On strong convergence of two numerical methods for singular initial value problems with multiplicative white noise

Abstract

Mean-square convergence of the Milstein method and the Euler–Maruyama (EM) method are investigated for stochastic singular initial value problems (SIVPs) driven by multiplicative white noise. For the first-order model equation, the existence, uniqueness, and moment boundedness of the exact solution are developed under some appropriate assumptions. On this basis, it is proved that the Milstein method is of 1/2−ϵ order convergence in the mean-square sense, and the convergence order can be increased to 1−ϵ when the diffusion coefficient vanishes at the origin, where ϵ is an arbitrarily small positive number. It is significantly different from the first-order mean-square convergence of the Milstein method to solve the classical stochastic ordinary differential equations. Meanwhile, whether the diffusion coefficient vanishes at the origin or not, the mean-square convergence order of the EM method is always 1/2−ϵ. Furthermore, a second-order stochastic SIVP with multiplicative noise is considered by converting it to a first-order system. When solving the system, it is shown that the EM method has the same form as the Milstein method, and thus enjoys the convergence order of the Milstein method. Numerical examples are provided to verify our theoretical prediction.