Variation conditionelle des processus stochastiques. (Conditional variation of random processes)
type: Item
instance of: scholarly article
title: Variation conditionelle des processus stochastiques. (Conditional variation of random processes)
zbMATH Open document ID: 0647.60053
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1988-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77327
review text: Let (\(\Omega,{\mathcal F},P,({\mathcal F}_ t)_{0\leq t\leq 1})\) be a stochastic basis and \((X_ t)_{0\leq t\leq 1}\) be a cadlag adapted process. For every partition \(\tau =(0=t_ 0<t_ 1<...<t_ n=1)\) of [0,1] let \[ V_{\tau}(X)=\sum^{n-1}_{i=0}| E(X_{t_{i+1}}-X_{t_ i}| {\mathcal F}_{t_ i\quad})| \text{ and } Q(X)=\sum^{n- 1}_{i=0}(X_{t_{i+1}}-X_{t_ i})^ 2. \] Then X is called: a (special) semimartingale) iff \(X=M+A\) where M is a local martingale and A is a (previsible) adapted cadlag process with bounded variation; a (special) Dirichlet process iff \(X=Z+A\) where Z is a (special) semimartingale and A adapted, continuous, having null quadratic variation (i.e. \(\lim_{| \tau | \to 0}Q_{\tau}(A)=0\) in probability); a quasimartingale iff \(\{V_{\tau}(X)\); \(\tau\) partition of [0,1]\(\}\) is bounded in \(L^ 1\). Moreover, the author calls X a prequasimartingale iff the convex hull of \(\{V_{\tau}(X)\); \(\tau\) partition\} is bounded in \(L^ 0\). The paper has two parts, which are independent. In the first one, the author proves that if X is a Dirichlet process such that the family \(\{Q_{\tau}(X)\); \(\tau\) partition with dyadic points\(\}\) is uniformly integrable, then X is special and its martingale component is in \(H^ 1,\) and applies this result to the study of a Gaussian-Dirichlet process. In the second part, he shows that, in the class of processes satisfying a supplementary technical condition, the concepts of semimartingale and prequasimartingale are the same, after giving an example that, in general, this is not the case (a prequasimartingale which is not a semimartingale). The reader is supposed to be familiar with \textit{C. Dellacherie} and \textit{P.-A. Meyer}'s book, Probabilité et potentiel, for the last part till now which appeared 1987 see Zbl 0624.60084, and so the proofs are only sketched (too sketchy, in our opinion). The reader is further frustrated by cryptical references such as ``It is enough to apply La Vallée- Poussin's lemma''; ``T…
Mathematics Subject Classification ID: 60G48
Mathematics Subject Classification ID: 60G07
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 4056696
zbMATH Keywords: semimartingale
zbMATH Keywords: local martingale
zbMATH Keywords: Dirichlet process
zbMATH Keywords: prequasimartingale
author: Ch. Stricker
reviewed by: Gheorghiţă Zbăganu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4056696