Martingale optimal transport with stopping

Erhan Bayraktar; Alexander M.G. Cox; Yavor Stoev

doi:10.2139/ssrn.2926552

Martingale optimal transport with stopping

Erhan Bayraktar, Alexander M.G. Cox, Yavor Stoev

University of Michigan

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Abstract

We solve the martingale optimal transport problem for cost functionals represented by optimal stopping problems. The measure-valued martingale approach developed in [A. M. G. Cox and S. Källblad, SIAM J. Control Optim., 55 (2017), pp. 3409–3436] allows us to obtain an equivalent infinite dimensional controller-stopper problem. We use the stochastic Perron’s method and characterize the finite dimensional approximation as a viscosity solution to the corresponding HJB equation. It turns out that this solution is the concave envelope of the cost function with respect to the atoms of the terminal law. We demonstrate the results by finding explicit solutions for a class of cost functions.

Original language	English
Pages (from-to)	417-433
Number of pages	17
Journal	SIAM Journal on Control and Optimization
Volume	56
Issue number	1
Early online date	13 Feb 2018
DOIs	https://doi.org/10.2139/ssrn.2926552 https://doi.org/10.1137/17M1114065
Publication status	Published - 2018

Bibliographical note

keywords: nonlinear Martingale optimal transport, dynamic programming, optimal stopping, stochastic Perron's method, viscosity solutions, state constraints, exit time problem, concave envelope, distribution constraints

Keywords

math.PR
60G40, 93E20, 91A10, 91A60, 60G07

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The final version is available at SIAM via https://doi.org/10.1137/17M1114065, and is © 2018, Society for Industrial and Applied Mathematics.
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M111406-1Final published version, 470 KB

Cite this

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AB - We solve the martingale optimal transport problem for cost functionals represented by optimal stopping problems. The measure-valued martingale approach developed in [A. M. G. Cox and S. Källblad, SIAM J. Control Optim., 55 (2017), pp. 3409–3436] allows us to obtain an equivalent infinite dimensional controller-stopper problem. We use the stochastic Perron’s method and characterize the finite dimensional approximation as a viscosity solution to the corresponding HJB equation. It turns out that this solution is the concave envelope of the cost function with respect to the atoms of the terminal law. We demonstrate the results by finding explicit solutions for a class of cost functions.

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Martingale optimal transport with stopping

Abstract

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