×

A propos d’une conjecture de Meyer. (On a conjecture of Meyer). (French) Zbl 0654.60036

Séminaire de probabilités XXII, Strasbourg/France, Lect. Notes Math. 1321, 144-146 (1988).
[For the entire collection see Zbl 0635.00013.]
Let \(X=(X_ t)\), \(t\geq 0\), be a square-integrable martingale, \(X_ 0=0\), which has the property of predictable representation (PR) and \(<X,X>_ t=t\). Denote by \({\mathcal M}\) the set of such martingales. The Brownian motion \(B=(B_ t)\) and the compensated Poisson process \(X_ t^{(\lambda)}=\sqrt{\lambda}(\pi_ t^{(\lambda)}-t/\lambda)\) belong to \({\mathcal M}\) where \(\pi^{(\lambda)}\) is the Poisson process with intensity 1/\(\lambda\). Note that \(X^{(\lambda)}\to B\), \(\lambda\) \(\to 0\). \(X^{(\lambda)}\) also has the PR-property when \(\lambda\) is a function of time. P. A. Meyer conjectured that each \(X\in {\mathcal M}\) has the form \(X=X^{(\rho)}\) where \(\rho\) is a Borelian function of time.
The paper contains a counter-example of this conjecture. The process of the counter-example is \(X_ t=B^ T_ t+Q_{(t-T)^+}\) where T is an \({\mathcal F}^ B\)-stopping time, Q is a compensated Poisson process with intensity 1 independent of B, and \((t-T)^+\) is the positive part of (t- T).
Reviewer: L.Gal’čuk

MSC:

60G44 Martingales with continuous parameter
60H05 Stochastic integrals

Citations:

Zbl 0635.00013
Full Text: Numdam EuDML