H 2 Approximation and calibration of laws of solutions to stochastic differential equations

Auteurs : Bion-Nadal, Jocelyne (Auteur de la Conférence)
CIRM (Editeur )

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coupling measure new Wasserstein type distance Hamilton-Jacobi-Bellman equation one dimensional case Hölder regular multidimensional case Kakutani fixed point method approximation by diffusion laws optimal coupling measure

Résumé : In many situations where stochastic modeling is used, one desires to choose the coefficients of a stochastic differential equation which represents the reality as simply as possible. For example one desires to approximate a diffusion model
with high complexity coefficients by a model within a class of simple diffusion models. To achieve this goal, we introduce a new Wasserstein type distance on the set of laws of solutions to d-dimensional stochastic differential equations.
This new distance $\widetilde{W}^{2}$ is defined similarly to the classical Wasserstein distance $\widetilde{W}^{2}$ but the set of couplings is restricted to the set of laws of solutions of 2$d$-dimensional stochastic differential equations. We prove that this new distance $\widetilde{W}^{2}$ metrizes the weak topology. Furthermore this distance $\widetilde{W}^{2}$ is characterized in terms of a stochastic control problem. In the case d = 1 we can construct an explicit solution. The multi-dimensional case, is more tricky and classical results do not apply to solve the HJB equation because of the degeneracy of the differential operator. Nevertheless, we prove that this HJB equation admits a regular solution.

Keywords : stochatic differential equation; Wasserstein distance

Codes MSC :
60H15 - Stochastic partial differential equations
60H30 - Applications of stochastic analysis (to PDE, etc.)
60J60 - Diffusion processes
91B70 - Stochastic models in economics
93E20 - Optimal stochastic control

    Informations sur la Vidéo

    Langue : Anglais
    Date de publication : 18/09/2018
    Date de captation : 04/09/2018
    Collection : Probability and Statistics
    Sous collection : Research talks
    Domaine : Probability & Statistics
    Format : MP4 (.mp4) - HD
    Durée : 00:29:37
    Audience : Chercheurs ; Doctorants , Post - Doctorants
    Download : https://videos.cirm-math.fr/2018-09-04_Bion_Nadal.mp4

Informations sur la rencontre

Nom de la rencontre : Innovative Research in Mathematical Finance / Recherche innovante en mathématiques financières
Organisateurs de la rencontre : Callegaro, Giorgia ; Jeanblanc, Monique ; Lépinette, Emmanuel ; Molchanov, Ilya ; Schweizer, Martin ; Touzi, Nizar
Dates : 03/09/2018 - 07/09/2018
Année de la rencontre : 2018
URL Congrès : https://conferences.cirm-math.fr/1816.html

Citation Data

DOI : 10.24350/CIRM.V.19442903
Cite this video as: Bion-Nadal, Jocelyne (2018). Approximation and calibration of laws of solutions to stochastic differential equations. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19442903
URI : http://dx.doi.org/10.24350/CIRM.V.19442903

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