The minimum maximum of a continuous martingale with given initial and terminal laws
Let \((M_t)_{0\leq t\leq 1}\) be a continuous martingale with initial law \(M_0\sim \mu_0\), and terminal law \(M_1\sim \mu_1\), and let \(S=\sup_{0\leq t\leq 1}M_t\). This paper proves that there exists a greatest lower bound with respect to stochastic ordering of probability measures, on the law of \(S\). An explicit construction of this bound is given. Furthermore a martingale is constructed which attains this minimum by solving a Skorokhod embedding problem. The form of this martingale is motivated by a simple picture. The result is applied to the robust hedging of a forward start digital option.
scientific article; zbMATH DE number 1906075
scientific article