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 Cĝ whose dynamic penalty functional αĝ is generated by a convex function ĝ. So the penalty Junctional takes the following form is the density process of an equivalent martingale measure (EMM) Q for S with respect to the fundamental EMM Q0. For a given ∈ L∞ (FT), we prove that (Cĝ(ξ) 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 (Y, θ1, θ2, L) such that θ2 is also bounded and 〈L〉T ∈ L∞ (FT), we prove that Cĝt(ξ) = Yt, Q0-a.s., for all t ∈ [0, T]. Finally, we introduce the concept of a bounded Cĝ-(super-)martingale and derive a decomposition for a Cĝ-supermartingale.