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June 2019 On a Wasserstein-type distance between solutions to stochastic differential equations
Jocelyne Bion–Nadal, Denis Talay
Ann. Appl. Probab. 29(3): 1609-1639 (June 2019). DOI: 10.1214/18-AAP1423


In this paper, we introduce a Wasserstein-type distance on the set of the probability distributions of strong solutions to stochastic differential equations. This new distance is defined by restricting the set of possible coupling measures. We prove that it may also be defined by means of the value function of a stochastic control problem whose Hamilton–Jacobi–Bellman equation has a smooth solution, which allows one to deduce a priori estimates or to obtain numerical evaluations. We exhibit an optimal coupling measure and characterize it as a weak solution to an explicit stochastic differential equation, and we finally describe procedures to approximate this optimal coupling measure.

A notable application concerns the following modeling issue: given an exact diffusion model, how to select a simplified diffusion model within a class of admissible models under the constraint that the probability distribution of the exact model is preserved as much as possible?


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Jocelyne Bion–Nadal. Denis Talay. "On a Wasserstein-type distance between solutions to stochastic differential equations." Ann. Appl. Probab. 29 (3) 1609 - 1639, June 2019.


Received: 1 November 2017; Revised: 1 May 2018; Published: June 2019
First available in Project Euclid: 19 February 2019

zbMATH: 07057462
MathSciNet: MR3914552
Digital Object Identifier: 10.1214/18-AAP1423

Primary: 60J60, 93E20

Rights: Copyright © 2019 Institute of Mathematical Statistics


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Vol.29 • No. 3 • June 2019
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