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Approximation and calibration of laws of solutions to stochastic differential equations

De Jocelyne Bion-Nadal

Apparaît dans la collection : Innovative Research in Mathematical Finance / Recherche innovante en mathématiques financières

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.

Informations sur la vidéo

Date de captation 04/09/2018
Date de publication 18/09/2018
Institut CIRM
Licence CC BY NC ND
Langue Anglais
Réalisateur(s) Guillaume Hennenfent
Format MP4

Données de citation

DOI 10.24350/CIRM.V.19442903
Citer cette vidéo Bion-Nadal, Jocelyne (04/09/2018). Approximation and calibration of laws of solutions to stochastic differential equations. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19442903
URL https://dx.doi.org/10.24350/CIRM.V.19442903

Domaine(s)

Probabilités

Bibliographie

C. Aliprantis and K. Border, Infinite Dimensional Analysis, 1999. - http://dx.doi.org/10.1007/3-540-29587-9
J. Claisse, D. Talay, and X. Tan, A Pseudo-Markov Property for Controlled Diffusion Processes, SIAM Journal on Control and Optimization, vol.54, issue.2, pp.1017-1029, 2016. - https://dx.doi.org/10.1137/151004252
W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, 1975. - https://dx.doi.org/10.1007/978-1-4612-6380-7
W. H. Fleming and M. Soner, Controlled Markov Processes and Viscosity Solutions, of Stochastic Modelling and Applied Probability, 2006. - https://doi.org/10.1007/978-3-662-03961-8
A. Friedman, Partial Differential Equations of Parabolic Type, 1964. - https://doi.org/10.1007/978-1-4612-0949-2
J. Hiriart-urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms I. Grundlehren der mathematische Wissenschaften, 1993. - https://dx.doi.org/10.1007/978-3-662-02796-7
I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, vol.113, 1991. - https://dx.doi.org/10.1007/978-1-4612-0949-2
N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of the Second Order, 1987 - https://dx.doi.org/10.1007/978-94-010-9557-0
H. Kunita, Stochastic differential equations and stochastic flows of diffeomorphisms. ´ Ecole d'´ eté de probabilités de Saint-Flour, p.1982, 1984. - https://dx.doi.org/10.1007/978-4-431-68532-6_13
A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, of Progress in Nonlinear Differential Equations and their Applications, 1995. - https://dx.doi.org/10.1007/978-3-0348-0557-5
D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Fundamental Principles of Mathematical Sciences Series, 1999. - https://dx.doi.org/10.1007/978-3-662-06400-9
D. W. Stroock and S. R. Varadhan, Multidimensional Diffusion Processes, Fundamental Principles of Mathematical Sciences, 1979. - https://dx.doi.org/10.1007/3-540-28999-2

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