### 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 language | English |
---|---|

Article number | 124669 |

Number of pages | 18 |

Journal | Applied Mathematics and Computation |

Volume | 364 |

DOIs | |

Publication status | Published - Jan 2020 |

### Keywords

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

### Cite this

*Applied Mathematics and Computation*,

*364*, [124669]. https://doi.org/10.1016/j.amc.2019.124669

}

*Applied Mathematics and Computation*, vol. 364, 124669. https://doi.org/10.1016/j.amc.2019.124669

**Closed-form approximate solutions for a class of coupled nonlinear stochastic differential equations.** / Meimaris, Antonios T.; Kougioumtzoglou, Ioannis A.; Pantelous, Athanasios A.

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

TY - JOUR

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

Y1 - 2020/1

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

UR - http://www.scopus.com/inward/record.url?scp=85070852718&partnerID=8YFLogxK

U2 - 10.1016/j.amc.2019.124669

DO - 10.1016/j.amc.2019.124669

M3 - Article

VL - 364

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

M1 - 124669

ER -