Positivity preserving stochastic θ-methods for selected SDEs

Several applications are modelled by stochastic differential equations with positive solutions. Numerical methods, able to preserve positivity, are absolutely needed in this case, in order to retain the intrinsic meaning of the solution in the spirit of the underlying application. This paper discusses the ability of stochastic θ-Milstein methods to generate positive sequences of numerical approximations, when they are applied to models with affine drift and square root diffusion. Moreover, we provide a class of θ-methods able to preserve the positivity of a jump extended nonlinear CIR model. Numerical experiments confirm the theoretical analysis.