Martingale Optimal Transport with Stopping
SIAM Journal on Control and Optimization, Forthcoming
18 Pages Posted: 3 Mar 2017 Last revised: 24 Nov 2017
Date Written: November 22, 2017
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. Kallblad. Model-independent bounds for Asian options: a dynamic programming approach. SIAM Journal on Control and Optimization, 55(6):3409–3436, 2017.] 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.
Keywords: martingale optimal transport, dynamic programming, optimal stopping, stochastic Perron method, viscosity solutions, concave envelope, distribution constraints
JEL Classification: C73
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