### 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.

Original language | English |
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Number of pages | 17 |

Journal | European Journal of Applied Mathematics |

DOIs | |

Publication status | Accepted/In press - 1 Sep 2018 |

### Keywords

- Cauchy-Schwarz inequality
- Error quantification
- Path Integral
- Stochastic Differential Equations
- Stochastic Dynamics

### Cite this

*European Journal of Applied Mathematics*. https://doi.org/10.1017/S0956792518000530

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*European Journal of Applied Mathematics*. https://doi.org/10.1017/S0956792518000530

**Approximate analytical solutions for a class of nonlinear stochastic differential equations.** / Meimaris, A. T.; Kougioumtzoglou, I. A.; Pantelous, A. A.

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

TY - JOUR

T1 - Approximate analytical solutions for a class of nonlinear stochastic differential equations

AU - Meimaris, A. T.

AU - Kougioumtzoglou, I. A.

AU - Pantelous, A. A.

PY - 2018/9/1

Y1 - 2018/9/1

N2 - 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.

AB - 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.

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=85053728670&partnerID=8YFLogxK

U2 - 10.1017/S0956792518000530

DO - 10.1017/S0956792518000530

M3 - Article

JO - European Journal of Applied Mathematics

JF - European Journal of Applied Mathematics

SN - 0956-7925

ER -