  We analyze exponential integrability properties of the Cox-Ingersoll-Ross
(CIR) process and its Euler discretizations with various types of truncation
and reflection at 0. These properties play a key role in establishing the
finiteness of moments and the strong convergence of numerical approximations
for a class of stochastic differential equations arising in finance. We prove
that both implicit and explicit Euler-Maruyama discretizations for the CIR
process preserve the exponential integrability of the exact solution for a wide
range of parameters, and find lower bounds on the explosion time.
