Variation conditionelle des processus stochastiques. (Conditional variation of random processes)
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''; ``Then we apply Yor's remark (a combination of Fefferman and Garsia-Neveu)'' etc. Even the example of a prequasimartingale not being a semimartingale is not clear. Why should the convex hull of \(\{Y_ t|;0\leq t\leq 1\}\) be bounded in \(L^ 0,\) \(Y_ t\) being a cadlag process? Yet, that is the author's argument that his \((Y_ t)_ t\) is a prequasimartingale. If the proofs were reasonably clear, the paper would have been longer, maybe, but readable.
scientific article; zbMATH DE number 4056696