# IJPAM: Volume 37, No. 3 (2007)

**THE -OPTIMAL MARTINGALE MEASURE WHEN**

THERE EXIST INACCESSIBLE JUMPS

THERE EXIST INACCESSIBLE JUMPS

Department of Mathematics and Statistics

University of Konstanz

Konstanz, D-78457, GERMANY

e-mail: michael.kohlmann@uni-konstanz.de

Department of Mathematics

Shanghai Jiaotong University

Shanghai, 200240, P.R. CHINA

e-mail: xiongdewen@sjtu.edu.cn

**Abstract.**We consider the -optimal martingale measure (the general
importance of these tools is shortly described in the conclusion)
in an incomplete financial market model with inaccessible jumps
described by a random jump measure. Using a dynamic programming
approach, we obtain a backward martingale equation (BME) with the
property that if the BME has a solution, then the -optimal
martingale measure is equivalent to the original measure.
Furthermore we give a description of the -optimal martingale
measure by the solution of the BME.

In a simple case similar to Jeanblanc, Kloeppel and Miyahara, see [#!Jeanblanc-Kloeppel-Miyahara2006!#], we give an explicit solution of the BME. As an application, we
consider the optimal utility of an investor with utility function
, and explicitly derive the optimal
strategy by the solution of the BME.

**Received: **March 21, 2007

**AMS Subject Classification: **90A09, 60H30, 60G44

**Key Words and Phrases: **optimality principle, backward martingale equations, stochastic Riccati equation, -optimal martingale measure

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2007

**Volume:** 37

**Issue:** 3