  In this paper we consider Roche accretion in an Extreme Mass-Ratio Inspiral
(EMRI) binary system formed by a star orbiting a massive black hole. The
ultimate goal is to detect the mass and spin of the black hole and provide a
test of general relativity in the strong-field regime from the resultant
quasi-periodic signals. Before accretion starts, the stellar orbit is presumed
to be circular and equatorial, and shrinks due to gravitational radiation. New
fitting formulae are presented for the inspiral time and the radiation-reaction
torque in the relativistic regime. If the inspiralling star fills its Roche
lobe outside the Innermost Stable Circular Orbit (ISCO) of the hole, gas will
flow through the inner Lagrange point (L1) to the hole. We give new
relativistic interpolation formulae for the volume enclosed by the Roche lobe.
If this mass-transfer happens on a time scale faster than the thermal time
scale but slower than the dynamical time scale, the star will evolve
adiabatically, and, in most cases, will recede from the hole while filling its
Roche lobe. We calculate how the stellar orbital period and mass-transfer rate
will change through the "Roche evolution" for various types of stars in the
relativistic regime. We envisage that the mass stream eventually hits the
accretion disc, where it forms a hot spot orbiting the hole and may ultimately
modulate the luminosity with the stellar orbital frequency. The observability
of such a modulation is discussed along with a possible interpretation of an
intermittent 1 hour period in the X-ray emission of RE J1034+396.
