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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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[20] BATH, BA2 7AY UNITED KINGDOM E-MAIL: dgh@maths.bath.ac.uk RISKLAB DEPARTMENT OF MATHEMATICS ETH ZÜRICH CH-8092 ZÜRICH SWITZERLAND E-MAIL: pedersen@math.ethz.ch
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