The Compatible Bond-Stock Market With Jumps

We construct a bond-stock market composed of d stocks and many bonds with jumps driven by general marked point process as well as by an ℝn-valued Wiener process. By composing these tools we introduce the concept of a compatible bond-stock market and give a necessary and sufficient condition for this property. We study no-arbitrage properties of the composed market where a compatible bond-stock market is arbitrage-free both for the bonds market and for the stocks market.We then turn to an incomplete compatible bond-stock market and give a necessary and sufficient condition for a compatible bond-stock market to be incomplete. In this market we consider the mean-variance hedging in the special situation where both B(u, T) and eG(u, y, T)-1 are quadratic functions of T - u. So, we need to extend the notion of a variance-optimal martingale (VOM) as in Xiong and Kohlmann (2009) to the more general market. By introducing two virtual stocks $\widetilde{S}_1, \widetilde{S}_2$, we prove that the VOM for the bond-stock market is the same as the VOM for the new stock market $\bar{S}, \widetilde{S}_1, \widetilde{S}_2$. The mean-variance hedging problem in this incomplete bond-stock market for a contingent claim $H \in L^2(\mathscr{F}_{T^*})$ is solved by deriving an explicit solution of the optimal measure-valued strategy and the optimal cost induced by the optimal strategy of MHV for the stocks $\bar{S}, \widetilde{S}_1, \widetilde{S}_2$ is computed.