Show that is a continuous strict local martingale, with the last element .
For any , because , is a well-defined Martingale on by Proposition 3.2.10 of [KS98].
Next, one can show that, on
is a strong solution of (1), and the uniquness follows by Theorem IX.3.5(ii) of [RY99]. Thus, is local martingale on .
With function of , consider time-changed process
One can check and is a continuous local martingale. By Levy’s characterization theorem, is a Brownian motion w.r.t. . Note that
as by Law of Iterated Logarithm. This implies is strict local martingale with