Continuous Martingales with Discontinuous Marginal Distributions

Isaacson, Dean

Isaacson, Dean

Ann. Math. Statist., Tome 42 (1971) no. 6, p. 2139-2142 / Harvested from Project Euclid

Résumé

We construct in this paper a continuous, nowhere constant, square integrable martingale such that $P\{M(\frac{1}{2})^k = 0\} \geqq \frac{7}{8}$ for $k \geqq 3$. This construction is used to show that in general, $\lim_{t\rightarrow 0}\int^t_0\Phi(s)dM(s, \omega)/M(t, \omega) \neq \Phi(0)$ where $\Phi(s)$ is nonrandom and right continuous, $M(t, \omega)$ is a continuous, nowhere constant, square integrable, martingale, and the limit is a limit in probability.

Publié le : 1971-12-14
Classification:

@article{1177693081,
     author = {Isaacson, Dean},
     title = {Continuous Martingales with Discontinuous Marginal Distributions},
     journal = {Ann. Math. Statist.},
     volume = {42},
     number = {6},
     year = {1971},
     pages = { 2139-2142},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1177693081}
}

Isaacson, Dean. Continuous Martingales with Discontinuous Marginal Distributions. Ann. Math. Statist., Tome 42 (1971) no. 6, pp.  2139-2142. http://gdmltest.u-ga.fr/item/1177693081/