The S-Related Dynamic Convex Valuation in the Brownian Motion Setting.

We consider the dynamic convex valuation (DCV) in an incomplete market of m stocks S = (S1,..., Sm) in the Brownian motion setting. In this framework, we continue our work in Xiong and Kohlmann [17] on S-related DCV by now considering the S-related DCV generated by a conditional g-expectation under an equivalent martingale measure Q0 for a given function g(t, y, z) satisfying a Lipschitz condition. We give a sufficient and necessary condition for g so that Eg is an S-related DCV. We mainly study the dynamics of an [image omitted]-dominated S-related DCV C = {(Ct(ξ)); ξ ∈L∞(FT)}. By applying Theorem 5 of Delbaen et al. [4], it is seen that the penalty functional α of C satisfies [image omitted] for a function [image omitted] with [image omitted], where k is a positive constant. Under the assumption that [image omitted] is continuous with respect to l, we prove that {Ct(ξ); t ∈ [0, T]} is the unique bounded solution of a BSDE generated by the function g(t, z2) with quadratic growth in z2. This main result generalizes Theorem 7.1 of Coquet et al. [2] about the “Eμ-dominated F-consistent expectation.” [ABSTRACT FROM AUTHOR]