Utility Maximization Under Partial Information in a Jump-Diffusion Model

This paper studies the question of maximization of expected utility from terminal wealth in a multidimensional jump-diffusion model of incomplete market. The special feature of our approach is that the investor only observes the vector of stock prices leading to a partial information framework. Resorting to non-linear filtering and change of measures techniques, we show that the optimization problem can be rewritten by one whose coefficients depend on past history of observed prices. Then, by using Malliavin calculus for jump-diffusion models, we fully characterize both the optimal wealth and investment policy processes through the duality approach of this problem. To this end, we derive an extension of Clark's formula for jump-diffusion models under an equivalent change of measure. Eventually, existence of solutions in the general case is studied under special hypotheses. As examples, special attention will be given to power, logarithmic and exponential utility functions for which the risk-premia processes are fully characterized. Thus, the question of hedging strategies for contingent claims will be examined through the case study of Asian options