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

Abstract

We consider the dynamic convex valuation (DCV) in an incomplete market of m stocks S = (S 1,…, S m ) 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 Q 0 for a given function g(t, y, z) satisfying a Lipschitz condition. We give a sufficient and necessary condition for g so that ℰ g is an S-related DCV. We mainly study the dynamics of an -dominated S-related DCV C = {(C t (ξ)); ξ ∈L ∞(ℱ T )}. By applying Theorem 5 of Delbaen et al. [4], it is seen that the penalty functional α of C satisfies for a function with , where k is a positive constant. Under the assumption that is continuous with respect to l, we prove that {C t (ξ); t ∈ [0, T]} is the unique bounded solution of a BSDE generated by the function g(t, z 2) with quadratic growth in z 2. This main result generalizes Theorem 7.1 of Coquet et al. [2] about the “ℰμ-dominated ℱ-consistent expectation.”