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

作者：Xiong, Dewen; Kohlmann, Michael^*

来源：Stochastic Analysis and Applications, 2010, 28(2): 202-225.

DOI：10.1080/07362990903546389

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

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 [image omitted] whose dynamic penalty functional [image omitted] is generated by a convex function [image omitted]. So the penalty functional takes the following form [image omitted] where [image omitted] 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 [image omitted] 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 [image omitted] such that 2 is also bounded and [image omitted], we prove that [image omitted], Q0-a.s., for all t[0, T]. Finally, we introduce the concept of a bounded [image omitted]-(super-)martingale and derive a decomposition for a [image omitted]-supermartingale.

• 出版日期2010
• 单位上海交通大学