McKean Feynman-Kac Probabilistic Representations of Non-linear Partial Differential Equations

Lucas Izydorczyk; Nadia Oudjane; Francesco Russo

doi:10.1007/978-3-030-87432-2_10

McKean Feynman-Kac Probabilistic Representations of Non-linear Partial Differential Equations

Lucas Izydorczyk
, Nadia Oudjane
, Francesco Russo

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

This paper presents a partial state of the art about the topic of representation of generalized Fokker-Planck Partial Differential Equations (PDEs) by solutions of McKean Feynman-Kac Equations (MFKEs) that generalize the notion of McKean Stochastic Differential Equations (MSDEs). While MSDEs can be related to non-linear Fokker-Planck PDEs, MFKEs can be related to non-conservative non-linear PDEs. Motivations come from modeling issues but also from numerical approximation issues in computing the solution of a PDE, arising for instance in the context of stochastic control. MFKEs also appear naturally in representing final value problems related to backward Fokker-Planck equations.

Original language	English
Title of host publication	Geometry and Invariance in Stochastic Dynamics
Editors	Stefania Ugolini, Marco Fuhrman, Elisa Mastrogiacomo, Paola Morando, Barbara Rüdiger
Publisher	Springer
Pages	187-212
Number of pages	26
ISBN (Print)	9783030874315
DOIs	https://doi.org/10.1007/978-3-030-87432-2_10
Publication status	Published - 1 Jan 2021
Event	International Conference on Random Transformations and Invariance in Stochastic Dynamics, RTISD 2019 - Verona, Italy Duration: 25 Mar 2019 → 29 Mar 2019

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	378
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	International Conference on Random Transformations and Invariance in Stochastic Dynamics, RTISD 2019
Country/Territory	Italy
City	Verona
Period	25/03/19 → 29/03/19

Keywords

Backward diffusion
Feynman-Kac measures
HJB equation
McKean stochastic differential equation
Probabilistic representation of PDEs
Time reversed diffusion

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10.1007/978-3-030-87432-2_10

Cite this

Izydorczyk, L., Oudjane, N., & Russo, F. (2021). McKean Feynman-Kac Probabilistic Representations of Non-linear Partial Differential Equations. In S. Ugolini, M. Fuhrman, E. Mastrogiacomo, P. Morando, & B. Rüdiger (Eds.), Geometry and Invariance in Stochastic Dynamics (pp. 187-212). (Springer Proceedings in Mathematics and Statistics; Vol. 378). Springer. https://doi.org/10.1007/978-3-030-87432-2_10

Izydorczyk, Lucas ; Oudjane, Nadia ; Russo, Francesco. / McKean Feynman-Kac Probabilistic Representations of Non-linear Partial Differential Equations. Geometry and Invariance in Stochastic Dynamics. editor / Stefania Ugolini ; Marco Fuhrman ; Elisa Mastrogiacomo ; Paola Morando ; Barbara Rüdiger. Springer, 2021. pp. 187-212 (Springer Proceedings in Mathematics and Statistics).

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title = "McKean Feynman-Kac Probabilistic Representations of Non-linear Partial Differential Equations",

abstract = "This paper presents a partial state of the art about the topic of representation of generalized Fokker-Planck Partial Differential Equations (PDEs) by solutions of McKean Feynman-Kac Equations (MFKEs) that generalize the notion of McKean Stochastic Differential Equations (MSDEs). While MSDEs can be related to non-linear Fokker-Planck PDEs, MFKEs can be related to non-conservative non-linear PDEs. Motivations come from modeling issues but also from numerical approximation issues in computing the solution of a PDE, arising for instance in the context of stochastic control. MFKEs also appear naturally in representing final value problems related to backward Fokker-Planck equations.",

keywords = "Backward diffusion, Feynman-Kac measures, HJB equation, McKean stochastic differential equation, Probabilistic representation of PDEs, Time reversed diffusion",

author = "Lucas Izydorczyk and Nadia Oudjane and Francesco Russo",

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Izydorczyk, L, Oudjane, N & Russo, F 2021, McKean Feynman-Kac Probabilistic Representations of Non-linear Partial Differential Equations. in S Ugolini, M Fuhrman, E Mastrogiacomo, P Morando & B Rüdiger (eds), Geometry and Invariance in Stochastic Dynamics. Springer Proceedings in Mathematics and Statistics, vol. 378, Springer, pp. 187-212, International Conference on Random Transformations and Invariance in Stochastic Dynamics, RTISD 2019, Verona, Italy, 25/03/19. https://doi.org/10.1007/978-3-030-87432-2_10

McKean Feynman-Kac Probabilistic Representations of Non-linear Partial Differential Equations. / Izydorczyk, Lucas; Oudjane, Nadia; Russo, Francesco.
Geometry and Invariance in Stochastic Dynamics. ed. / Stefania Ugolini; Marco Fuhrman; Elisa Mastrogiacomo; Paola Morando; Barbara Rüdiger. Springer, 2021. p. 187-212 (Springer Proceedings in Mathematics and Statistics; Vol. 378).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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T1 - McKean Feynman-Kac Probabilistic Representations of Non-linear Partial Differential Equations

AU - Izydorczyk, Lucas

AU - Oudjane, Nadia

AU - Russo, Francesco

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N2 - This paper presents a partial state of the art about the topic of representation of generalized Fokker-Planck Partial Differential Equations (PDEs) by solutions of McKean Feynman-Kac Equations (MFKEs) that generalize the notion of McKean Stochastic Differential Equations (MSDEs). While MSDEs can be related to non-linear Fokker-Planck PDEs, MFKEs can be related to non-conservative non-linear PDEs. Motivations come from modeling issues but also from numerical approximation issues in computing the solution of a PDE, arising for instance in the context of stochastic control. MFKEs also appear naturally in representing final value problems related to backward Fokker-Planck equations.

AB - This paper presents a partial state of the art about the topic of representation of generalized Fokker-Planck Partial Differential Equations (PDEs) by solutions of McKean Feynman-Kac Equations (MFKEs) that generalize the notion of McKean Stochastic Differential Equations (MSDEs). While MSDEs can be related to non-linear Fokker-Planck PDEs, MFKEs can be related to non-conservative non-linear PDEs. Motivations come from modeling issues but also from numerical approximation issues in computing the solution of a PDE, arising for instance in the context of stochastic control. MFKEs also appear naturally in representing final value problems related to backward Fokker-Planck equations.

KW - Backward diffusion

KW - Feynman-Kac measures

KW - HJB equation

KW - McKean stochastic differential equation

KW - Probabilistic representation of PDEs

KW - Time reversed diffusion

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SN - 9783030874315

T3 - Springer Proceedings in Mathematics and Statistics

SP - 187

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BT - Geometry and Invariance in Stochastic Dynamics

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A2 - Mastrogiacomo, Elisa

A2 - Morando, Paola

A2 - Rüdiger, Barbara

PB - Springer

T2 - International Conference on Random Transformations and Invariance in Stochastic Dynamics, RTISD 2019

Y2 - 25 March 2019 through 29 March 2019

ER -

McKean Feynman-Kac Probabilistic Representations of Non-linear Partial Differential Equations

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