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The Dynamic Convex Valuation Related to the Price Process in a Market with General Jumps

Abstract We consider an incomplete market with general jumps in the given price process S of a risky asset. We define the S-related dynamic convex valuation (S-related DCV) which is time-consistent. We discuss the representation for a given S-related DCV C in terms of a ‘penalty functional’ α and give some characteristics of α, which are the sufficient conditions for a given C to be an S-related DCV. Finally, we give two special forms of α satisfying those conditions to describe the dynamics of the corresponding S-related DCV by a backward semimartingale equation.