On CiR equations with general factors

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) = cx1/\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 from [I. Cox, J. Ingersoll, and S. Ross, Econometrica, 53 (2004), pp. 385--408] and extends the Vasi\v cek result (see [O. Vasi\v cek, J. Financial Econom., 5 (1977), pp. 177--188]) to nonnegative short rates.