Optimal Exponential Utility in a Jump Bond Market

We consider the optimal exponential utility in a bond market with jumps basing on a model similar to Björk et al. 4 Björk , T. , Kabanov , Y. , and Runggaldier , W. 1997 . Bond market structure in the presence of marked point processes . Math. Finance 7 : 211 – 223 .Crossref], Web of Science ®] , Google Scholar]], which is arbitrage free. Similar to the normalized integral with respect to the cylindrical martingale first introduced in Mikulevicius and Rozovskii 13 Mikulevicius , R. , and Rozovskii , B.L. 1998 . Normalized stochastic integrals in topological vector spaces . In : Séminaire de probabilités XXXII (Lecture Notes in Mathematics) . Springer , Berlin , pp. 137 – 165 . Google Scholar]], we introduce the (𝕄, Q ⁰)-normalized martingale and local (𝕄, Q ⁰)-normalized martingale. For a given maturity T ₀ ∈ 0, T*], we describe the minimal entropy martingale (MEM) based on T ₀, T*] by a backward semimartingale equation (BSE) w.r.t. the (𝕄, Q ⁰)-normalized martingale. Then we give an explicit form of the optimal approximate wealth to the optimal exp-utility problem by making use of the solution of the BSE. Finally, we describe the dynamics of the exp utility indifference valuation of a bounded contingent claim H ∈ L ^∞(? _{T 0} ) by another BSE under the minimal entropy martingale measure in the incomplete market.