- Connaître la Bibliothèque
- Trouver un livre
- Trouver une revue
- Faire une recherche bibliographique
- Trouver un document électronique
0
Post-edited
Approximation and calibration of laws of solutions to stochastic differential equations
Auteurs : Bion-Nadal, Jocelyne (Auteur de la Conférence)
CIRM (Editeur )
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
CIRM (Editeur )
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 : 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
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 : AnglaisDate de publication : 18/09/2018 Date de captation : 04/09/2018 Collection : Research talks ; Probability and Statistics Format : MP4 (.mp4) - HD Durée : 00:29:37 Domaine : Probability & Statistics 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 financièresOrganisateurs 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.19442903Cite 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 |
Voir aussi
- [Multi angle] Set-valued risk measures of non-convex portfolios / Auteur de la Conférence Molchanov, Ilya.
- [Multi angle] Discounting invariant FTAP for large financial markets / Auteur de la Conférence Balint, Daniel.
- [Multi angle] From multivariate quantiles to set-valued risk measures: a set optimization - approach to financial models with frictions / Auteur de la Conférence Hamel, Andreas H..
- [Multi angle] Pricing without martingale measure / Auteur de la Conférence Carassus, Laurence.
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
Z