Martingale laws, densities and decomposition of Föllmer-Schweizer
type: Item
instance of: scholarly article
title: Martingale laws, densities and decomposition of Föllmer-Schweizer
zbMATH Open document ID: 0772.60033
author: Jean-Pascal Ansel
author: Ch. Stricker
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1993-01-17T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77438
review text: The paper continues topics in [second author, ibid. 26, No. 3, 451-460 (1990; Zbl 0704.60045)]. Consider a probability space \((\Omega,{\mathcal F}_ 1,P)\), a filtration \(({\mathcal F}_ t)_{t\in[0,1]}\) on it etc. Consider first a continuous \(R^ d\)-valued process \(X=M+B\), where \(M\) is a continuous local martingale and \(B\) is previsible with finite variation. Let \(N^ i\) be orthogonal \(R\)-valued continuous local martingales with \(M=\sigma N\), \(\sigma\) previsible, let \(A\) be continuous ascending adapted with \(d\langle N^ i\), \(N^ i\rangle=\zeta^ iA\). If there exists a probability \(Q\), equivalent to \(P\), such that \(X\) is a \(Q\)-martingale, then one may choose previsible \(\nu,b\) such that \(b=\sigma\nu\), \(X=\sigma N+bA\) and \(\theta^ i=(\nu^ i/\zeta^ i)1_{(\zeta^ i\neq 0)}\) are \(N^ i\)-integrable. Moreover, all possible \(Q\) are \({\mathcal E}(-\theta N+L)P\), where \(L\) is a local martingale orthogonal to all \(N^ i\) (and \({\mathcal E}\) is the ``stochastic exponential''). \(Q\) is unique iff \(N\) has the previsible representation property and \(\sigma\) may be chosen invertible. The authors consider then discontinuous \(R\)-valued processes. They first give a correct proof of a result of the second author [Stochastic differential systems, Proc. IFIP-WG 7/1 Work. Conf., Eisenach/GDR 1986, Lect. Notes Control Inf. Sci. 96, 373-380 (1987; Zbl 0655.60038)], namely: if \(D\) is ascending right continuous previsible, \(A\) is right continuous previsible with finite variation, \(p\in(1,\infty)\), \(p^{- 1}+q^{-1}=1\) and \(K=\{\int HdA\); \(H\) previsible bounded, \(\int| H|^ pdD\leq 1\}\) is bounded in probability, then there exists a previsible \(\alpha\) with \(dA=\alpha dD\), \(\int|\alpha|^ qdD<\infty\). Then they prove, via four lemmas etc., that, if \(X\) is adapted, cadlag, \(X_ 0=0\), \(\sup_ s| X_ s|\in L^ 2\) and for the closure \(\overline K\) of \(K\) in \(L^ 1\) we have \(\overline K\cap L^ 1_ +=\{0\}\), then \(X\) is a s…
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60H05
zbMATH DE Number: 97824
zbMATH Keywords: arbitrage
zbMATH Keywords: stochastic exponential
zbMATH Keywords: minimal martingale law
zbMATH Keywords: local martingale
zbMATH Keywords: previsible representation property
reviewed by: Ion Cuculescu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 97824