## The Annals of Applied Probability

### The maximum maximum of a martingale with given $\mathbf{n}$ marginals

#### Abstract

We obtain bounds on the distribution of the maximum of a martingale with fixed marginals at finitely many intermediate times. The bounds are sharp and attained by a solution to $n$-marginal Skorokhod embedding problem in Obłój and Spoida [An iterated Azéma-Yor type embedding for finitely many marginals (2013) Preprint]. It follows that their embedding maximizes the maximum among all other embeddings. Our motivating problem is superhedging lookback options under volatility uncertainty for an investor allowed to dynamically trade the underlying asset and statically trade European call options for all possible strikes and finitely-many maturities. We derive a pathwise inequality which induces the cheapest superhedging value, which extends the two-marginals pathwise inequality of Brown, Hobson and Rogers [Probab. Theory Related Fields 119 (2001) 558–578]. This inequality, proved by elementary arguments, is derived by following the stochastic control approach of Galichon, Henry-Labordère and Touzi [Ann. Appl. Probab. 24 (2014) 312–336].

#### Article information

Source
Ann. Appl. Probab., Volume 26, Number 1 (2016), 1-44.

Dates
Revised: September 2014
First available in Project Euclid: 5 January 2016

https://projecteuclid.org/euclid.aoap/1452003233

Digital Object Identifier
doi:10.1214/14-AAP1084

Mathematical Reviews number (MathSciNet)
MR3449312

Zentralblatt MATH identifier
1337.60078

#### Citation

Henry-Labordère, Pierre; Obłój, Jan; Spoida, Peter; Touzi, Nizar. The maximum maximum of a martingale with given $\mathbf{n}$ marginals. Ann. Appl. Probab. 26 (2016), no. 1, 1--44. doi:10.1214/14-AAP1084. https://projecteuclid.org/euclid.aoap/1452003233

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