XV: 40, 590-603, LNM 850 (1981)
STROOCK, Daniel W.; YOR, Marc
Some remarkable martingales (Martingale theory)
This is a sequel to a well-known paper by the authors (Ann. ENS, 13, 1980) on the subject of pure martingales. A continuous martingale $(M_t)$ with $<M,M>_{\infty}=\infty$ is pure if the time change which reduces it to a Brownian motion $(B_t)$ entails no loss of information, i.e., if $M$ is measurable w.r.t. the $\sigma$-field generated by $B$. The first part shows the purity of certain stochastic integrals. Among the striking examples considered, the stochastic integrals $\int_0^t B^n_sdB_s$ are extremal for every integer $n$, pure for $n$ odd, but nothing is known for $n$ even. A beautiful result unrelated to purity is the following: complex Brownian motion $Z_t$ starting at $z_0$ and its (Lévy) area integral generate the same filtration if and only if $z_0\neq0$
Keywords: Pure martingales, Previsible representation
Nature: Original
Retrieve article from Numdam

---

XV: 41, 604-617, LNM 850 (1981)
LÉPINGLE, Dominique; MEYER, Paul-André; YOR, Marc
Extrémalité et remplissage de tribus pour certaines martingales purement discontinues (General theory of processes, Martingale theory)
This paper consists roughly of two parts. First, the study of a filtration where all martingales are purely discontinuous, and jump on a given well-ordered optional set. Then under a simple separability assumption, one can construct one single martingale which generates the filtration. The second part deals with the same problem as in 1540, but replacing continuous martingales by purely discontinuous martingales with unit jumps, and Brownian motion by a Poisson process. It is shown that the situation is much simpler, purity and extremality being equivalent in this case
Keywords: Poisson processes, Pure martingales, Previsible representation, Jumps
Nature: Original
Retrieve article from Numdam

---