A Robust Markowitz Mean-Variance Portfolio Selection Model with an Intractable Claim

This paper studies a robust Markowitz mean-variance model where an intractable claim is involved in the terminal wealth. The term “intractable claim” refers to claims (rewards or losses) that are completely irrelevant to the underlying market. The payoffs of such claims cannot be predicted or hedged based on the underlying financial market even if the information of the financial market is increasingly available to the investor over time. The target of the investor is to minimize the variance in the worst scenario over all the possible realizations of the underlying intractable claim. Because of the time-inconsistent nature of the problem, both the standard penalization approach and the duality method used to tackle robust stochastic control problems fail in solving our problem. Instead, the quantile formulation approach is adopted to tackle the problem and an explicit closed-form solution is obtained. The properties of the mean-variance frontier are also discussed.