Topology Induced By Convex Order And Peacock Geodesics

In recent decades,the theory of optimal transport has been extensively developed.It is widely used in artificial intelligence,meteorology,image matching,network design,mathematical finance and other fields,and has attracted more and more researchers.Inspired by the robust pricing of exotic options in mathematical finance,researchers introduced additional constraints in optimal transportation and established the famous theory of martingale optimal transport.And,various of concrete martingale optimal transport problems for particular payoffs have been studied by virtue of stochastic control or Skorokhod embedding techniques.A sequence of probability measures satisfying the convex order relationship is called peacock.Studies have shown that any peacock has a related martingale,and the distribution of the martingale is this peacock.Because of the close correspondence between peacock and martingale,research on peacock has become an active area of option pricing.However,the topological properties of convex order relationship,the geometry and the properties of geodesic dynamics of peacock have yet to be systematically studied,and many basic problems have not been solved.This is the research motivation of this article.Specifically:1.Topology induced by convex order relationship.Convex order relationship is a kind of partial order relationship between probability measures.Two probability measures have convex order relationship,which means that they maintain a consistent magnitude relationship for any convex function integral.For two arbitrarily given probability measures,the convex order relationship determines a set between the two given probability measures.If there is a peacock with them as endpoints,it must take values from this set.By constructing a special control function to estimate the conditions,the tightness and uniform integrability are obtained successively.We prove the linear convexity of this set and the compactness with respect to the Wasserstein distance.In addition,for two probability measures with a bilateral convex order relationship,we use the smooth bump function to construct the test convex function,and prove that they are equal.This ruled out the possibility of the existence of the peacock loop.We discussed the stability of the convex order relationship: even if the expected value is not changed,we have constructed a sufficiently small perturbation to break the original convex order relationship.2.The geometric properties of the peacock geodesic.We mainly study the properties of the peacock geodesic in the Wasserstein space with base space R.At the same time,the geodesicity of the Wasserstein space also ensures the existence of the peacock geodesic.The value of the geodesic at different times is a measure,and the inverse of its distribution function can also be represented by the inverse of endpoints' distribution functions.On the premise that the endpoints satisfy the convex order relationship,we prove that it is a peacock by verifying that the call function of each point on the geodesic line increases monotonically.Ray and its defined Busemann function is an important tool for studying a wide range of geometries.The negative(weak)gradient line of the Busemann function is co-ray.We give a necessary and sufficient condition for the ray to be peacock.According to the expression formula of the inverse of the geodesic distribution function,we have obtained the expression formula of the inverse of the co-ray's distribution function.From this we also proved that when ray is peacock,its co-ray is also peacock.