Shortfall risk measure for general semimartingales

In this talk, we focus on a convex risk measure which is induced by the shortfall risk, which is defined as the weighted expectation by some loss function of the positive part of the difference between a claim and a portfolio value at the maturity of our market. The convex risk measure concerned in this talk expresses the least cost to suppress its shortfall risk below the threshold being determined beforehand. Consider an incomplete financial market being composed of one riskless asset and d risky assets. The fluctuation of the risky assets is described by an RCLL semimartingale S, which is possibly non-locally bounded. Let l be a loss function, that is, an R+-valued continuous non-decreasing convex function defined on R. We assume that l(x) = 0 if x ≤ 0, and l(x) > 0 if x > 0. Let Θ be a predictably convex subset of S-integrable predictable processes which means the set of all admissible strategies, and GT (θ) := ∫ T 0 θsdSs. Recall that, when the price of a claim H and the hedging strategy are given by x ∈ R and θ ∈ Θ respectively, the shortfall risk with a loss function l for sellers is expressed by E[l(−x − GT (θ) + H)]. If we denote by X the collection of all claims, the acceptance set A0 on X with a level δ > 0 should be defined as