For the CIR equation dXt = (θ − kXt) dt + σ √ Xt dBt, we propose positive weak firstand second-order approximations that use, at each step, generation of discrete (respectively twoand three-valued) random variables (Theorems 3 and 4). The equation is split into deterministic part dDt = (θ − kDt) dt, which is solved exactly, and stochastic part dSt = σ √ St dBt, which is actually approximated in...

This note derives the shapes of the yield curve as a function of the current spot rate under Cox, Ingersoll, and Ross (CIR) (1985b) single factor model. Corresponding results reported in CIR are shown to be incorrect.

For positive diffusions, we construct split-step second-order weak approximations preserving the positivity property. For illustration, we apply the construction to some popular stochastic differential equations such as Verhulst, CIR, and CKLS equations.