On generalized CIR equations

Abstract

The paper is concerned with stochastic equations for the short rate process $R$ $$ dR(t)=F(R(t))dt+G(R(t-))dZ(t), $$ in the affine model of the bond prices. The equation is driven by a L\'evy martingale $Z$. It is shown that the discounted bond prices are local martingales if either $Z$ is a stable process of index $\alpha\in(1,2]$,\,$F(x)= ax +b, b\geq 0$, $G(x)=cx^{1/\alpha}, c>0$ or $Z$ must be a L\'evy martingale with positive jumps and trajectories of bounded variation, $F(x)= ax +b, b\geq 0$ and G is a constant. The result generalizes the well known Cox-Ingersoll-Ross result and extends the Vasicek result to non-negative short rates.