An S-Related DCV Generated by a Convex Function in a Jump Market

Abstract

We consider an incomplete market with general jumps, in which the discounted price process S of a risky asset is a given bounded semimartingale. We continue our work on the S-related dynamic convex valuation (DCV) initiated in Xiong and Kohlmann [23] by considering here an S-related DCV whose dynamic penalty functional is generated by a convex function . So the penalty functional takes the following form

where

is the density process of an equivalent martingale measure (EMM) Q for S with respect to the fundamental EMM Q ⁰. For a given ξ ∊L ^∞(ℱ _T ), we prove that is the first component of the minimal bounded solution of a backward semimartingale equation (BSE) generated by a convex, possibly non-Lipschitz g. If this BSE has a bounded solution such that θ₂ is also bounded and , we prove that , Q ⁰-a.s., for all t ∊ [0, T]. Finally, we introduce the concept of a bounded -(super-)martingale and derive a decomposition for a -supermartingale.

Keywords:

• Backward semimartingale equation (BSE)
• Dynamic convex risk measure
• Dynamic convex valuation (DCV)
• Time-consistent property

Mathematics Subject Classification:

• 90A09
• 60H30
• 60G44

Dewen Xiong was supported by the National Natural Science Foundation of China under Grant No. 70671069 and Grant No. 10801097 and National Basic Research Program of China (973 Program) under Grant No. 2007CB814903. He thanks the University of Konstanz for its hospitality and both authors gratefully acknowledge support from the Ausbauprogramm 2012 at the University of Konstanz.