Abstract
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.
Citation
Dean Isaacson. "Continuous Martingales with Discontinuous Marginal Distributions." Ann. Math. Statist. 42 (6) 2139 - 2142, December, 1971. https://doi.org/10.1214/aoms/1177693081
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