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.
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