The Minimal Entropy and the Convergence of the p-Optimal Martingale Measures in a General Jump Model

Abstract

We first consider the minimal entropy martingale measure in a general jump model introduced in Kohlmann and Xiong (International Journal of Pure and Applied Mathematics, 37(3):321–348) and give a description of this measure (MEMM) as the solution of a backward martingale equation (BME). To relate the (MEMM) to the p-optimal martingale measure (p-OMM) we consider the convergence of the solution of the BME associated with the (p-OMM) to the solution of the BME associated with the MEMM. Under some assumptions, we prove the convergence of the p-OMM to the MEMM both in entropy and strongly in L ¹(P). As an application, we consider the exp-optimal utility of an investor with utility function U _exp(x) = −exp(− k ₀ x), and as q↑∞, we show that the q-optimal terminal wealth of an investor with utility converges to the exp-optimal terminal wealth of an investor with utility function U _exp(x) strongly in L ^r (P) for a large enough r.

Keywords:

• Backward martingale equations
• Minimal entropy martingale measure
• Optimality principle
• p-Optimal martingale measure

Mathematics Subject Classification:

• 90A09
• 60H30
• 60G44

Supported by National Natural Science Foundation of China under Grant No. 70671069 and National Basic Research Program of China (973 Program) under Grant No. 2007CB814903. The second author thanks the University of Konstanz for its support and hospitality.