IJPAM: Volume 37, No. 3 (2007)


Michael Kohlmann$^1$, Dewen Xiong$^2$
$^1$Department of Mathematics and Statistics
University of Konstanz
Konstanz, D-78457, GERMANY
e-mail: michael.kohlmann@uni-konstanz.de
$^2$Department of Mathematics
Shanghai Jiaotong University
Shanghai, 200240, P.R. CHINA
e-mail: xiongdewen@sjtu.edu.cn

Abstract.We consider the $p$-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 $p$-optimal martingale measure is equivalent to the original measure. Furthermore we give a description of the $p$-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 $U(x)=-\vert 1-\frac{x}{k_0}\vert^q$, 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, $p$-optimal martingale measure

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 37
Issue: 3