No arbitrage of the first kind and local martingale numéraires

A supermartingale deflator (resp. local martingale deflator) multiplicatively transforms nonnegative wealth processes into supermartingales (resp. local martingales). A supermartingale numéraire (resp. local martingale numéraire) is a wealth process whose reciprocal is a supermartingale deflator (resp. local martingale deflator). It has been established in previous works that absence of arbitrage of the first kind ( NA1NA1 ) is equivalent to the existence of the (unique) supermartingale numéraire, and further equivalent to the existence of a strictly positive local martingale deflator; however, under NA1NA1 , a local martingale numéraire may fail to exist. In this work, we establish that under NA1NA1 , a supermartingale numéraire under the original probability PP becomes a local martingale numéraire for equivalent probabilities arbitrarily close to PP in the total variation distance.