The \(p\)-optimal martingale measure and its asymptotic relation with the minimal-entropy martingale measure
type: Item
instance of: scholarly article
title: The \(p\)-optimal martingale measure and its asymptotic relation with the minimal-entropy martingale measure
zbMATH Open document ID: 0923.60045
DOI: 10.2307/3318433
author: Peter Grandits
published in: Bernoulli
publication date: 1999-10-17T00:00:00Z
review text: Let \(S=(S_k)_{k=0,1,\ldots,N}\) be an \(\mathbb{R}^d\)-valued stochastic process in finite discrete time and assume that \(S\) admits an equivalent martingale measure. The \(p\)-optimal martingale measure \(Q^{(p)}\) is that signed martingale measure for \(S\) whose density with respect to the original measure \(P\) has minimal \(L^p(P)\)-norm, and the minimal-entropy martingale measure \(Q^E\) is that martingale measure for \(S\) which minimizes relative entropy with respect to \(P\). The main result shows that if \(S\) is bounded, the densities \(Z^{(p)}\) converge for \(p\to 1\) to \(Z^E\) in \(L^1(P)\). The proof uses a mixture of probabilistic and functional-analytic arguments and exploits in particular the explicit structure of \(Z^{(p)}\) and \(Z^E\). A counterexample shows that the assertion fails in general if \(S\) is unbounded. An auxiliary result shows that the space of all final values of stochastic integrals of \(S\) is closed in \(L^0(P)\); this requires no boundedness assumptions.
reviewed by: Martin Schweizer
Mathematics Subject Classification ID: 60G42
Mathematics Subject Classification ID: 91B28
Mathematics Subject Classification ID: 60G48
zbMATH DE Number: 1294458
zbMATH Keywords: martingale measures
zbMATH Keywords: relative entropy
zbMATH Keywords: convergence
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/3318433
OpenAlex ID: W2021721819
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description: scientific article; zbMATH DE number 1294458