as the fundamental theorem of supereplication establish.