Discretisation and Duality of Optimal Skorokhod Embedding Problems

Alexander M. G. Cox, Sam M. Kinsley

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Abstract

We prove a strong duality result for a linear programming problem which has the interpretation of being a discretised optimal Skorokhod embedding problem, and we recover this continuous time problem as a limit of the discrete problems. With the discrete setup we show that for a suitably chosen objective function, the optimiser takes the form of a hitting time for a random walk. In the limiting problem we then reprove the existence of the Root, Rost, and cave embedding solutions of the Skorokhod embedding problem. The main strength of this approach is that we can derive properties of the discrete problem more easily than in continuous time, and then prove that these properties hold in the limit. For example, a consequence of the strong duality result is that dual optimisers exist, and our limiting arguments can be used to derive properties of the continuous time dual functions. These arguments are applied in Cox and Kinsley (0000), where the existence of dual solutions is required to prove characterisation results for optimal barriers in a financial application.

Original languageEnglish
Pages (from-to)2376-2405
JournalStochastic Processes and their Applications
Volume129
Issue number7
DOIs
Publication statusPublished - 31 Jul 2018

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Skorokhod Problem
Embedding Problem
Duality
Discretization
Caves
Continuous Time
Strong Duality
Linear programming
Limiting
Hitting Time
Random walk
Objective function
Roots

ASJC Scopus subject areas

  • Statistics and Probability
  • Modelling and Simulation
  • Applied Mathematics

Cite this

Discretisation and Duality of Optimal Skorokhod Embedding Problems. / Cox, Alexander M. G.; Kinsley, Sam M.

In: Stochastic Processes and their Applications, Vol. 129, No. 7, 31.07.2018, p. 2376-2405.

Research output: Contribution to journalArticle

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