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An analytical-numerical method for fast evaluation of probability densities for transient solutions of nonlinear Itô’s stochastic differential equations

Journal article
Authors Eugen Mamontov
Arvid Naess
Published in International Journal of Engineering Science
Volume 47
Issue 1
Pages 116-130
ISSN 0020-7225
Publication year 2009
Published at Department of Physics (GU)
Pages 116-130
Language en
Keywords Itô’s stochastic differential equation, transition probability density, analytical-numerical method, damping matrix, motion of ship in stochastic sea, stochastic rolling of ship
Subject categories Mathematical Analysis, Mathematical physics, Computational physics, Non-linear dynamics, chaos, Fluid mechanics, Other engineering mechanics, Other engineering physics


Probability densities for solutions of nonlinear Itô’s stochastic differential equations are described by the corresponding Kolmogorov-forward/Fokker-Planck equations. The densities provide the most complete information on the related probability distributions. This is an advantage crucial in many applications such as modelling floating structures under the stochastic load due to wind or sea waves. Practical methods for numerical solution of the probability-density equations are combined, analytical-numerical techniques. The present work develops a new analytical-numerical approach, the successive-transition (ST) method, which is a version of the path-integration (PI) method. The ST technique is based on an analytical approximation for the transition-probability density. It enables the PI approach to explicitly allow for the damping matrix in the approximation. This is achieved by extending another method, introduced earlier for bistable nonlinear reaction-diffusion equations, to the probability-density equations. The ST method also includes a control for the size of the time step. The overall accuracy of the ST method can be tested on various nonlinear examples. One such example is proposed. It is one-dimensional nonlinear Itô’s equation describing the velocity of a ship maneuvering along a straight line under the action of the stochastic drag due to wind or sea waves. Another problem in marine engineering, the rolling of a ship up to its possible capsizing is also discussed in connection with the complicated damping-matrix picture. The work suggests a few directions for future research.

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