TY - JOUR
T1 - The Root solution to the multi-marginal embedding problem
T2 - an optimal stopping and time-reversal approach
AU - Cox, Alexander M.G.
AU - Obłój, Jan
AU - Touzi, Nizar
N1 - Funding Information:
Acknowledgements The authors are grateful to many who have commented on this research project. In particular, we thank the participants of the BIRS Workshop Mathematical Finance: Arbitrage and Portfolio Optimization in May 2014 and HIM Workshop Optimal Transport and Stochastics in March 2015 for their helpful comments and remarks. Jan Obłój’s research has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 335421. He is also grateful to the Oxford-Man Institute of Quantitative Finance and St John’s College in Oxford for their support. Nizar Touzi’s research has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 321111. He also gratefully acknowledges the financial support from the Chair Financial Risks of the Risk Foundation sponsored by Société Générale, and the Chair Finance and Sustainable Development sponsored by EDF and CA-CIB.
Funding Information:
The authors are grateful to many who have commented on this research project. In particular, we thank the participants of the BIRS Workshop Mathematical Finance: Arbitrage and Portfolio Optimization in May 2014 and HIM Workshop Optimal Transport and Stochastics in March 2015 for their helpful comments and remarks. Jan Ob??j?s research has received funding from the European Research Council under the European Union?s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 335421. He is also grateful to the Oxford-Man Institute of Quantitative Finance and St John?s College in Oxford for their support. Nizar Touzi?s research has received funding from the European Research Council under the European Union?s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 321111. He also gratefully acknowledges the financial support from the Chair Financial Risks of the Risk Foundation sponsored by Soci?t? G?n?rale, and the Chair Finance and Sustainable Development sponsored by EDF and CA-CIB.
Publisher Copyright:
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2019/2/4
Y1 - 2019/2/4
N2 - We provide a complete characterisation of the Root solution to the Skorokhod embedding problem (SEP) by means of an optimal stopping formulation. Our methods are purely probabilistic and the analysis relies on a tailored time-reversal argument. This approach allows us to address the long-standing question of a multiple marginals extension of the Root solution of the SEP. Our main result establishes a complete solution to the n-marginal SEP using first hitting times of barrier sets by the time–space process. The barriers are characterised by means of a recursive sequence of optimal stopping problems. Moreover, we prove that our solution enjoys a global optimality property extending the one-marginal Root case. Our results hold for general, one-dimensional, martingale diffusions.
AB - We provide a complete characterisation of the Root solution to the Skorokhod embedding problem (SEP) by means of an optimal stopping formulation. Our methods are purely probabilistic and the analysis relies on a tailored time-reversal argument. This approach allows us to address the long-standing question of a multiple marginals extension of the Root solution of the SEP. Our main result establishes a complete solution to the n-marginal SEP using first hitting times of barrier sets by the time–space process. The barriers are characterised by means of a recursive sequence of optimal stopping problems. Moreover, we prove that our solution enjoys a global optimality property extending the one-marginal Root case. Our results hold for general, one-dimensional, martingale diffusions.
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U2 - 10.1007/s00440-018-0833-1
DO - 10.1007/s00440-018-0833-1
M3 - Article
AN - SCOPUS:85041904536
SN - 0178-8051
VL - 173
SP - 211
EP - 259
JO - Probability Theory and Related Fields
JF - Probability Theory and Related Fields
IS - 1-2
ER -