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 Lévy martingale $Z$. It is shown that the discounted bond prices are local martingales if either $Z$ is a stable process of index $\alpha \in (1,2]$,\,$F(x)=ax+b,b\ge 0$, $G(x)=c{x}^{1/\alpha },c>0$ or $Z$ must be a Lévy martingale with positive jumps and trajectories of bounded variation, $F(x)=ax+b,b\ge 0$ 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.