Quand l’inégalité de Kunita-Watanabe est-elle une égalite? (When is the Kunita-Watanabe inequality an equality?). (French) Zbl 0622.60048

Lin, Cheng-De

Probabilités XX, Proc. Sémin., Strasbourg 1984/85, Lect. Notes Math. 1204, 40-47 (1986).

[For the entire collection see Zbl 0593.00014.]
Let \((X_ t;t\geq 0)\) and \((Y_ t;t\geq 0)\) be two semimartingales. In order to give an answer to the question formulated in the title, the author gives conditions under which (1) \([X,Y]=[X][Y]\) and (2) \(<X,Y>=<X><Y>\). If X and Y are local martingales, (1) holds iff \(Y=\gamma X\), where \(\gamma\) is a random variable satisfying a measurability condition. In general (1) holds iff \(Y-B^ c=\gamma (X-A^ c)\) where \(A^ c\) and \(B^ c\) are the continuous parts of the finite variation processes in the decompositions of X and Y \((X=M+A\) and \(Y=N+B).\)
For (2) the conditions mentioned above are necessary, but not sufficient unless the underlying filtration is quasi-leftcontinuous. In order to get a sufficient condition for (2), the measurability assumption for \(\gamma\) has to be strengthened. By a counterexample the author shows that the strengthened condition is not necessary in general.

Reviewer: M.Dozzi

MSC:

60G44

Martingales with continuous parameter

Keywords:

inequality of Kunita-Watanabe; semimartingales; local martingales; quasi- leftcontinuous

Citations:

Zbl 0593.00014