From an Itô type calculus for Gaussian processes to integrals of log-normal processes increasing in the convex order
type: Item
instance of: scholarly article
title: From an Itô type calculus for Gaussian processes to integrals of log-normal processes increasing in the convex order
zbMATH Open document ID: 1233.60008
author: Francis Hirsch
author: Bernard Roynette
author: Marc Yor
published in: Journal of the Mathematical Society of Japan
publication date: 2011-09-27T00:00:00Z
review text: \textit{P. P. Carr, C.-O. Ewald} and \textit{Y. Xiao} [``On the qualitative effect of volatility and duration on prices of Asian options'', Finance Research Letters, 5, 162--171 (2008), \url{doi:10.2139/ssrn.1086927}] showed that the process \[ A_t=\frac1t\int_0^t\exp\bigg(B_s-\frac{s}2\bigg)\,ds,\quad t\geq0, \] where \((B_s,\;s\geq0)\) is a standard Brownian motion, is a process that is increasing in the convex order. The authors generalize this to a Gaussian framework. First, they present an Itô type formula for a Gaussian process that is well adapted to the study of processes that are increasing in the convex order in a Gaussian framework. Then, they show that some processes that are associated with \(A_t\) are increasing in the convex order. Finally, they construct Gaussian sheets allowing to exhibit martingales with the same one-marginals as the processes mentioned above.
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60G44
zbMATH DE Number: 5950726
zbMATH Keywords: convex order
zbMATH Keywords: 1-martingale
zbMATH Keywords: Gaussian process
zbMATH Keywords: Gaussian sheet
zbMATH Keywords: log-normal process
zbMATH Keywords: Itô type formula
zbMATH Keywords: Itô's calculus
reviewed by: Moshe Shaked
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/06330887
OpenAlex ID: W4297820203
cites work: A Brownian sheet martingale with the same marginals as the arithmetic average of geometric Brownian motion
cites work: Generalized sweeping-out and probability
cites work: Stopping distributions for right processes
cites work: Peacocks and associated martingales, with explicit constructions
cites work: Unifying constructions of martingales associated with processes increasing in the convex order, via Lévy and Sato sheets
cites work: A construction of processes with one dimensional martingale marginals, based upon path-space Ornstein-Uhlenbeck processes and the Brownian sheet
cites work: A construction of processes with one-dimensional martingale marginals, associated with a Lévy process, via its Lévy sheet
cites work: Looking for martingales associated to a self-decomposable law
cites work: Gaussian Hilbert Spaces
cites work: Une inegalité du type de Slepian et Gordon sur les processus gaussiens. (An inequality of Slepian and Gordon type for Gaussian processes)
cites work: Markov-Komposition und eine Anwendung auf Martingale. (Markov compositions and an application to martingales)
cites work: On a one-parameter generalization of the Brownian bridge and associated quadratic functionals
cites work: Q5587587
cites work: The stopping distributions of a Markov process
cites work: Q4000424
cites work: The Existence of Probability Measures with Given Marginals
cites work: Q4413910
cites work: Construction of pure states in mean field models for spin glasses
cites work: Q4693741
DOI: 10.2969/JMSJ/06330887
description: scientific article; zbMATH DE number 5950726