Continuous Martingales with Discontinuous Marginal Distributions

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.