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Probability and Statistics

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. 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 ...

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Documents 60J60 | enregistrements trouvés : 122

Approximation and calibration of laws of solutions to stochastic differential equations Bion-Nadal, Jocelyne | CIRM H Nouveau

Probability measures on groups ixproceedings of a conference held in oberwolfach,frg,jan. 17-23,1988 Heyer, H. | Springer-Verlag 1989 Nouveau

Stochastic analysis and applicationspapers from the first lisbon conference on...,held sept.,1989 Cruzeiro, A. B. ; Zambrini, J. C. | Birkhäuser 1991 Nouveau

Séminaire Bourbaki. Vol. 1984-85exposés 633-650 | Société Mathématique de France 1986 Nouveau

Diffusion processes and related problems in analysis. Vol. 1:diffusions in analysis and geometryproceddings of an international conferenceEvanston # october 23-27, 1989 Pinsky, Mark A. | Birkhäuser 1990 Nouveau

Diffusion processes and related problems in analysis, Vol. IIconference on the subject of stochastic flows, during the weekend of March 16-18, 1990 the university of North Carolina at Charlotte Pinsky, Mark A. ; Wihstutz, Volker | Birkhäuser 1992 Nouveau

Séminaire de probabilités IV | Springer-Verlag 1970 Nouveau

Ecole d'été de probabilités de Saint-Flour III Meyer, P. A. ; Priouret, P. ; Spitzer , F. ; Badrikian, Albert ; Hennequin, P. L. | Springer-Verlag 1974 Nouveau

Mathematics of random media :proceedings of the AMS-SIAM summer seminar held at Virginia Polytechnic Institute and State University#May 29 - June 9 Kohler, Werner E. ; White, Benjamin S. | American Mathematical Society 1991 Nouveau

II Simposio de probabilidad y procesos estocasticosI Encuentro Mexico-Chile de analysis estocastico Caballero, M. E. ; Gorostiza, L. G. | Sociedad Mathematica Mexicana 1992 Nouveau

Lectures on probability theory and statisticsEcole d'été de probabilité de saint-flour XXV - 1995 Barlow, M. t. ; Nualart, D. ; Bernard, P. | Springer 1998 Nouveau

Stochastic PDE's and Kolmogorov equations in infinite dimensions :lectures given at the 2nd session of the Centro Internazionale Matematico Estivo#(C.I.M.E.)#Aug. 24 - Sept. 1 Krylov, N. V. ; Rockner, Michael ; Zabczyk, Jerzy ; Da Prato, G. | Springer 1999 Nouveau

Lectures on probability theory and statistics :Ecole d'été de probabilité de Saint-Flour XXIV -1994 Dobrushin, R. L. ; Groeneboom, P. ; Ledoux, M. ; Bernard, P. | Springer 1996 Nouveau

Stochastic models :seventh symposium on probability and stochastic processes held at Mexico City#June 23-28 Gonzales-Barrios, José M. ; Leon, Jorge A. ; Meda, Ana | American Mathematical Society 2003 Nouveau

Mathematics of finance :proceedings of an AMS-IMS-SIAM joint summer research conference on ...#June 22-26 Yin, G. George ; Zhang, Qing | American Mathematical Society 2004 Nouveau

Quantum graphs and their applications :proceedings of an AMS-IMS-SIAM joint summer research conference on quantum graphs and their applications#June 19-23 Berkolaiko, Gregory ; Carlson, Robert ; Fulling, Stephen A. ; Kuchment, Peter | American Mathematical Society 2006 Nouveau

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Probability measures on groups ix
proceedings of a conference held in oberwolfach,frg,jan. 17-23,1988 Heyer, H. | Springer-Verlag 1989 Nouveau

Stochastic analysis and applications
papers from the first lisbon conference on...,held sept.,1989 Cruzeiro, A. B. ; Zambrini, J. C. | Birkhäuser 1991 Nouveau

Séminaire Bourbaki. Vol. 1984-85
exposés 633-650 | Société Mathématique de France 1986 Nouveau

Diffusion processes and related problems in analysis. Vol. 1:
diffusions in analysis and geometry
proceddings of an international conference
Evanston # october 23-27, 1989 Pinsky, Mark A. | Birkhäuser 1990 Nouveau

Diffusion processes and related problems in analysis, Vol. II
conference on the subject of stochastic flows, during the weekend of March 16-18, 1990 the university of North Carolina at Charlotte Pinsky, Mark A. ; Wihstutz, Volker | Birkhäuser 1992 Nouveau

Mathematics of random media :
proceedings of the AMS-SIAM summer seminar held at Virginia Polytechnic Institute and State University#May 29 - June 9 Kohler, Werner E. ; White, Benjamin S. | American Mathematical Society 1991 Nouveau

II Simposio de probabilidad y procesos estocasticos
I Encuentro Mexico-Chile de analysis estocastico Caballero, M. E. ; Gorostiza, L. G. | Sociedad Mathematica Mexicana 1992 Nouveau

Lectures on probability theory and statistics
Ecole d'été de probabilité de saint-flour XXV - 1995 Barlow, M. t. ; Nualart, D. ; Bernard, P. | Springer 1998 Nouveau

Stochastic PDE's and Kolmogorov equations in infinite dimensions :
lectures given at the 2nd session of the Centro Internazionale Matematico Estivo#(C.I.M.E.)#Aug. 24 - Sept. 1 Krylov, N. V. ; Rockner, Michael ; Zabczyk, Jerzy ; Da Prato, G. | Springer 1999 Nouveau

Lectures on probability theory and statistics :
Ecole d'été de probabilité de Saint-Flour XXIV -1994 Dobrushin, R. L. ; Groeneboom, P. ; Ledoux, M. ; Bernard, P. | Springer 1996 Nouveau

Stochastic models :
seventh symposium on probability and stochastic processes held at Mexico City#June 23-28 Gonzales-Barrios, José M. ; Leon, Jorge A. ; Meda, Ana | American Mathematical Society 2003 Nouveau

Mathematics of finance :
proceedings of an AMS-IMS-SIAM joint summer research conference on ...#June 22-26 Yin, G. George ; Zhang, Qing | American Mathematical Society 2004 Nouveau

Quantum graphs and their applications :
proceedings of an AMS-IMS-SIAM joint summer research conference on quantum graphs and their applications#June 19-23 Berkolaiko, Gregory ; Carlson, Robert ; Fulling, Stephen A. ; Kuchment, Peter | American Mathematical Society 2006 Nouveau