During the early history of the soar system, it is likely that the outer planets changed their distance from the sun, and hence, their influence on the asteroid belt evolved with time. The gravitational influence of Jupiter and Saturn on the orbital evolution of asteroids in the outer asteroid belt was calculated. The results show that the sweeping of mean motion resonances associated with planetary migration efficiently destabilizes orbits in the outer asteroid belt on a time scale of 10 million years. This mechanism provides an explanation for the observed depletion of asteroids in that region.