https://hal-enpc.archives-ouvertes.fr/hal-01963507Alfonsi, AurélienAurélienAlfonsiCERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique - ENPC - École des Ponts ParisTechMATHRISK - Mathematical Risk Handling - UPEM - Université Paris-Est Marne-la-Vallée - ENPC - École des Ponts ParisTech - Inria de Paris - Inria - Institut National de Recherche en Informatique et en AutomatiqueCorbetta, JacopoJacopoCorbettaCERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique - ENPC - École des Ponts ParisTechJourdain, BenjaminBenjaminJourdainCERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique - ENPC - École des Ponts ParisTechMATHRISK - Mathematical Risk Handling - UPEM - Université Paris-Est Marne-la-Vallée - ENPC - École des Ponts ParisTech - Inria de Paris - Inria - Institut National de Recherche en Informatique et en AutomatiqueSampling of one-dimensional probability measures in the convex order and computation of robust option price boundsHAL CCSD2019Martingale optimal transportRobust option price boundsLinear programmingSampling techniquesConvex order[MATH.MATH-PR] Mathematics [math]/Probability [math.PR]Alfonsi, Aurélien2018-12-21 12:50:292022-07-08 10:06:032018-12-21 12:52:52enJournal articleshttps://hal-enpc.archives-ouvertes.fr/hal-01963507/document10.1142/S021902491950002Xapplication/pdf1For µ and ν two probability measures on the real line such that µ is smaller than ν in the convex order, this property is in general not preserved at the level of the empirical measures µI = 1 I I i=1 δX i and νJ = 1 J J j=1 δY j , where (Xi) 1≤i≤I (resp. (Yj) 1≤j≤J) are independent and identically distributed according to µ (resp. ν). We investigate modifications of µI (resp. νJ) smaller than νJ (resp. greater than µI) in the convex order and weakly converging to µ (resp. ν) as I, J → ∞. According to Kertz and Rösler (1992), the set of probability measures on the real line with a finite first order moment is a complete lattice for the increasing and the decreasing convex orders. For µ and ν in this set, this enables us to define a probability measure µ ∨ ν (resp. µ ∧ ν) greater than µ (resp. smaller than ν) in the convex order. We give efficient algorithms permitting to compute µ ∨ ν and µ ∧ ν (and therefore µI ∨ νJ and µI ∧ νJ) when µ and ν have finite supports. Last, we illustrate by numerical experiments the resulting sampling methods that preserve the convex order and their application to approximate martingale optimal transport problems and in particular to calculate robust option price bounds.