Deterministic quadratic variation characterizes a Gaussian continuous local martingale
= Deterministic quadratic variation characterizes a Gaussian continuous local martingale
If a continuous local martingale starts at zero and has deterministic continuous <quadratic variation> $[M]_t=f(t)$, the <Dambis-Dubins-Schwarz theorem> gives $M_t=B_{f(t)}$. It is therefore a centered <Gaussian process>. The converse follows from <independent increments of a Gaussian martingale>.