A predictable decomposition in an infinite assets model with jumps. Application to hedging and optimal investment

Abstract

Motivated by the theory of bond markets, we consider an infinite assets model driven by marked point process and Wiener process. The self-financed wealth processes are defined by using measure-valued strategies. Going further on the works of Bjork et al. [“Bond market structure in the presence of marked point processes”, Mathematical Finance, 7 (1997a) pp. 211–239; “Towards a general theory of bond markets”, Finance and Stochastics, 1 (1997b) pp. 141–174] who focus on the existence of martingale measures and market completeness questions, we study here the incompleteness case. Our main result is a predictable decomposition theorem for supermartingales in this infinite assets model context. The concept of approximate wealth processes is introduced, and we show in an example that the space of measure-valued strategies is not complete with respect to the semimartingale topology. As in the case of stock markets, one can then derive a dual representation of the super-replication cost and study the problem of utility maximization by duality methods.

Keywords:

• Bond markets
• Measure-valued portfolio
• Jump-diffusion model
• Stochastic integral
• Incomplete market
• Predictable decomposition
• G10
• E43
• 60H05
• 91B28

Acknowledgements

I would like to thank Jerzy Zabczyk for discussions. I am also grateful to Maurizio Pratelli for useful remarks.