  We estimated the rate of comet and asteroid collisions with the terrestrial
planets by calculating the orbits of 13000 Jupiter-crossing objects (JCOs) and
1300 resonant asteroids and computing the probabilities of collisions based on
random-phase approximations and the orbital elements sampled with a 500 yr
step. The Bulirsh-Stoer and a symplectic orbit integrator gave similar results
for orbital evolution, but sometimes give different collision probabilities
with the Sun. A small fraction of former JCOs reached orbits with aphelia
inside Jupiter's orbit, and some reached Apollo orbits with semi-major axes
less than 2 AU, Aten orbits, and inner-Earth orbits (with aphelia less than
0.983 AU) and remained there for millions of years. Though less than 0.1% of
the total, these objects were responsible for most of the collision probability
of former JCOs with Earth and Venus. Some Jupiter-family comets can reach
inclinations i>90 deg. We conclude that a significant fraction of near-Earth
objects could be extinct comets that came from the trans-Neptunian region.
