We report on lattice-matched InAlN/GaN MOSHEMTs with an oxide-filled recess, self-aligned to the gate prepared by thermal oxidation at 800 degrees C in oxygen atmosphere. The device delivered a maximum current density of 2.4 A/mm. Pulse measurements showed no apparent lag effects, indicating a high-quality native oxide. This was confirmed by monitoring the radio-frequency load lines in the time domain. The MOSHEMT yielded a power density of 6 W/mm at a drain voltage as low as 20 V and at 4 GHz, a power added efficiency of 32% and an f(t) and f(max) of 61 and 112 GHz, respectively, illustrating the capability of such MOSHEMT to operate at high frequencies.