Suppose M is a continuous local martingale such that is deterministic. Prove that M is a Gaussian process. Suppose M is a continuous local martingale with M0 = 0, a.s. Show that there exists a...

Suppose M is a continuous local martingale such that

is deterministic. Prove that M is a Gaussian process.

Suppose M is a continuous local martingale with M₀
= 0, a.s. Show that there exists a Brownian motion W , an increasing process τ_t, and a stopping time T such that

for all t.

Let Mt be a continuous local martingale. Show that the events

differ by at most a null set.