### Abstract

A closed-form analytical approximation is derived for the response transition probability density function (PDF) of a certain class of stochastic differential equations with nonlinear drift and constant diffusion coefficients. This is done by resorting to a recently developed Wiener path integral based technique (WPI) in conjunction with a Cauchy-Schwarz inequality treatment of the problem. The derived approximation can be used, due to its analytical nature, as a direct SDE response PDF estimate that requires zero computational effort for its determination. Further, it facilitates an error quantification analysis, which yields an a priori estimate of the anticipated accuracy obtained by applying the approximate methodology. The reliability of the approximation is demonstrated via several engineering mechanics/dynamics related numerical examples pertaining to the stochastic beam bending problem, as well as to the response determination of stochastically excited nonlinear oscillators.

Language | English |
---|---|

Pages | 87-94 |

Number of pages | 8 |

Journal | Probabilistic Engineering Mechanics |

Volume | 54 |

DOIs | |

Publication status | Published - 20 Jun 2018 |

Externally published | Yes |

### Keywords

- Cauchy-Schwarz inequality
- Error quantification
- Path integral
- Stochastic differential equations
- Stochastic dynamics

### Cite this

*Probabilistic Engineering Mechanics*,

*54*, 87-94. https://doi.org/10.1016/j.probengmech.2017.07.005

}

*Probabilistic Engineering Mechanics*, vol. 54, pp. 87-94. https://doi.org/10.1016/j.probengmech.2017.07.005

**A closed form approximation and error quantification for the response transition probability density function of a class of stochastic differential equations.** / Meimaris, Antonios T.; Kougioumtzoglou, Ioannis A.; Pantelous, Athanasios A.

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

TY - JOUR

T1 - A closed form approximation and error quantification for the response transition probability density function of a class of stochastic differential equations

AU - Meimaris, Antonios T.

AU - Kougioumtzoglou, Ioannis A.

AU - Pantelous, Athanasios A.

PY - 2018/6/20

Y1 - 2018/6/20

N2 - A closed-form analytical approximation is derived for the response transition probability density function (PDF) of a certain class of stochastic differential equations with nonlinear drift and constant diffusion coefficients. This is done by resorting to a recently developed Wiener path integral based technique (WPI) in conjunction with a Cauchy-Schwarz inequality treatment of the problem. The derived approximation can be used, due to its analytical nature, as a direct SDE response PDF estimate that requires zero computational effort for its determination. Further, it facilitates an error quantification analysis, which yields an a priori estimate of the anticipated accuracy obtained by applying the approximate methodology. The reliability of the approximation is demonstrated via several engineering mechanics/dynamics related numerical examples pertaining to the stochastic beam bending problem, as well as to the response determination of stochastically excited nonlinear oscillators.

AB - A closed-form analytical approximation is derived for the response transition probability density function (PDF) of a certain class of stochastic differential equations with nonlinear drift and constant diffusion coefficients. This is done by resorting to a recently developed Wiener path integral based technique (WPI) in conjunction with a Cauchy-Schwarz inequality treatment of the problem. The derived approximation can be used, due to its analytical nature, as a direct SDE response PDF estimate that requires zero computational effort for its determination. Further, it facilitates an error quantification analysis, which yields an a priori estimate of the anticipated accuracy obtained by applying the approximate methodology. The reliability of the approximation is demonstrated via several engineering mechanics/dynamics related numerical examples pertaining to the stochastic beam bending problem, as well as to the response determination of stochastically excited nonlinear oscillators.

KW - Cauchy-Schwarz inequality

KW - Error quantification

KW - Path integral

KW - Stochastic differential equations

KW - Stochastic dynamics

UR - http://www.scopus.com/inward/record.url?scp=85026294938&partnerID=8YFLogxK

U2 - 10.1016/j.probengmech.2017.07.005

DO - 10.1016/j.probengmech.2017.07.005

M3 - Article

VL - 54

SP - 87

EP - 94

JO - Probabilistic Engineering Mechanics

T2 - Probabilistic Engineering Mechanics

JF - Probabilistic Engineering Mechanics

SN - 0266-8920

ER -