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Approximation and calibration of laws of solutions to stochastic differential equations
Post-edited
Authors : Bion-Nadal, Jocelyne (Author of the conference)
CIRM (Publisher )
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 (Publisher )
Abstract :
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
MSC Codes : 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
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 18/09/2018 Conference Date : 04/09/2018 Subseries : Research talks arXiv category : Probability Mathematical Area(s) : Probability & Statistics Format : MP4 (.mp4) - HD Video Time : 00:29:37 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2018-09-04_Bion_Nadal.mp4 
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Information on the Event
Event Title : Innovative Research in Mathematical Finance / Recherche innovante en mathématiques financièresEvent Organizers : Callegaro, Giorgia ; Jeanblanc, Monique ; Lépinette, Emmanuel ; Molchanov, Ilya ; Schweizer, Martin ; Touzi, Nizar Dates : 03/09/2018 - 07/09/2018 Event Year : 2018 Event URL : 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 |
See Also
- [Multi angle] Set-valued risk measures of non-convex portfolios / Author of the conference Molchanov, Ilya.
- [Multi angle] Discounting invariant FTAP for large financial markets / Author of the conference Balint, Daniel.
- [Multi angle] From multivariate quantiles to set-valued risk measures: a set optimization - approach to financial models with frictions / Author of the conference Hamel, Andreas H..
- [Multi angle] Pricing without martingale measure / Author of the conference Carassus, Laurence.
Bibliography
- 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
