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Approximate analytical solutions for a class of nonlinear stochastic differential equations

Part of: Stochastic analysis Functional-differential and differential-difference equations Combinatorics, graph theory, probability theory, statistics

Published online by Cambridge University Press:  18 September 2018

A. T. MEIMARIS ,
I. A. KOUGIOUMTZOGLOU  and
A. A. PANTELOUS
A. T. MEIMARIS
Affiliation:
Department of Econometrics and Business Statistics, Monash Business School, Monash University, 20 Chancellors Walk, Wellington Road, Clayton, Victoria 3800, Australia emails: Antonios.Meimaris@monash.edu; Athanasios.Pantelous@monash.edu
I. A. KOUGIOUMTZOGLOU
Affiliation:
Department of Civil Engineering and Engineering Mechanics, The Fu Foundation School of Engineering and Applied Science, Columbia University, 500 West 120th Street, New York, NY 10027, USA email: ikougioum@columbia.edu
A. A. PANTELOUS
Affiliation:
Department of Econometrics and Business Statistics, Monash Business School, Monash University, 20 Chancellors Walk, Wellington Road, Clayton, Victoria 3800, Australia emails: Antonios.Meimaris@monash.edu; Athanasios.Pantelous@monash.edu
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Abstract

An approximate analytical solution is derived for a certain class of stochastic differential equations with constant diffusion, but nonlinear drift coefficients. Specifically, a closed form expression is derived for the response process transition probability density function (PDF) based on the concept of the Wiener path integral and on a Cauchy–Schwarz inequality treatment. This is done in conjunction with formulating and solving an error minimisation problem by relying on the associated Fokker–Planck equation operator. The developed technique, which requires minimal computational cost for the determination of the response process PDF, exhibits satisfactory accuracy and is capable of capturing the salient features of the PDF as demonstrated by comparisons with pertinent Monte Carlo simulation data. In addition to the mathematical merit of the approximate analytical solution, the derived PDF can be used also as a benchmark for assessing the accuracy of alternative, more computationally demanding, numerical solution techniques. Several examples are provided for assessing the reliability of the proposed approximation.

Keywords

Stochastic Differential EquationsStochastic DynamicsPath IntegralError quantificationCauchy-Schwarz inequality

MSC classification

Primary: 34K50: Stochastic functional-differential equations
Secondary: 60H10: Stochastic ordinary differential equations 60H35: Computational methods for stochastic equations 81P20: Stochastic mechanics (including stochastic electrodynamics) 97K60: Distributions and stochastic processes
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Special Issue 5: Stochastic Analysis in Applications , October 2019 , pp. 928 - 944
Copyright
© Cambridge University Press 2018 

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