Stochastic Runge-Kutta method for stochastic delay differential equations

Abstract

Random e�ect and time delay are inherent properties of many real phenomena around us, hence it is required to model the system via stochastic delay di�erential equations (SDDEs). However, the complexity arises due to the presence of both randomness and time delay. The analytical solution of SDDEs is hard to be found. In such a case, a numerical method provides a way to solve the problem. Nevertheless, due to the lacking of numerical methods available for solving SDDEs, a wide range of researchers among the mathematicians and scientists have not incorporated the important features of the real phenomena, which include randomness and time delay in modeling the system. Hence, this research aims to generalize the convergence proof of numerical methods for SDDEs when the drift and di�usion functions are Taylor expansion and to develop a stochastic Runge{Kutta for solving SDDEs. Motivated by the relative paucity of numerical methods accessible in simulating the strong solution of SDDEs, the numerical schemes developed in this research is hoped to bridge the gap between the evolution of numerical methods in ordinary di�erential equations (ODEs), delay di�erential equations (DDEs), stochastic di�erential equations (SDEs) and SDDEs. The extension of numerical methods of SDDEs is far from complete. Rate of convergence of recent numerical methods available in approximating the solution of SDDEs only reached the order of 1.0. One of the important factors of the rapid progression of the development of numerical methods for ODEs, DDEs and SDEs is the convergence proof of the approximation methods when the drift and di�usion coe�cients are Taylor expansion that had been generalized. The convergence proof of numerical schemes for SDDEs has yet to be generalized. Hence, this research is carried out to solve this problem. Furthermore, the derivative-free method has not yet been established. Hence, development of a derivative{free method with 1.5 order of convergence, namely stochastic Runge{ Kutta, to approximate the solution of SDDEs with a constant time lag, r > 0, is also included in this thesis.