No arbitrage of the first kind and local martingale numéraires
No arbitrage of the first kind and local martingale numéraires
复制标题
没有第一类套利和当地鞅法
作者：
Y. Kabanov;C. Kardaras;Shiqi Song
上鞅平减因子(即局部鞅平减因子)将非负财富过程乘法变换为上鞅(即局部鞅)。一个上鞅numéraire(对应于局部鞅numéraire)是一个财富过程，它的倒数是一个上鞅平减指数(对应于局部鞅平减指数)。在前人的工作中，第一类套利(NA1$Mbox{na}_{1}$)的不存在等价于(唯一的)上鞅Numéraire的存在，并进一步等价于存在严格正的局部鞅平减指数；然而，在NA1$Mbox{na}_{1}$下，局部鞅Numéraire可能不存在。在这项工作中，我们证明了在NA1$Mbox{na}_{1}$下，原概率P$P$下的上鞅数变成了在全变差距离中任意接近P$P$的等价概率的局部鞅数.
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 (NA1$mbox{NA}_{1}$) 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 NA1$mbox{NA}_{1}$, a local martingale numéraire may fail to exist. In this work, we establish that under NA1$mbox{NA}_{1}$, a supermartingale numéraire under the original probability P$P$ becomes a local martingale numéraire for equivalent probabilities arbitrarily close to P$P$ in the total variation distance.