Positivity preserving truncated scheme for the stochastic Lotka-Volterra model with small moment convergence

This work concerns with the numerical approximation for the stochastic Lotka- Volterra model originally studied by Mao et al. (Stoch Process Appl 97(1):95-110, 2002). The natures of the model including multi-dimension, super-linearity of both the drift and diffusion coefficients and the positivity of the solution make most of the existing numerical methods fail. In particular, the super-linearity of the diffusion coef-ficient results in the explosion of the 1st moment of the analytical solution at a finite time. This becomes one of our main technical challenges. As a result, the convergence framework is to be set up under the ?th moment with 0 < ? < 1. The idea developed in this paper will not only be able to cope with the stochastic Lotka-Volterra model but also work for a large class of multi-dimensional super-linear SDE models.