### 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 language | English |
---|---|

Pages (from-to) | 2376-2405 |

Journal | Stochastic Processes and their Applications |

Volume | 129 |

Issue number | 7 |

DOIs | |

Publication status | Published - 31 Jul 2018 |

### Fingerprint

### ASJC Scopus subject areas

- Statistics and Probability
- Modelling and Simulation
- Applied Mathematics

### Cite this

*Stochastic Processes and their Applications*,

*129*(7), 2376-2405. https://doi.org/10.1016/j.spa.2018.07.008

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

Research output: Contribution to journal › Article

*Stochastic Processes and their Applications*, vol. 129, no. 7, pp. 2376-2405. https://doi.org/10.1016/j.spa.2018.07.008

}

TY - JOUR

T1 - Discretisation and Duality of Optimal Skorokhod Embedding Problems

AU - Cox, Alexander M. G.

AU - Kinsley, Sam M.

PY - 2018/7/31

Y1 - 2018/7/31

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85050465318&partnerID=8YFLogxK

U2 - 10.1016/j.spa.2018.07.008

DO - 10.1016/j.spa.2018.07.008

M3 - Article

VL - 129

SP - 2376

EP - 2405

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 7

ER -