Propagation of convexity by Markovian and martingalian semigroups
type: Item
instance of: scholarly article
title: Propagation of convexity by Markovian and martingalian semigroups
zbMATH Open document ID: 0974.60057
DOI: 10.1023/A:1008699315785
author: Claude Martini
published in: Potential Analysis
publication date: 2001-04-02T00:00:00Z
review text: This paper is an extract from the author's Ph. D. thesis ``Propagation de la convexité par des semigroupes markoviens and martingaliens sur la demi-droite'' (Université d'Evry, 1995). The central question is to find criteria guaranteeing that a Markovian operator \(Q\) maps convex functions on the half-line \((0,\infty)\) into convex functions. This question originates from the study of volatility risks in mathematical finance. The present paper is of a purely theoretical nature; for connections with mathematical finance the reader is referred to the author's thesis. Let \(Q\) be a linear operator on a vector space \(G\) of functions over \((0,\infty)\) which contains all piecewise linear functions. A particular choice will be \[ G:= \Biggl\{g\in C((0,\infty)):\|g\|:= \sup{|g(x)\over 1+ x}< \infty\Biggr\}. \] The operator \(Q\) is called positive if \(f\geq 0\Rightarrow Qf\geq 0\), Markovian if \(Q1=1\) and martingalian if \(Qx=x\). \(Q\) is said to propagate convexity if the associated call function \(C_Q(x,y):= Q(\cdot-y)^+(x)\) is convex as a function of \(x\) (or, equivalently, if the put function \(P_Q(x, y):= Q(y-\cdot)^+(x)\) is convex in \(x\)); the convexity of \(C_Q(x,\cdot)\) is always fulfilled. If \(Q\) propagates convexity, the second (distributional) derivative \(\partial^2_z C_Q(dz, y)\) exists and induces a kernel \(\widehat Q(y, dz)\) which, in turn, allows to associate with \(Q\) the call-put dual operator via \[ \widehat Q f(y):= \text{boundary values at }0 \& \infty+ \int f(z) \widehat Q(y,dz). \] The relation between \(Q\) and \(\widehat Q\) is given by \[ \widehat Q(\cdot-x)^+(y)= Q(y-\cdot)^+(x).\tag{\(*\)} \] The main result of the first section is that a positive, Markovian and martingalian \(Q\) propagates convexity if and only if \(\widehat Q\) has the same properties. In the second section the author studies semigroups of positive, Markovian and martingalian operators \(\{Q_t\}_{t\geq 0}\) defined on the space \(G\). Notice that \((*)\) …
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 46A20
Mathematics Subject Classification ID: 47D03
Mathematics Subject Classification ID: 91B28
zbMATH DE Number: 1286663
zbMATH Keywords: convexity
zbMATH Keywords: Markovian semigroup
zbMATH Keywords: martingalian semigroup
zbMATH Keywords: \(h\)-duality
zbMATH Keywords: Lévy-Khinchin formula
zbMATH Keywords: dual semigroup
zbMATH Keywords: option pricing
reviewed by: R. L. Schilling
MaRDI profile type: Publication
full work available at URL: a:1008699315785
OpenAlex ID: W114152411
description: scientific article; zbMATH DE number 1286663