An efficient Monte Carlo scheme for Zakai equations

Christian Beck; Sebastian Becker; Patrick Cheridito; Arnulf Jentzen; Ariel Neufeld

doi:10.1016/j.cnsns.2023.107438

An efficient Monte Carlo scheme for Zakai equations

Christian Beck, Sebastian Becker, Patrick Cheridito^*, Arnulf Jentzen, Ariel Neufeld

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we develop a numerical method for efficiently approximating solutions of certain Zakai equations in high dimensions. The key idea is to transform a given Zakai SPDE into a PDE with random coefficients. We show that under suitable regularity assumptions on the coefficients of the Zakai equation, the corresponding random PDE admits a solution random field which, for almost all realizations of the random coefficients, can be written as a classical solution of a linear parabolic PDE. This makes it possible to apply the Feynman–Kac formula to obtain an efficient Monte Carlo scheme for computing approximate solutions of Zakai equations. The approach achieves good results in up to 25 dimensions with fast run times.

Original language	English
Article number	107438
Journal	Communications in Nonlinear Science and Numerical Simulation
Volume	126
DOIs	https://doi.org/10.1016/j.cnsns.2023.107438
Publication status	Published - Nov 2023
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2023 The Author(s)

ASJC Scopus Subject Areas

Numerical Analysis
Modelling and Simulation
Applied Mathematics

Keywords

Doss–Sussmann transformation
Feynman–Kac representation
Nonlinear filtering problems
Stochastic partial differential equations
Zakai equation

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10.1016/j.cnsns.2023.107438

Cite this

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title = "An efficient Monte Carlo scheme for Zakai equations",

abstract = "In this paper we develop a numerical method for efficiently approximating solutions of certain Zakai equations in high dimensions. The key idea is to transform a given Zakai SPDE into a PDE with random coefficients. We show that under suitable regularity assumptions on the coefficients of the Zakai equation, the corresponding random PDE admits a solution random field which, for almost all realizations of the random coefficients, can be written as a classical solution of a linear parabolic PDE. This makes it possible to apply the Feynman–Kac formula to obtain an efficient Monte Carlo scheme for computing approximate solutions of Zakai equations. The approach achieves good results in up to 25 dimensions with fast run times.",

keywords = "Doss–Sussmann transformation, Feynman–Kac representation, Nonlinear filtering problems, Stochastic partial differential equations, Zakai equation",

author = "Christian Beck and Sebastian Becker and Patrick Cheridito and Arnulf Jentzen and Ariel Neufeld",

note = "Publisher Copyright: {\textcopyright} 2023 The Author(s)",

year = "2023",

month = nov,

doi = "10.1016/j.cnsns.2023.107438",

language = "English",

volume = "126",

journal = "Communications in Nonlinear Science and Numerical Simulation",

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T1 - An efficient Monte Carlo scheme for Zakai equations

AU - Beck, Christian

AU - Becker, Sebastian

AU - Cheridito, Patrick

AU - Jentzen, Arnulf

AU - Neufeld, Ariel

PY - 2023/11

Y1 - 2023/11

N2 - In this paper we develop a numerical method for efficiently approximating solutions of certain Zakai equations in high dimensions. The key idea is to transform a given Zakai SPDE into a PDE with random coefficients. We show that under suitable regularity assumptions on the coefficients of the Zakai equation, the corresponding random PDE admits a solution random field which, for almost all realizations of the random coefficients, can be written as a classical solution of a linear parabolic PDE. This makes it possible to apply the Feynman–Kac formula to obtain an efficient Monte Carlo scheme for computing approximate solutions of Zakai equations. The approach achieves good results in up to 25 dimensions with fast run times.

AB - In this paper we develop a numerical method for efficiently approximating solutions of certain Zakai equations in high dimensions. The key idea is to transform a given Zakai SPDE into a PDE with random coefficients. We show that under suitable regularity assumptions on the coefficients of the Zakai equation, the corresponding random PDE admits a solution random field which, for almost all realizations of the random coefficients, can be written as a classical solution of a linear parabolic PDE. This makes it possible to apply the Feynman–Kac formula to obtain an efficient Monte Carlo scheme for computing approximate solutions of Zakai equations. The approach achieves good results in up to 25 dimensions with fast run times.

KW - Doss–Sussmann transformation

KW - Feynman–Kac representation

KW - Nonlinear filtering problems

KW - Stochastic partial differential equations

KW - Zakai equation

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U2 - 10.1016/j.cnsns.2023.107438

DO - 10.1016/j.cnsns.2023.107438

M3 - Article

AN - SCOPUS:85168094285

SN - 1007-5704

VL - 126

JO - Communications in Nonlinear Science and Numerical Simulation

JF - Communications in Nonlinear Science and Numerical Simulation

M1 - 107438

ER -

An efficient Monte Carlo scheme for Zakai equations

Abstract

Bibliographical note

ASJC Scopus Subject Areas

Keywords

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Researchers from Swiss Federal Institute of Technology Report Recent Findings in Nonlinear Science and Numerical Simulation (An Efficient Monte Carlo Scheme for Zakai Equations)

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An efficient Monte Carlo scheme for Zakai equations

Abstract

Bibliographical note

ASJC Scopus Subject Areas

Keywords

Access to Document

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Fingerprint

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Researchers from Swiss Federal Institute of Technology Report Recent Findings in Nonlinear Science and Numerical Simulation (An Efficient Monte Carlo Scheme for Zakai Equations)

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