Computing best bounds for nonlinear risk measures with partial information

Extreme events occur rarely, but these are often the circumstances where an insurance coverage is demanded. Given the first, say, n moments of the risk(s) of the events, one is able to compute or approximate the tight bounds for risk measures in the form of E(ψ(x)) through semidefinite programmings (SDP), via distributional robust optimization formulations. Existing results in the literature have already demonstrated the power of this technique when ψ(x) is linear or piecewise linear. In this paper, we extend the technique in the case where ψ(x) is a polynomial or fractional polynomial.