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

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.