## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 29, Number 3 (2019), 1609-1639.

### On a Wasserstein-type distance between solutions to stochastic differential equations

Jocelyne Bion–Nadal and Denis Talay

#### Abstract

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?

#### Article information

**Source**

Ann. Appl. Probab., Volume 29, Number 3 (2019), 1609-1639.

**Dates**

Received: November 2017

Revised: May 2018

First available in Project Euclid: 19 February 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1550566838

**Digital Object Identifier**

doi:10.1214/18-AAP1423

**Mathematical Reviews number (MathSciNet)**

MR3914552

**Subjects**

Primary: 60J60: Diffusion processes [See also 58J65] 93E20: Optimal stochastic control

**Keywords**

Stochastic differential equations Wasserstein distance stochastic control

#### Citation

Bion–Nadal, Jocelyne; Talay, Denis. On a Wasserstein-type distance between solutions to stochastic differential equations. Ann. Appl. Probab. 29 (2019), no. 3, 1609--1639. doi:10.1214/18-AAP1423. https://projecteuclid.org/euclid.aoap/1550566838