Sampling of Probability Measures in the Convex Order and Approximation of Martingale Optimal Transport Problems

2017 ◽  
Author(s):  
Aurrlien Alfonsi ◽  
Jacopo Corbetta ◽  
Benjamin Jourdain
Keyword(s):  
Optimal Transport ◽  
Probability Measures ◽  
Convex Order ◽  
Martingale Optimal Transport ◽  
Transport Problems

scholarly journals SAMPLING OF ONE-DIMENSIONAL PROBABILITY MEASURES IN THE CONVEX ORDER AND COMPUTATION OF ROBUST OPTION PRICE BOUNDS

2019 ◽  
Vol 22 (03) ◽  
pp. 1950002 ◽  
Author(s):  
AURÉLIEN ALFONSI ◽  
JACOPO CORBETTA ◽  
BENJAMIN JOURDAIN
Keyword(s):  
Real Line ◽  
Optimal Transport ◽  
Option Price ◽  
Order Moment ◽  
Probability Measures ◽  
Convex Order ◽  
Empirical Measures ◽  
The Real ◽  
Martingale Optimal Transport ◽  
Transport Problems

For [Formula: see text] and [Formula: see text] two probability measures on the real line such that [Formula: see text] is smaller than [Formula: see text] in the convex order, this property is in general not preserved at the level of the empirical measures [Formula: see text] and [Formula: see text], where [Formula: see text] (resp., [Formula: see text]) are independent and identically distributed according to [Formula: see text] (resp., [Formula: see text]). We investigate modifications of [Formula: see text] (resp., [Formula: see text]) smaller than [Formula: see text] (resp., greater than [Formula: see text]) in the convex order and weakly converging to [Formula: see text] (resp., [Formula: see text]) as [Formula: see text]. According to  Kertz & 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 [Formula: see text] and [Formula: see text] in this set, this enables us to define a probability measure [Formula: see text] (resp., [Formula: see text]) greater than [Formula: see text] (resp., smaller than [Formula: see text]) in the convex order. We give efficient algorithms permitting to compute [Formula: see text] and [Formula: see text] (and therefore [Formula: see text] and [Formula: see text]) when [Formula: see text] and [Formula: see text] 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.

scholarly journals OPTIMAL PATHS RELATED TO TRANSPORT PROBLEMS

2003 ◽  
Vol 05 (02) ◽  
pp. 251-279 ◽  
Author(s):  
QINGLAN XIA
Keyword(s):  
Probability Measure ◽  
Weak Topology ◽  
Optimal Transport ◽  
Vector Measure ◽  
Probability Measures ◽  
Actual Cost ◽  
Total Cost ◽  
Optimal Paths ◽  
Transport Path ◽  
Transport Problems

In transport problems of Monge's types, the total cost of a transport map is usually an integral of some function of the distance, such as |x - y|p. In many real applications, the actual cost may naturally be determined by a transport path. For shipping two items to one location, a "Y shaped" path may be preferable to a "V shaped" path. Here, we show that any probability measure can be transported to another probability measure through a general optimal transport path, which is given by a vector measure in our setting. Moreover, we define a new distance on the space of probability measures which in fact metrizies the weak * topology of measures. Under this distance, the space of probability measures becomes a length space. Relations as well as related problems about transport paths and transport plans are also discussed in the end.

Schrödinger dynamics and optimal transport of measures

2021 ◽  
pp. 2150016
Author(s):  
Lorenzo Zanelli
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Transport Theory ◽  
Optimal Transport ◽  
Time Dependent ◽  
Probability Measures ◽  
Flat Torus ◽  
Optimal Transport Theory

In this paper, we recover a class of displacement interpolations of probability measures, in the sense of the Optimal Transport theory, by means of semiclassical measures associated with solutions of Schrödinger equation defined on the flat torus. Moreover, we prove the completing viewpoint by proving that a family of displacement interpolations can always be viewed as a path of time-dependent semiclassical measures.

scholarly journals Martingale Optimal Transport with Stopping

2018 ◽  
Vol 56 (1) ◽  
pp. 417-433 ◽  
Author(s):  
Erhan Bayraktar ◽  
Alexander M. G. Cox ◽  
Yavor Stoev
Keyword(s):  
Optimal Transport ◽  
Martingale Optimal Transport

scholarly journals Scaling algorithms for unbalanced optimal transport problems

2018 ◽  
Vol 87 (314) ◽  
pp. 2563-2609 ◽  
Author(s):  
Lénaïc Chizat ◽  
Gabriel Peyré ◽  
Bernhard Schmitzer ◽  
François-Xavier Vialard
Keyword(s):  
Optimal Transport ◽  
Transport Problems ◽  
Scaling Algorithms
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