1. A host-vehicle-travel-position specification apparatus comprising: a position calculation section that calculates a current vehicle position of a vehicle; a map matching section that map matches the current vehicle position to a road in map data and specifies a map-matching reference trajectory; a first range specification section that specifies an estimation error range of estimated absolute positions containing an estimated absolute trajectory; a second range specification section that specifies an estimation error range of map matching reference positions based on relation between the map-matching reference trajectory and the estimation error range of estimated absolute positions on condition that the map matching section continues to map match the current vehicle position to one road longer than or equal to a specified distance; a correction section that corrects the map-matching reference trajectory in accordance with the estimation error range of map matching reference positions; and a travel position specification section that specifies a vehicle travel position of the vehicle based on the map-matching reference trajectory corrected by the correction section.