Utility Optimization Problem And Generalization Of P-Optimal Martingale Measure With Jumps

We consider the utility optimization problem with the utility function U(x) =xαα, x > 0,α∈(0, 1) in an incomplete financial market model with jumps. Hence,the aim is to maximize the expected utility over terminal values that are attainable byadmissible strategies. Firstly, we transform the primal problem into its dual problemwith the help of dual approach. Furthermore, we turn the problem to theα-optimalmartingale measure problem.In fact, theα-optimal martingale measure problem is a generalization of the p-optimal martingale measure in Michael Kohlmann and Dewen Xiong (2007)[1] whoconsidered the condition of p≥1 while we consider p =αα?1 < 0. Moreover, we alsoderive the similar results for p < 0 and 0 < p≤1. Similar to Michael Kohlmannand Dewen Xiong (2007)[1], we first introduce a new martingale measure Q0∈Mαe.Then we treat the problem under Q0 as the basic underlying measure. By using thedynamic programming approach we establish a backward martingale equation (BME)to show that if the BME has a solution, then theα-optimal martingale measure Q? isequivalent to P. We give a description of theα-optimal martingale measure Q? andthe maximal expected utility in terms of the solution of the BME. Also, we give anexplicit solution of the BME under the financial market model with jumps which cannumerically be easily obtained. Furthermore, we derive the value of maximal expectedutility, the optimal portfolio strategy and the terminal wealth process with the primalutility problem by the solution of the BME.