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 coefficient 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 \(\theta \)th moment with \(0<\theta <1\). 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.