## The minimum maximum of a continuous martingale with given initial and terminal laws.(English)Zbl 1016.60047

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.

### MSC:

 60G44 Martingales with continuous parameter 60G40 Stopping times; optimal stopping problems; gambling theory 91B28 Finance etc. (MSC2000) 60G30 Continuity and singularity of induced measures
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### References:

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