The Annals of Applied Probability

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

Jocelyne Bion–Nadal and Denis Talay

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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


Export citation

References

  • [1] Aliprantis, C. D. and Border, K. C. (1999). Infinite-Dimensional Analysis, 2nd ed. Springer, Berlin.
  • [2] Claisse, J., Talay, D. and Tan, X. (2016). A pseudo-Markov property for controlled diffusion processes. SIAM J. Control Optim. 54 1017–1029.
  • [3] Fleming, W. H. and Rishel, R. W. (1975). Deterministic and Stochastic Optimal Control. Springer, Berlin.
  • [4] Fleming, W. H. and Soner, H. M. (2006). Controlled Markov Processes and Viscosity Solutions, 2nd ed. Stochastic Modelling and Applied Probability 25. Springer, New York.
  • [5] Friedman, A. (1964). Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ.
  • [6] Hiriart-Urruty, J.-B. and Lemaréchal, C. (1993). Convex Analysis and Minimization Algorithms. I. Grundlehren der Mathematischen Wissenschaften 305. Springer, Berlin.
  • [7] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [8] Krylov, N. V. (1987). Nonlinear Elliptic and Parabolic Equations of the Second Order. Mathematics and Its Applications (Soviet Series) 7. Reidel, Dordrecht.
  • [9] Kunita, H. (1984). Stochastic differential equations and stochastic flows of diffeomorphisms. In École D’été de Probabilités de Saint-Flour, XII—1982. Lecture Notes in Math. 1097 143–303. Springer, Berlin.
  • [10] Ladyženskaja, O. A., Solonnikov, V. A. and Ural’ceva, N. N. (1968). Linear and Quasilinear Equations of Parabolic Type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. Amer. Math. Soc., Providence, RI.
  • [11] Lunardi, A. (1995). Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications 16. Birkhäuser, Basel.
  • [12] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften 293. Springer, Berlin.
  • [13] Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften 233. Springer, Berlin.