Closed-form approximate solutions for a class of coupled nonlinear stochastic differential equations

Antonios T. Meimaris, Ioannis A. Kougioumtzoglou, Athanasios A. Pantelous

Research output: Contribution to journalArticleResearchpeer-review

Abstract

An approximate solution technique is developed for a class of coupled multi-dimensional stochastic differential equations with nonlinear drift and constant diffusion coefficients. Relying on a Wiener path integral formulation and employing the Cauchy–Schwarz inequality, an approximate closed-form expression for the joint response process transition probability density function is determined. Next, the accuracy of the approximation is further enhanced by proposing a more versatile closed-form expression with additional “degrees of freedom”; that is, parameters to be determined. To this aim, an error minimization problem related to the corresponding Fokker–Planck equation is formulated and solved. Several diverse numerical examples are considered for demonstrating the reliability of the herein developed solution technique, which requires minimal computational cost for determining the joint response transition probability density function and exhibits satisfactory accuracy as compared with pertinent Monte Carlo simulation data.

Original languageEnglish
Article number124669
Number of pages18
JournalApplied Mathematics and Computation
Volume364
DOIs
Publication statusPublished - Jan 2020

Keywords

  • Cauchy–Schwarz inequality
  • Error quantification
  • Fokker–Planck equation
  • Nonlinear stochastic dynamics
  • Path Integral
  • Stochastic differential equations

Cite this

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abstract = "An approximate solution technique is developed for a class of coupled multi-dimensional stochastic differential equations with nonlinear drift and constant diffusion coefficients. Relying on a Wiener path integral formulation and employing the Cauchy–Schwarz inequality, an approximate closed-form expression for the joint response process transition probability density function is determined. Next, the accuracy of the approximation is further enhanced by proposing a more versatile closed-form expression with additional “degrees of freedom”; that is, parameters to be determined. To this aim, an error minimization problem related to the corresponding Fokker–Planck equation is formulated and solved. Several diverse numerical examples are considered for demonstrating the reliability of the herein developed solution technique, which requires minimal computational cost for determining the joint response transition probability density function and exhibits satisfactory accuracy as compared with pertinent Monte Carlo simulation data.",
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Closed-form approximate solutions for a class of coupled nonlinear stochastic differential equations. / Meimaris, Antonios T.; Kougioumtzoglou, Ioannis A.; Pantelous, Athanasios A.

In: Applied Mathematics and Computation, Vol. 364, 124669, 01.2020.

Research output: Contribution to journalArticleResearchpeer-review

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T1 - Closed-form approximate solutions for a class of coupled nonlinear stochastic differential equations

AU - Meimaris, Antonios T.

AU - Kougioumtzoglou, Ioannis A.

AU - Pantelous, Athanasios A.

PY - 2020/1

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N2 - An approximate solution technique is developed for a class of coupled multi-dimensional stochastic differential equations with nonlinear drift and constant diffusion coefficients. Relying on a Wiener path integral formulation and employing the Cauchy–Schwarz inequality, an approximate closed-form expression for the joint response process transition probability density function is determined. Next, the accuracy of the approximation is further enhanced by proposing a more versatile closed-form expression with additional “degrees of freedom”; that is, parameters to be determined. To this aim, an error minimization problem related to the corresponding Fokker–Planck equation is formulated and solved. Several diverse numerical examples are considered for demonstrating the reliability of the herein developed solution technique, which requires minimal computational cost for determining the joint response transition probability density function and exhibits satisfactory accuracy as compared with pertinent Monte Carlo simulation data.

AB - An approximate solution technique is developed for a class of coupled multi-dimensional stochastic differential equations with nonlinear drift and constant diffusion coefficients. Relying on a Wiener path integral formulation and employing the Cauchy–Schwarz inequality, an approximate closed-form expression for the joint response process transition probability density function is determined. Next, the accuracy of the approximation is further enhanced by proposing a more versatile closed-form expression with additional “degrees of freedom”; that is, parameters to be determined. To this aim, an error minimization problem related to the corresponding Fokker–Planck equation is formulated and solved. Several diverse numerical examples are considered for demonstrating the reliability of the herein developed solution technique, which requires minimal computational cost for determining the joint response transition probability density function and exhibits satisfactory accuracy as compared with pertinent Monte Carlo simulation data.

KW - Cauchy–Schwarz inequality

KW - Error quantification

KW - Fokker–Planck equation

KW - Nonlinear stochastic dynamics

KW - Path Integral

KW - Stochastic differential equations

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