Are there long-living Hilda-like asteroids between Jupiter and Saturn?

To clarify the question of asteroids between Jupiter and Saturn a numerical study of the stability of orbits of fictitious bodies in this region has been undertaken. There are almost no minor planets moving there with the exception of 61 faint asteroids. After determining the mean motion resonances and the three-body resonances, we have studied orbits for inclinations from 1 to 91 degrees to find out stable regions from 6.8 to 8.0 au. This was done up to 100 Myrs in the dynamical model Sun-Jupiter-Saturn and massless fictitious asteroids. In a study with a finer grid in inclination from 1 to 31 degrees and semi-major axes from 6.88 to 7.30 au, we integrated ten thousand orbits up to the moment of their escape. These results showed a region in semi-major axes with stable orbits in the interval from 7.12 to 7.23 au for moderate inclinations. Nevertheless, being a temporary capture there could be these objects much larger than the tiny ones discovered up to now in the region between Jupiter and Saturn.