On the Existence of Minimax Martingale Measures

Abstract

Embedding asset pricing in a utility maximization framework leads naturally to the concept of minimax martingale measures. We consider a market model where the price process is assumed to be an d ‐semimartingale X and the set of trading strategies consists of all predictable, X ‐integrable, d ‐valued processes H for which the stochastic integral ( H.X ) is uniformly bounded from below. When the market is free of arbitrage, we show that a sufficient condition for the existence of the minimax measure is that the utility function u : → is concave and nondecreasing. We also show the equivalence between the no free lunch with vanishing risk condition, the existence of a separating measure, and a properly defined notion of viability.