A novel efficient technique for solving nonlinear stochastic Itô–Volterra integral equations

Abstract

There is a growing need of stochastic integral equations (SIEs) to investigate the behavior of complex dynamical systems. Since real-world phenomena frequently dependent on noise sources, modeling them naturally necessitates the use of SIEs. As most SIEs cannot be solved explicitly, thus the behaviors of the studied systems are investigated using approximate solutions of their SIEs. Despite the fact that this problem has been soundly investigated and numerous methods have been presented, the practice demonstrated that obtaining satisfied approximations is not always guaranteed, necessitating the development of new effective techniques. This paper gives a new technique for solving nonlinear Itô–Volterra SIEs by reducing them to linear or nonlinear algebraic systems via the power of a combination of generalized Lagrange functions and Jacobi–Gauss collocation points. The accuracy and reliability of the new technique are evaluated and compared with the existing techniques. Moreover, sufficient conditions to make the estimate error tends to zero are given. The new technique shows surprisingly high efficiency over the existing techniques in terms of computational efficiency and approximation capability. The accuracy of the solution based on the new technique is much higher than that via the existing techniques. The required time of the new technique is much less than that of the existing techniques, where, in some circumstances, the existing techniques take more than 20 times as long as the new technique.