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

ISSN： 1045-1129

Source： Stochastics and Stochastics Reports, Vol.75, Iss.5, 2003-10, pp. : 343-368

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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.