The Mean-Variance Hedging and Exponential Utility in a Bond Market With Jumps

We construct a market of bonds with jumps driven by a general marked point

process as well as by an Rn-valued Wiener process, in which there exists at least one equivalent

martingale measure Q0. In this market we consider the mean-variance hedging of a contingent

claim H 2 L2(FT0) based on the self-financing portfolios on the given maturities T1, · · · , Tn

with T0 T. We introduce the concept of variance-optimal martingale

(VOM) and describe the VOM by a backward semimartingale equation (BSE). We derive an

explicit solution of the optimal strategy and the optimal cost of the mean-variance hedging by

the solutions of two BSEs.

The setting of this problem is a bit unrealistic as we restrict the available bonds to those

with a a pregiven finite number of maturities. So we extend the model to a bond market with

jumps and a continuum of maturities and strategies which are Radon measure valued processes.

To describe the market we consider the cylindrical and normalized martingales introduced by

Mikulevicius et al.. In this market we the consider the exp-utility problem and derive some

results on dynamic indifference valuation.

The talk bases on recent common work with Dewen Xiong.